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y(Rule) 37 b(applied) e(to) j(the) g(Liouvillean|an) d(argumen) m(t) j(that) g(w) m(as) g(used) f(in) f(recen) m(t) j(pap) s (ers) e(on) 244 4320 y(the) h(return) e(to) j(equilibrium;) d(\(3\)) j (the) f(F) -8 b(ermi) 37 b(Golden) g(Rule) g(applied) e(to) j(the) g (so-called) g(C-) 244 4433 y(Liouvillean.) 50 b(These) 35 b(three) g(applications) d(lead) j(to) g(three) g(Lev) m(el) g(Shift) e (Op) s(erators.) 52 b(As) 35 b(our) 244 4545 y(main) k(result,) k(w) m (e) e(pro) m(v) m(e) h(that) f(if) e(the) i(reserv) m(oir) p Fp 41 w(R) p Fq 40 w(is) f(thermal) g(\(if) g(it) g(satis\014es) g(the) h(KMS) 244 4658 y(condition\),) 35 b(then) f(the) h(Lev) m(el) f(Shift) f(Op) s(erator) h(obtained) g(in) f(\(1\)) j(\(often) f(called) f(the) h (Da) m(vies) 244 4771 y(generator\)) 47 b(and) d(the) h(Lev) m(el) h (Shift) d(Op) s(erator) i(constructed) g(in) f(\(2\)) j(are) e (connected) h(b) m(y) f(a) 244 4884 y(similarit) m(y) 32 b(transformation.) 55 b(W) -8 b(e) 36 b(also) f(sho) m(w) g(that) g (the) h(Da) m(vies) g(generator) g(coincides) e(with) 244 4997 y(the) c(Lev) m(el) h(Shift) e(Op) s(erator) h(obtained) f(in) g (\(3\)) j(for) e(a) h(general) p Fp 30 w(R) p Fq(.) p Fo 1865 5753 a(1) p 90 rotate dyy eop %%Page: 2 2 2 1 bop Fo 3731 100 a(2) p Fn 0 407 a(1) 161 b(In) l(tro) t(duction) p Fo 0 626 a(In) 40 b(his) e(1949) h(Chigaco) f(lecture) i(notes) f([F],) i(F) -8 b(ermi) 38 b(called) g(the) h(form) m(ulas) f(for) h(the) g (2nd) g(order) h(p) s(er-) 0 746 y(turbativ) m(e) 31 b(calculations) e(of) i(energy) h(lev) m(els) f(the) p Fs 32 w(Golden) i(R) n(ule) p Fo(.) 42 b(There) 33 b(exists) f(a) e(n) m (um) m(b) s(er) i(of) e(mathe-) 0 867 y(matically) e(rigorous) h (implemen) m(tations) f(of) j(the) p Fs 31 w(F) -7 b(ermi) 32 b(Golden) h(R) n(ule) p Fo 30 w(\(F) m(GR\).) d(One) h(of) f(them) h (is) f(the) 0 987 y(so-called) p Fs 31 w(van) k(Hove) h(\(or) g(we) -5 b(ak) 34 b(c) -5 b(oupling\)) 34 b(limit) p Fo(.) 146 1107 y(T) -8 b(o) 30 b(describ) s(e) g(the) g(general) f(structure) h (of) f(the) h(v) -5 b(an) 30 b(Ho) m(v) m(e) g(limit,) d(consider) j(a) f(family) e(of) i(op) s(erators) p Fm 0 1228 a(L) p Fl 66 1243 a(\025) p Fo 155 1228 a(:=) p Fm 42 w(L) p Fk 366 1243 a(0) p Fo 434 1228 a(+) p Fm 28 w(\025Q) p Fo(.) 71 b(Let) p Fm 42 w(P) p Fo 54 w(b) s(e) 42 b(a) f(pro) 5 b(jection) 42 b(comm) m(uting) d(with) i(the) h(unp) s(erturb) s(ed) h (op) s(erator) p Fm 41 w(L) p Fk 3740 1243 a(0) p Fo 0 1348 a(satisfying) p Fm 44 w(P) 14 b(QP) p Fo 62 w(=) 48 b(0.) 81 b(Under) 45 b(appropriate) f(assumptions) h([Da1) o(,) g(Da3) o (],) j(one) d(can) h(sho) m(w) g(that) 0 1469 y(there) 33 b(exists) h(an) e(op) s(erator) g(\000) h(suc) m(h) h(that) 1273 1689 y(lim) p Fl 1267 1751 a(\025) p Fj(!) p Fk(0) p Fm 1431 1689 a(P) p Fo 14 w(e) p Fj 1551 1647 a(\000) p Fk(i) p Fl(tL) p Fi 1699 1656 a(0) p Fl 1733 1647 a(=\025) p Fi 1809 1624 a(2) p Fo 1848 1689 a(e) p Fk 1891 1647 a(i) p Fl(tL) p Fh 1984 1659 a(\025) p Fl 2026 1647 a(=\025) p Fi 2102 1624 a(2) p Fm 2141 1689 a(P) p Fo 41 w(=) 28 b(e) p Fk 2392 1647 a(i) p Fl(t) p Fk(\000) p Fm 2485 1689 a(:) p Fo 1067 w(\(1.1\)) 0 1944 y(W) -8 b(e) 33 b(will) d(call) h(\000) i(the) p Fs 33 w(L) -5 b(evel) 34 b(Shift) h(Op) -5 b(er) g(ator) p Fo 32 w(\(LSO\).) 146 2064 y(In) 48 b(the) g(literature) e(one) h(can) h(\014nd) f(other) h (rigorous) e(forms) h(of) f(F) m(GR.) h(They) i(usually) d(express) 0 2185 y(the) g(idea) f(that) g(LSO) g(describ) s(es) h(the) g(shift) f (of) g(eigen) m(v) -5 b(alues) 45 b(and) h(resonances) h(at) e(the) h (2nd) f(order) 0 2305 y(of) h(p) s(erturbation) g(theory) -8 b(.) 86 b(Some) 46 b(of) g(these) h(applications) e(are) h(discussed) j (in) c([DJ2,) i(DJ3) o(].) 85 b(F) -8 b(or) 0 2425 y(shortness,) 35 b(in) e(this) f(note) i(w) m(e) g(will) d(restrict) i(ourselv) m(es) i (to) d(the) i(dynamical) d(form) h(of) h(F) m(GR|the) f(v) -5 b(an) 0 2546 y(Ho) m(v) m(e) 34 b(limit.) 146 2666 y(There) f(exist) e (n) m(umerous) h(pap) s(ers) g(studying) g(a) f(\\small) e(quan) m(tum) i(system) p Fg 33 w(S) p Fo 39 w(in) m(teracting) f(with) h(a) 0 2787 y(reserv) m(oir) p Fg 32 w(R) p Fo(".) 43 b(In) 32 b(man) m(y) f(of) g(them) g(the) h(F) -8 b(ermi) 29 b(Golden) i(Rule) g (pla) m(ys) g(a) g(cen) m(tral) h(role.) 42 b(Among) 30 b(these) 0 2907 y(applications) k(of) i(F) m(GR) g(to) g(the) h(study) h (of) p Fg 35 w(S) p Fo 33 w(+) p Fg 25 w(R) p Fo 36 w(w) m(e) g(w) m (ould) e(lik) m(e) g(to) g(distinguish) f(the) i(follo) m(wing) d(2) 0 3027 y(t) m(yp) s(es:) 41 3179 y(\(1\)) p Ff 48 w(V) -9 b(an) 47 b(Ho) m(v) m(e) f(limit) c(for) 47 b(the) f(reduced) h (dynamics.) p Fo 67 w(W) -8 b(e) 40 b(assume) h(that) f(the) h(reserv) m (oir) g(is) 214 3299 y(initially) 21 b(in) i(a) i(stationary) e(state) i (for) f(the) h(unp) s(erturb) s(ed) h(dynamics.) 40 b(W) -8 b(e) 25 b(lo) s(ok) e(at) h(the) h(ev) m(olution) 214 3420 y(of) 36 b(observ) -5 b(ables) 37 b(of) e(the) i(small) d(system.) 54 b(One) 37 b(can) f(then) h(sho) m(w) g(that) f(under) h(mild) c (conditions) 214 3540 y(the) c(reduced) g(dynamics) f(in) f(the) h(v) -5 b(an) 28 b(Ho) m(v) m(e) h(limit) 24 b(is) k(a) f(completely) g(p) s (ositiv) m(e) g(semigroup) g([Da1) o(,) 214 3660 y(LeSp) q(].) 75 b(The) 43 b(op) s(erator) g(\000) f(obtained) h(in) f(this) g(w) m(a) m (y) i(\(the) f(generator) g(of) g(this) f(semigroup\)) g(is) 214 3781 y(often) e(called) e(the) h(Da) m(vies) g(generator.) 64 b(This) 39 b(construction) g(is) g(regarded) h(as) f(an) g(example) g (of) 214 3901 y(ho) m(w) 34 b(irrev) m(ersible) e(b) s(eha) m(vior) g (can) h(emerge) g(from) e(a) h(rev) m(ersible) h(Hamiltonian) c (dynamics.) 41 4063 y(\(2\)) p Ff 48 w(F) -9 b(ermi) 47 b(Golden) h(Rule) f(used) i(in) f(recen) m(t) f(w) m(orks) i(on) f(the) g(return) g(to) g(equilibrium.) p Fo 214 4183 a(The) 41 b(main) d(goal) g(of) i(a) f(n) m(um) m(b) s(er) h(of) f(recen) m(t) j (pap) s(ers) e([DJ2) o(,) g(BFS,) g(JP1,) g(M]) g(is) f(to) h(pro) m(v) m(e) h(that) 214 4304 y(if) h(the) h(reserv) m(oir) h(is) e(in) g (thermal) f(state,) 46 b(then) e(the) f(coupled) g(system) p Fg 44 w(S) p Fo 37 w(+) p Fg 29 w(R) p Fo 43 w(has) g(only) f(one) 214 4424 y(normal) c(stationary) h(state.) 65 b(This) 40 b(problem) e(can) i(b) s(e) g(reform) m(ulated) f(in) m(to) g(a) g (question) h(ab) s(out) 214 4545 y(p) s(oin) m(t) 31 b(sp) s(ectrum) g(of) g(a) f(certain) h(naturally) e(de\014ned) k (self-adjoin) m(t) c(op) s(erator|the) h(Liouvillean.) 214 4665 y(An) 38 b(argumen) m(t) f(based) h(on) f(F) m(GR) f(leads) h(to) g (an) g(appropriate) f(LSO.) h(Analysis) g(of) f(this) h(LSO) g(is) 214 4785 y(the) c(k) m(ey) h(step) f(in) f(the) g(pro) s(of) g(of) f(a) h (n) m(um) m(b) s(er) h(of) f(results) g(related) g(to) g(the) h(return) g(to) e(equilibrium) 214 4906 y([DJ2].) 146 5057 y(Let) 38 b(us) f(stress) i(that) e(b) s(oth) f(in) h(\(1\)) f(and) i(\(2\)) e(w) m(e) i(consider) g(the) f(same) g(ph) m(ysical) g(system) p Fg 38 w(S) p Fo 33 w(+) p Fg 25 w(R) p Fo(.) 0 5178 y(Nev) m (ertheless,) e(these) f(t) m(w) m(o) f(applications) e(are) h(quite) h (di\013eren) m(t.) p 90 rotate dyy eop %%Page: 3 3 3 2 bop Fo 3731 100 a(3) 146 407 y(The) 31 b(di\013erence) f(that) g (is) f(visible) f(at) i(the) g(\014rst) g(sigh) m(t) f(is) g(that) h (in) f(\(1\)) g(w) m(e) i(use) f(the) g(v) -5 b(an) 30 b(Ho) m(v) m(e) h(limit,) 0 527 y(whereas) 41 b(in) d(\(2\)) h(w) m(e) h (use) h(the) e(sp) s(ectral) g(form) f(of) h(F) m(GR.) g(This) g (di\013erence) h(is) f(due) h(to) e(our) i(ph) m(ysical) 0 648 y(motiv) -5 b(ation.) 39 b(Mathematically) -8 b(,) 27 b(one) i(can) h(also) e(consider) h(the) g(v) -5 b(an) 29 b(Ho) m(v) m(e) h(limit) c(for) i(the) h(Liouvillean,) 0 768 y(ev) m(en) 34 b(though) f(to) f(our) g(kno) m(wledge) i(it) d(do) s (es) j(not) e(ha) m(v) m(e) i(a) e(clear) g(ph) m(ysical) h (signi\014cance.) 146 888 y(The) 28 b(more) e(imp) s(ortan) m(t) e (di\013erence) j(is) f(that) g(in) g(\(1\)) g(and) h(\(2\)) p Fm 26 w(P) p Fo 40 w(and) p Fm 26 w(L) p Fl 2706 903 a(\025) p Fo 2778 888 a(are) g(di\013eren) m(t) f(mathemat-) 0 1009 y(ical) 37 b(ob) 5 b(jects.) 61 b(In) 39 b(\(1\)) f(exp) q(\(i) p Fm(tL) p Fl 1180 1024 a(\025) p Fo 1225 1009 a(\)) g(is) g(the) g (Heisen) m(b) s(erg) h(dynamics) f(of) g(the) h(algebra) e(of) h (observ) -5 b(ables,) 0 1129 y(whereas) 31 b(in) e(\(2\)) p Fm 29 w(L) p Fl 697 1144 a(\025) p Fo 772 1129 a(is) g(the) h (so-called) e(standard) i(Liouvillean.) 40 b(In) 30 b(\(1\)) p Fm 29 w(P) p Fo 43 w(is) f(the) h(conditional) d(exp) s(ec-) 0 1249 y(tation) 33 b(on) m(to) i(the) g(observ) -5 b(ables) 35 b(of) p Fg 35 w(S) p Fo 7 w(,) h(whereas) g(in) e(\(2\)) p Fm 34 w(P) p Fo 48 w(is) g(the) h(orthogonal) e(pro) 5 b(jection) 34 b(on) m(to) h(the) 0 1370 y(v) -5 b(acuum) 38 b(sector.) 59 b(The) 39 b(LSO) f(obtained) f(in) g(\(1\)) h(is) f (di\013eren) m(t) h(from) e(the) i(LSO) g(obtained) f(in) g(\(2\).) 59 b(In) 0 1490 y(particular,) 32 b(the) i(t) m(w) m(o) g(LSO's) g(ha) m (v) m(e) h(di\013eren) m(t) f(sp) s(ectra.) 46 b(Note,) 34 b(ho) m(w) m(ev) m(er,) j(that) c(b) s(oth) g(LSO's) h(act) g(on) 0 1611 y(the) f(same) f(space:) 45 b(the) 33 b(space) h(of) e(matrices) g (describing) g(the) h(observ) -5 b(ables) 33 b(of) f(the) h(small) d (systems) p Fg 34 w(S) p Fo 7 w(.) 146 1731 y(The) 39 b(main) c(result) i(of) g(our) g(pap) s(er) g(is) g(the) h(pro) s(of) e (of) h(the) h(follo) m(wing) c(fact:) 53 b(if) 36 b(the) i(reserv) m (oir) f(is) g(in) 0 1851 y(thermal) 27 b(equlibrium,) g(then) i(the) g (t) m(w) m(o) g(LSO's) g(are) f(related) g(b) m(y) h(a) f(similarit) m (y) d(transformation.) 40 b(Th) m(us,) 0 1972 y(in) 32 b(particular,) f(in) h(the) h(thermal) e(case,) i(they) h(are) e(isosp) s(ectral.) 146 2092 y(Our) 37 b(result) f(is) g(an) g(example) g(of) g (sp) s(ecial) f(prop) s(erties) h(enjo) m(y) m(ed) i(b) m(y) f(thermal) e(equilibrium) f(states) 0 2213 y([BR2].) 52 b(In) 36 b(order) f(to) h(form) m(ulate) e(it) g(w) m(e) j(need) f(to) f(use) i (some) e(\(relativ) m(ely) f(few\)) i(concepts) h(b) s(elonging) 0 2333 y(to) g(the) h(area) f(of) g(op) s(erator) f(algebras.) 57 b(In) 38 b(particular,) f(the) h(fact) f(that) g(the) h(reserv) m(oir) g (is) e(in) h(thermal) 0 2453 y(equilibrium) 26 b(is) i(expressed) k(b) m (y) d(the) h(KMS) f(condition) e(of) h(the) i(reserv) m(oir) f(state) g (wrt) g(the) g(unp) s(erturb) s(ed) 0 2574 y(dynamics.) 146 2694 y(In) j(this) f(note) h(w) m(e) g(also) e(consider) i(a) f(3rd) g (application) e(of) i(the) h(F) -8 b(ermi) 29 b(Golden) i(Rule) f(to) h (the) h(study) 0 2814 y(of) 26 b(small) d(systems) 28 b(coupled) e(to) f(a) h(reserv) m(oir.) 42 b(In) 26 b(this) g (application) d(the) k(main) d(ob) 5 b(ject) 27 b(is) e(the) i (so-called) 0 2935 y(C-Liouvillean) g(in) m(tro) s(duced) j(in) f ([JP5].) 43 b(W) -8 b(e) 30 b(sho) m(w) h(that) e(this) h(application) d (is) i(essen) m(tially) h(equiv) -5 b(alen) m(t) 0 3055 y(to) 32 b(\(1\).) p Ff 0 3272 a(Ac) m(kno) m(wledgmen) m(t.) p Fo 75 w(W) -8 b(e) 44 b(are) g(grateful) e(to) h(H.) h(Sp) s(ohn) g (and) g(C.-A.) f(Pillet) f(useful) i(discussions.) 0 3393 y(A) h(part) h(of) f(this) g(w) m(ork) h(has) g(b) s(een) g(done) g (during) f(the) g(visit) g(of) g(J.D.) g(to) g(the) h(McGill) d(Univ) m (ersit) m(y) -8 b(,) 0 3513 y(whic) m(h) 41 b(w) m(as) g(supp) s(orted) h(b) m(y) f(NSER) m(C,) h(and) e(during) g(the) h(visit) e(to) h(Aarh) m (us) h(Univ) m(ersit) m(y) h(supp) s(orted) 0 3633 y(b) m(y) i(MaPh) m (ySto) h(funded) f(b) m(y) h(the) e(Danish) g(National) e(Researc) m(h) k(F) -8 b(oundation.) 74 b(His) 44 b(researc) m(h) h(w) m(as) 0 3754 y(also) 39 b(partly) g(supp) s(orted) h(b) m(y) h(the) f(P) m (ostdo) s(ctoral) e(T) -8 b(raining) 38 b(Program) h(HPRN-CT-2002-0277) e(and) 0 3874 y(b) m(y) 26 b(KBN) f(\(gran) m(ts) g(SPUB127) g(and) g (2) g(P03A) g(027) f(25\).) 40 b(The) 26 b(researc) m(h) h(of) d(V.J.) h (w) m(as) h(partly) f(supp) s(orted) 0 3994 y(b) m(y) 33 b(NSER) m(C.) p Fn 0 4326 a(2) 161 b(Small) 46 b(quan) l(tum) d(system) h(in) l(teracting) g(with) g(a) i(reserv) l(oir) p Fo 0 4546 a(Consider) 39 b(a) f(small) f(quan) m(tum) i(system) p Fg 39 w(S) p Fo 47 w(in) m(teracting) e(with) h(a) h(reserv) m(oir) p Fg 39 w(R) p Fo(.) 62 b(The) 39 b(Hilb) s(ert) e(space) 0 4666 y(of) 42 b(the) h(system) p Fg 44 w(S) p Fo 51 w(is) p Fg 42 w(K) p Fo 44 w(and) g(its) f(Hamiltonian) d(is) j(a) h (self-adjoin) m(t) e(op) s(erator) p Fm 42 w(K) p Fo 7 w(.) 74 b(Its) 43 b(algebra) f(of) 0 4786 y(observ) -5 b(ables) 38 b(is) p Fg 37 w(B) p Fo 3 w(\() p Fg(K) p Fo 1 w(\),) h(the) f(Banac) m(h) g(space) g(of) f(all) f(b) s(ounded) i (op) s(erators) f(on) p Fg 37 w(K) p Fo 1 w(.) 58 b(Throughout) 38 b(the) 0 4907 y(pap) s(er) 33 b(w) m(e) g(will) e(assume) i(that) f (dim) p Fg 15 w(K) p Fm 29 w(<) p Fg 28 w(1) p Fo(.) 146 5027 y(The) 43 b(system) p Fg 42 w(R) p Fo 42 w(is) f(describ) s(ed) g (b) m(y) g(a) p Fm 42 w(W) p Fj 1703 4991 a(\003) p Fo 1742 5027 a(-dynamical) d(system) k(\() p Fe(M) p Fj 2728 5042 a(R) p Fm 2792 5027 a(;) 17 b(\034) p Fj 2878 5042 a(R) p Fo 2942 5027 a(\).) 71 b(W) -8 b(e) 42 b(assume) g(that) p Fe 0 5147 a(M) p Fj 105 5162 a(R) p Fo 205 5147 a(is) 36 b(giv) m(en) g(in) f(the) i(standard) f(form) f(on) h(the) g(Hilb) s (ert) f(space) p Fg 37 w(H) p Fj 2496 5162 a(R) p Fo 2561 5147 a(,) i(and) f(w) m(e) h(denote) g(b) m(y) p Fg 37 w(H) p Fk 3507 5106 a(+) p Fj 3506 5174 a(R) p Fo 3570 5147 a(,) p Fm 37 w(J) p Fj 3688 5162 a(R) p Fo 3752 5147 a(,) 0 5268 y(and) p Fm 33 w(L) p Fj 256 5283 a(R) p Fo 354 5268 a(the) c(corresp) s(onding) g(natural) f(cone,) i(mo) s(dular) d(conjugation,) h(and) h(standard) g(Liouvillean.) p 90 rotate dyy eop %%Page: 4 4 4 3 bop Fo 3731 100 a(4) 0 407 y(W) -8 b(e) 31 b(also) e(assume) i (that) f(\() p Fe(M) p Fj 1046 422 a(R) p Fm 1110 407 a(;) 17 b(\034) p Fj 1196 422 a(R) p Fo 1261 407 a(\)) 30 b(has) g(a) h(distingushed) f(normal) e(stationary) i(state) h(and) f (w) m(e) i(denote) 0 527 y(b) m(y) h(\012) p Fj 205 542 a(R) p Fo 303 527 a(its) f(\(unique\)) h(v) m(ector) g(represen) m (tativ) m(e) i(in) p Fg 32 w(H) p Fk 1950 486 a(+) p Fj 1949 554 a(R) p Fo 2013 527 a(.) p Fg 44 w(j) p Fo(\012) p Fj 2182 542 a(R) p Fo 2246 527 a(\)\(\012) p Fj 2392 542 a(R) p Fg 2457 527 a(j) p Fo 32 w(denotes) f(pro) 5 b(jection) 32 b(on) h(\012) p Fj 3539 542 a(R) p Fo 3604 527 a(.) 146 648 y(The) h(coupled) f(system) p Fg 33 w(S) p Fo 30 w(+) p Fg 22 w(R) p Fo 33 w(is) f(describ) s(ed) h(as) g (follo) m(ws.) 42 b(Its) 34 b(algebra) d(of) h(observ) -5 b(ables) 33 b(is) p Fe 1502 868 a(M) p Fo 27 w(=) p Fg 28 w(B) p Fo 3 w(\() p Fg(K) p Fo 1 w(\)) p Fg 23 w(\012) p Fe 22 w(M) p Fj 2186 883 a(R) p Fm 2250 868 a(;) p Fo 0 1088 a(and) g(its) f(free) p Fm 33 w(W) p Fj 619 1051 a(\003) p Fo 658 1088 a(-dynamics) g(is) p Fm 1098 1308 a(\034) p Fl 1151 1266 a(t) p Fk 1140 1332 a(0) p Fo 1181 1308 a(\() p Fm(A) p Fo(\)) c(=) f(e) p Fk 1504 1266 a(i) p Fl(tL) p Fi 1597 1243 a(semi) 1597 1287 y(0) p Fm 1725 1308 a(A) p Fo(e) p Fj 1841 1266 a(\000) p Fk(i) p Fl(tL) p Fi 1989 1243 a(semi) 1989 1287 y(0) p Fm 2116 1308 a(;) 212 b(A) p Fg 28 w(2) p Fe 28 w(M) p Fm(;) p Fo 897 w(\(2.2\)) 0 1528 y(where) p Fm 1365 1648 a(L) p Fk 1431 1607 a(semi) 1431 1673 y(0) p Fo 1601 1648 a(=) p Fm 28 w(K) p Fg 29 w(\012) p Fo 23 w(1) 22 b(+) g(1) p Fg 22 w(\012) p Fm 22 w(L) p Fj 2322 1663 a(R) p Fm 2387 1648 a(:) p Fo 1165 w(\(2.3\)) 146 1822 y(Let) p Fm 40 w(V) p Fg 62 w(2) p Fe 41 w(M) p Fo 39 w(b) s(e) 40 b(a) g(selfadjoin) m (t) f(p) s(erturbation) g(and) p Fm 40 w(\025) p Fo 40 w(a) g(real) g(parameter.) 65 b(The) 41 b(assumption) 0 1943 y(that) p Fm 28 w(V) p Fo 50 w(is) 28 b(b) s(ounded) h(is) e(made) h(only) g(for) g(simplicit) m(y) d(of) j(exp) s(osition|the) g (discussion) g(of) g(un) m(b) s(ounded) 0 2063 y(p) s(erturbations) 35 b(a\016liated) f(to) p Fe 36 w(M) p Fo 35 w(is) h(v) m(ery) i(similar) 32 b(except) 38 b(for) d(a) g(n) m(um) m(b) s(er) h(of) f(additional) e (tec) m(hnical) 0 2183 y(assumptions) f(\(see) i(Section) e(5\).) 44 b(Let) p Fm 1393 2402 a(L) p Fk 1459 2361 a(semi) p Fl 1459 2427 a(\025) p Fo 1629 2402 a(=) p Fm 28 w(L) p Fk 1799 2361 a(semi) 1799 2427 y(0) p Fo 1963 2402 a(+) p Fm 22 w(\025V) 5 b(;) 1365 2640 y(\034) p Fl 1418 2599 a(t) 1407 2665 y(\025) p Fo 1452 2640 a(\() p Fm(A) p Fo(\)) 28 b(=) g(e) p Fk 1776 2599 a(i) p Fl(tL) p Fi 1869 2576 a(semi) p Fh 1869 2622 a(\025) p Fm 1996 2640 a(A) p Fo(e) p Fj 2112 2599 a(\000) p Fk(i) p Fl(tL) p Fi 2260 2576 a(semi) p Fh 2260 2622 a(\025) p Fm 2388 2640 a(:) p Fo 0 2859 a(The) p Fm 28 w(W) p Fj 301 2823 a(\003) p Fo 340 2859 a(-dynamical) e(system) i(\() p Fe(M) p Fm(;) 17 b(\034) p Fl 1384 2874 a(\025) p Fo 1429 2859 a(\)) 27 b(describ) s(es) i(the) e(in) m(teracting) g(system) p Fg 28 w(S) p Fo 19 w(+) p Fg 12 w(R) p Fo 28 w(in) f(the) i(so) g (called) 0 2980 y(semistandard) 36 b(represen) m(tation.) 55 b(This) 37 b(represen) m(tation) g(is) f(commonly) e(used) j(in) f(the) h(literature) e(on) 0 3100 y(Mark) m(o) m(vian) e(semigroups) f(of) g (op) s(en) h(quan) m(tum) g(systems.) 146 3220 y(F) -8 b(ollo) m(wing) 34 b(the) k(terminology) c(of) j([DJ3) o(],) i(the) e (op) s(erators) p Fm 37 w(L) p Fk 2387 3184 a(semi) 2387 3245 y(0) p Fo 2566 3220 a(and) p Fm 37 w(L) p Fk 2826 3184 a(semi) p Fl 2826 3246 a(\025) p Fo 3005 3220 a(are) g(called) f (the) h(free) 0 3341 y(and) c(full) d(semi-Liouvilleans) f(resp) s (ectiv) m(ely) -8 b(.) 146 3461 y(A) 48 b(t) m(ypical) e(example) g(of) h(the) h(reserv) m(oir) f(system) i(is) d(a) h(free) h(F) -8 b(ermi) 45 b(or) i(Bose) g(gas) h(in) e(thermal) 0 3581 y(equilibrium) d(at) i(in) m(v) m(erse) j(temp) s(erature) p Fm 45 w(\014) 56 b(>) p Fo 50 w(0.) 83 b(The) 47 b(reserv) m(oir) f(ma) m(y) g(also) f(ha) m(v) m(e) i(a) f(comp) s(osite) 0 3702 y(structure) 39 b(and) f(consist) h(of) p Fm 37 w(N) p Fo 10 w(-subreserv) m(oirs) h(at) e(di\013eren) m(t) g(temp) s (eratures) h(\(suc) m(h) g(reserv) m(oirs) g(ha) m(v) m(e) 0 3822 y(b) s(een) 27 b(studied) f(in) f(the) i(literature) d(on) i (non-equilibrium) d(quan) m(tum) j(statistical) e(mec) m(hanics,) k (see) f([JP4,) 0 3943 y(JP5,) 36 b(LeSp) q(,) f(Ru]\).) 53 b(F) -8 b(or) 34 b(our) i(purp) s(oses,) h(it) e(is) g(natural) f(to) h (k) m(eep) j(the) e(reserv) m(oir) g(system) g(as) g(general) 0 4063 y(as) d(p) s(ossible.) 146 4183 y(The) d(e\013ect) f(of) f(the) h (reserv) m(oir) g(on) g(the) g(dynamics) f(of) p Fg 28 w(S) p Fo 36 w(in) g(the) h(w) m(eak) h(coupling) d(regime) h(\() p Fm(\025) p Fo 28 w(small\)) 0 4304 y(has) 39 b(b) s(een) h(sub) 5 b(ject) 40 b(of) e(man) m(y) h(studies.) 63 b(A) 39 b(traditional) c (approac) m(h) k(to) g(this) f(question) h(has) g(b) s(een) h(to) 0 4424 y(in) m(tegrate) 22 b(the) i(v) -5 b(ariables) 21 b(of) i(the) g(reserv) m(oir) g(and) g(follo) m(w) e(the) i(reduced) i (dynamics) d(of) h(the) g(small) e(system) 0 4545 y(on) 26 b(the) h(V) -8 b(an) 26 b(Ho) m(v) m(e) i(time) d(scale) p Fm 26 w(t=\025) p Fk 1307 4508 a(2) p Fo 1346 4545 a(.) 42 b(In) 26 b(the) h(V) -8 b(an) 26 b(Ho) m(v) m(e) i(w) m(eak) f (coupling) e(limit) p Fm 23 w(\025) p Fg 28 w(!) p Fo 27 w(0,) i(the) g(reduced) 0 4665 y(dynamics) 40 b(of) p Fg 40 w(S) p Fo 49 w(b) s(ecomes) g(Mark) m(o) m(vian) h(and) g(irrev) m (ersible.) 67 b(Its) 41 b(generator|often) f(computed) h(b) m(y) 0 4785 y(a) f(formal) d(F) -8 b(ermi) 38 b(Golden) i(Rule) f (calculation|captures) f(the) j(basic) f(ph) m(ysical) f(pro) s(cesses) j(\(energy) 0 4906 y(emission/absorption\)) 27 b(of) i(op) s(en) h (quan) m(tum) f(system) p Fg 31 w(S) p Fo 37 w(to) g(the) g(2nd) h (order) f(of) g(p) s(erturbation) g(theory) -8 b(.) 146 5026 y(This) 36 b(approac) m(h) g(can) g(b) s(e) g(traced) h(bac) m(k) f (to) g(the) g(w) m(orks) h(of) e(P) m(auli,) h(Wigner-W) -8 b(eissk) m(opf) 36 b(and) g(V) -8 b(an) 0 5146 y(Ho) m(v) m(e) 39 b([P) m(a1,) f(P) m(a2) q(,) g(W,) g(VH],) h(and) f(has) h(b) s(ecome) e (a) h(source) h(of) e(man) m(y) h(w) m(orks) h(in) f(ph) m(ysics) h (literature) 0 5267 y(\(see) 34 b(e.g.) 43 b([Haa,) 33 b(KTH]) g(for) f(references) i(and) f(additional) d(information\).) p 90 rotate dyy eop %%Page: 5 5 5 4 bop Fo 3731 100 a(5) 146 407 y(On) 35 b(the) g(mathematical) d (side,) j(the) g(\014rst) h(complete) e(results) h(concerning) g (existence) h(of) e(the) i(V) -8 b(an) 0 527 y(Ho) m(v) m(e) 43 b(limit) c(and) j(form) f(of) h(the) h(Mark) m(o) m(vian) f(generator) g (w) m(ere) i(obtained) e(b) m(y) h(Da) m(vies) f([Da1) o(,) g(Da2) o (].) 0 648 y(These) 54 b(pap) s(ers) e(w) m(ere) h(follo) m(w) m(ed) e (b) m(y) h(a) g(large) e(b) s(o) s(dy) i(of) f(mathematical) e(ph) m (ysics) k(literature) d(\(see) 0 768 y([GFVKS) o(]) 28 b(for) e(a) h(review) h(of) f(early) g(results\).) 42 b(The) 28 b(Da) m(vies) f(theory) h(and) f(early) g(mathematical) d (results) 0 888 y(in) 37 b(the) h(theory) g(of) f(Mark) m(o) m(vian) h (generators) f(of) g(op) s(en) h(quan) m(tum) g(systems) h(are) e (discussed) i(in) e(detail) 0 1009 y(in) 32 b(the) h(forthcoming) d (article) i([DJ3) o(].) 146 1129 y(The) f(in) m(tegration) e(of) g(the) i(reserv) m(oir) g(v) -5 b(ariables) 28 b(is) i(formalized) e(as) i (follo) m(ws.) 41 b(W) -8 b(e) 31 b(will) d(w) m(ork) j(in) e(the) 0 1249 y(Heisen) m(b) s(erg) k(picture.) 44 b(F) -8 b(or) p Fm 32 w(B) p Fg 27 w(\012) p Fm 23 w(C) p Fg 34 w(2) 28 b(B) p Fo 3 w(\() p Fg(K) c(\012) f(H) p Fj 1829 1264 a(R) p Fo 1893 1249 a(\)) 33 b(let) p Fm 1224 1453 a(P) p Fk 1287 1468 a(H) p Fo 1344 1453 a(\() p Fm(B) p Fg 27 w(\012) p Fm 23 w(C) p Fo 7 w(\)) 27 b(=) h(\(\012) p Fj 1937 1468 a(R) p Fg 2001 1453 a(j) p Fm(C) p Fo 7 w(\012) p Fj 2176 1468 a(R) p Fo 2241 1453 a(\)) p Fm(B) p Fg 27 w(\012) p Fo 23 w(1) p Fm(:) p Fo 0 1656 a(The) 45 b(map) p Fm 43 w(P) p Fk 503 1671 a(H) p Fo 604 1656 a(uniquely) e(extends) j(to) e(a) f(pro) 5 b(jection) 44 b(on) g(the) g(Banac) m(h) g(space) p Fg 45 w(B) p Fo 3 w(\() p Fg(K) 32 b(\012) e(H) p Fj 3437 1671 a(R) p Fo 3502 1656 a(\).) 77 b(W) -8 b(e) 0 1776 y(iden) m(tify) p Fg 32 w(B) p Fo 3 w(\() p Fg(K) p Fo 1 w(\)) 33 b(with) f(Ran) p Fm(P) p Fk 1069 1791 a(H) p Fo 1159 1776 a(b) m(y) p Fg 33 w(B) p Fo 3 w(\() p Fg(K) p Fo 1 w(\)) p Fg 28 w(3) p Fm 28 w(B) p Fg 33 w(7!) p Fm 28 w(B) p Fg 27 w(\012) p Fo 22 w(1.) 44 b(Ob) m(viously) -8 b(,) 32 b(for) p Fm 32 w(X) r(;) 17 b(B) p Fg 33 w(2) 28 b(B) p Fo 3 w(\() p Fg(K) p Fo 1 w(\),) 613 1979 y(T) -8 b(r) p Fj 713 1994 a(K\012H) p Fd 882 2005 a(R) p Fc 943 1899 a(\000) p Fm 989 1979 a(X) p Fg 8 w(\012j) p Fo(\012) p Fj 1253 1994 a(R) p Fo 1317 1979 a(\)\(\012) p Fj 1463 1994 a(R) p Fg 1528 1979 a(j) p Fm 32 w(\034) p Fj 1641 1938 a(\000) p Fl(t) p Fk 1630 2004 a(0) p Fm 1726 1979 a(\034) p Fl 1779 1938 a(t) 1768 2004 y(\025) p Fo 1814 1979 a(\() p Fm(B) p Fg 5 w(\012) p Fo 1 w(1\)) p Fc 2096 1899 a(\001) p Fo 2169 1979 a(=) 27 b(T) -8 b(r) p Fj 2372 1994 a(K) p Fo 2431 1979 a(\() p Fm(X) 8 b(P) p Fk 2621 1994 a(H) p Fm 2677 1979 a(\034) p Fj 2730 1938 a(\000) p Fl(t) p Fk 2719 2004 a(0) p Fm 2815 1979 a(\034) p Fl 2868 1938 a(t) 2857 2004 y(\025) p Fm 2903 1979 a(P) p Fk 2966 1994 a(H) p Fm 3023 1979 a(B) p Fo 5 w(\)) p Fm(:) p Fo 0 2183 a(The) 34 b(maps) p Fm 1214 2303 a(T) p Fl 1285 2262 a(t) 1271 2328 y(\025) p Fo 1344 2303 a(:=) p Fm 28 w(P) p Fk 1538 2318 a(H) p Fm 1595 2303 a(\034) p Fj 1648 2262 a(\000) p Fl(t) p Fk 1637 2328 a(0) p Fm 1733 2303 a(\034) p Fl 1786 2262 a(t) 1775 2328 y(\025) p Fm 1820 2303 a(P) p Fk 1883 2318 a(H) p Fo 1968 2303 a(:) p Fg 28 w(B) p Fo 3 w(\() p Fg(K) p Fo 1 w(\)) p Fg(!B) p Fo 3 w(\() p Fg(K) p Fo 1 w(\)) 1014 b(\(2.4\)) 0 2470 y(describ) s(e) 32 b(the) h(reduced) g(dynamics) f(of) p Fg 31 w(S) p Fo 39 w(in) f(the) i(Heisen) m(b) s(erg) f(picture.) 44 b(The) 32 b(family) p Fg 30 w(f) p Fm(T) p Fl 3309 2434 a(t) 3295 2496 y(\025) p Fg 3340 2470 a(g) p Fl 3390 2485 a(t) p Fj(\025) p Fk(0) p Fo 3541 2470 a(is) g(not) 0 2611 y(a) j(semigroup.) 49 b(Ho) m(w) m(ev) m(er,) 38 b(one) d(exp) s(ects) i(that) p Fm 34 w(T) p Fl 1833 2560 a(t=\025) p Fi 1934 2537 a(2) p Fl 1819 2639 a(\025) p Fo 2008 2611 a(con) m(v) m(erges) h(to) c(a) h(semigroup) f(as) p Fm 35 w(\025) p Fg 31 w(!) p Fo 31 w(0.) 50 b(This) 0 2732 y(limiting) 33 b(semigroup) j(describ) s(es) i(the) f(dynamics) g (of) g(op) s(en) g(quan) m(tum) h(system) p Fg 38 w(S) p Fo 44 w(in) f(the) g(V) -8 b(an) 37 b(Ho) m(v) m(e) 0 2852 y(w) m(eak) d(coupling) d(limit.) 146 2972 y(F) -8 b(or) 32 b(our) g(purp) s(oses,) i(the) f(only) f(imp) s(ortan) m(t) f (thing) h(is) g(that) g(the) h(V) -8 b(an) 32 b(Ho) m(v) m(e) i(w) m (eak) g(coupling) d(limit) 0 3093 y(exists) e(and) g(particular) e (conditions) h(whic) m(h) h(quaran) m(tee) h(the) f(existence) h(of) e (the) h(limit) c(are) k(inessen) m(tial.) 0 3213 y(Hence,) 34 b(w) m(e) g(p) s(ostulate:) p Ff 0 3402 a(Assumption) i(2.A) p Fs 49 w(Ther) -5 b(e) 34 b(exists) h(an) f(op) -5 b(er) g(ator) p Fo 35 w(\000) p Fk 1973 3417 a(H) p Fo 2057 3402 a(:) p Fg 28 w(B) p Fo 3 w(\() p Fg(K) p Fo 1 w(\)) p Fg(!B) p Fo 3 w(\() p Fg(K) p Fo 1 w(\)) p Fs 36 w(such) 35 b(that) g(for) p Fm 35 w(t) p Fg 27 w(\025) p Fo 29 w(0) p Fs(,) p Fo 1300 3625 a(lim) p Fl 1295 3687 a(\025) p Fj(!) p Fk(0) p Fm 1458 3625 a(P) p Fk 1521 3640 a(H) p Fm 1578 3625 a(\034) p Fj 1631 3574 a(\000) p Fl(t=\025) p Fi 1787 3550 a(2) p Fk 1620 3649 a(0) p Fm 1827 3625 a(\034) p Fl 1880 3574 a(t=\025) p Fi 1981 3550 a(2) p Fl 1869 3652 a(\025) p Fm 2021 3625 a(P) p Fk 2084 3640 a(H) p Fo 2169 3625 a(=) 27 b(e) p Fk 2315 3583 a(i) p Fl(t) p Fk(\000) p Fi 2404 3594 a(H) p Fm 2458 3625 a(:) p Fo 1094 w(\(2.5\)) 146 4017 y(W) -8 b(e) 32 b(will) c(call) h(the) j(op) s (erator) e(\000) p Fk 1291 4032 a(H) p Fo 1379 4017 a(the) h(Da) m (vies) g(generator) g(in) f(the) i(Heisen) m(b) s(erg) f(picture.) 43 b(A) 31 b(F) -8 b(ermi) 0 4137 y(Golden) 32 b(Rule) g(computation) f (yields) h(that) 288 4352 y(\000) p Fk 349 4367 a(H) p Fo 434 4352 a(=) 27 b(lim) p Fl 538 4415 a(\017) p Fj(\045) p Fk(0) p Fc 820 4258 a(X) p Fl 722 4473 a(e) p Fj(2) p Fk(sp\([) p Fl(K) q(;) p Fj(\001) p Fk(]\)) p Fo 1079 4352 a(1) p Fl 1128 4367 a(e) p Fo 1165 4352 a(\([) p Fm(K) r(;) p Fg 17 w(\001) p Fo 17 w(]\)\([) p Fm(V) 5 b(;) p Fg 17 w(\001) p Fo 17 w(]\)\() p Fm(e) p Fo 20 w(+) 22 b(i) p Fm(\017) p Fg 22 w(\000) p Fo 22 w([) p Fm(L) p Fk 2232 4311 a(semi) 2232 4377 y(0) p Fm 2375 4352 a(;) p Fg 17 w(\001) p Fo 17 w(]\)) p Fj 2529 4311 a(\000) p Fk(1) p Fo 2622 4352 a(\([) p Fm(V) 5 b(;) p Fg 17 w(\001) p Fo 17 w(]\)1) p Fl 2952 4367 a(e) p Fo 2988 4352 a(\([) p Fm(K) r(;) p Fg 17 w(\001) p Fo 17 w(]\)) 287 b(\(2.6\)) 0 4666 y(\(sp) q(\() p Fm(A) p Fo(\)) 37 b(stands) i(for) e(the) h(sp) s(ectrum) g(of) f(the) h(op) s (erator) p Fm 37 w(A) p Fo 38 w(and) g(1) p Fl 2427 4681 a(e) p Fo 2463 4666 a(\() p Fm(A) p Fo(\)) g(for) f(the) h(sp) s (ectral) g(pro) 5 b(jection) 0 4786 y(on) m(to) p Fm 27 w(e) p Fg 28 w(2) p Fo 28 w(sp) q(\() p Fm(A) p Fo(\)\),) 28 b(and) f(indeed) h(one) f(can) h(pro) m(v) m(e) g(this) f(form) m(ula) e (under) j(v) m(ery) h(general) d(conditions) g(\(see) 0 4907 y([Da1) o(,) 33 b(DJ3) o(]\).) 44 b(Ho) m(w) m(ev) m(er,) 35 b(the) e(sp) s(eci\014c) g(form) e(of) h(\000) p Fk 1897 4922 a(H) p Fo 1987 4907 a(will) e(not) i(concern) i(us) f(here.) 146 5027 y(In) 42 b(the) g(last) f(sev) m(eral) h(y) m(ears) g(there) g (has) g(b) s(een) g(a) f(reviv) -5 b(al) 41 b(of) g(in) m(terest) h(in) e(rigorous) g(mathemat-) 0 5147 y(ical) 51 b(study) j(of) e(the) h(mo) s (dels) e(\() p Fe(M) p Fm(;) 17 b(\034) p Fl 1384 5162 a(\025) p Fo 1429 5147 a(\).) 104 b(These) 54 b(studies) f(w) m(ere) h (based) g(on) e(mathematical) e(tec) m(h-) 0 5268 y(niques) 27 b(\(T) -8 b(omita-T) g(ak) m(esaki) 24 b(mo) s(dular) g(theory) -8 b(,) 28 b(quan) m(tum) e(Ko) s(opmanism,) f(Mourre) i(theory) -8 b(,) 28 b(sp) s(ectral) p 90 rotate dyy eop %%Page: 6 6 6 5 bop Fo 3731 100 a(6) 0 407 y(complex) 38 b(deformations\)) g(whic) m (h) h(allo) m(w) m(ed) g(for) f(detailed) g(understanding) h(of) g(the) g(dynamics.) 63 b(The) 0 527 y(emerging) 41 b(picture) i(is) f(that) g (ergo) s(dic) g(prop) s(erties) g(and) h(thermo) s(dynamics) e(of) h (the) h(system) p Fg 44 w(S) p Fo 36 w(+) p Fg 29 w(R) p Fo 0 648 a(are) 38 b(con) m(trolled) g(b) m(y) p Fs 39 w(sp) -5 b(e) g(ctr) g(al) 40 b(r) -5 b(esonanc) g(es) p Fo 37 w(of) 37 b(t) m(w) m(o) i(op) s(erators,) h(the) f(standard) g (Liouvillean) c(and) k(C-) 0 768 y(Liouvillean,) 32 b(canonically) g (asso) s(ciated) i(to) f(the) i(pair) e(\() p Fe(M) p Fm(;) 17 b(\034) p Fl 2227 783 a(\025) p Fo 2272 768 a(\)) 34 b(b) m(y) h(T) -8 b(omita-T) g(ak) m(esaki) 33 b(mo) s(dular) f(the-) 0 888 y(ory) f([JP1) q(,) g(JP2,) g(JP4) q(,) g (JP5].) 43 b(A) 31 b(natural) f(and) i(imp) s(ortan) m(t) d(question) j (is) e(ho) m(w) i(is) f(the) g(sp) s(ectral) g(F) -8 b(ermi) 0 1009 y(Golden) 30 b(Rule) g(for) g(these) i(resonances) g (related) f(to) f(the) h(generator) g(\000) p Fk 2538 1024 a(H) p Fo 2595 1009 a(.) 42 b(T) -8 b(o) 31 b(describ) s(e) g(the) g(answ) m(er) h(w) m(e) 0 1129 y(will) e(consider) j(separately) g(the) g(thermal) e(equilibrium) f(and) i(the) h(nonequilibrium) d(case.) p Fn 0 1462 a(3) 161 b(Thermal) 54 b(equilibrium) h(case) p Fo 0 1681 a(W) -8 b(e) 35 b(will) e(freely) i(use) g(the) h(language) d (and) i(notation) f(of) g(algebraic) f(quan) m(tum) i(statistical) e (mec) m(hanics) 0 1801 y(and) g(T) -8 b(omita-T) g(ak) m(esaki) 31 b(mo) s(dular) f(theory) -8 b(.) 45 b(The) 33 b(b) s(o) s(oks) g([BR1) o (,) g(BR2,) f(Ha,) h(St,) g(StZs]) f(are) h(standard) 0 1922 y(references.) 46 b(A) 32 b(mo) s(dern) g(exp) s(osition) g(can) h (b) s(e) f(also) g(found) h(in) f(the) h(recen) m(t) h(article) d ([DJP].) 146 2042 y(In) 39 b(this) e(section) h(the) h(distinguished) e (in) m(v) -5 b(arian) m(t) 37 b(state) p Fe 38 w(M) p Fj 2335 2057 a(R) p Fg 2436 2042 a(3) p Fm 37 w(A) p Fg 37 w(7!) p Fo 37 w(\(\012) p Fj 2894 2057 a(R) p Fg 2958 2042 a(j) p Fm(A) p Fo(\012) p Fj 3129 2057 a(R) p Fo 3194 2042 a(\)) h(is) f(a) h(\() p Fm(\034) p Fj 3540 2057 a(R) p Fm 3605 2042 a(;) 17 b(\014) p Fo 6 w(\)-) 0 2162 y(KMS) 27 b(state) h(for) e(some) p Fm 27 w(\014) 34 b(>) p Fo 27 w(0) 27 b(\(in) f(other) i(w) m(ords,) h(the) e(reserv) m (oir) h(is) f(initially) c(in) j(thermal) f(equilibrium) 0 2283 y(at) 32 b(in) m(v) m(erse) i(temp) s(erature) p Fm 33 w(\014) p Fo 6 w(\).) 146 2403 y(The) h(inner) e(pro) s(duct) g (\() p Fm(X) p Fg 8 w(j) p Fm(B) p Fo 5 w(\)) c(=) g(T) -8 b(r\() p Fm(X) p Fj 1597 2367 a(\003) p Fm 1636 2403 a(B) p Fo 5 w(\)) 33 b(mak) m(es) p Fg 34 w(B) p Fo 3 w(\() p Fg(K) p Fo 1 w(\)) h(in) m(to) f(Hilb) s(ert) f(space,) i (denoted) p Fm 35 w(l) p Fk 3560 2367 a(2) p Fo 3599 2403 a(\() p Fg(K) p Fo 1 w(\).) 0 2524 y(Note) g(that) p Fg 34 w(B) p Fo 3 w(\() p Fg(K) p Fo 1 w(\)) g(acts) h(naturally) d(on) p Fm 34 w(l) p Fk 1497 2487 a(2) p Fo 1537 2524 a(\() p Fg(K) p Fo 1 w(\)) i(b) m(y) h(righ) m(t) e(m) m(ultiplication.) 43 b(This) 34 b(de\014nes) i(a) d(represen-) 0 2644 y(tation) p Fm 33 w(\031) p Fj 344 2659 a(S) p Fo 428 2644 a(:) p Fg 31 w(B) p Fo 3 w(\() p Fg(K) p Fo 1 w(\)) p Fg 32 w(!) e(B) p Fo 3 w(\() p Fm(l) p Fk 1007 2608 a(2) p Fo 1047 2644 a(\() p Fg(K) p Fo 1 w(\)\).) 50 b(Let) p Fm 35 w(J) p Fj 1546 2659 a(S) p Fo 1629 2644 a(:) p Fm 31 w(l) p Fk 1718 2608 a(2) p Fo 1758 2644 a(\() p Fg(K) p Fo 1 w(\)) p Fg(!) p Fm(l) p Fk 2042 2608 a(2) p Fo 2082 2644 a(\() p Fg(K) p Fo 1 w(\)) 34 b(b) s(e) h(de\014ned) h(b) m (y) p Fm 36 w(J) p Fj 2934 2659 a(S) p Fo 2986 2644 a(\() p Fm(X) p Fo 8 w(\)) 31 b(=) p Fm 31 w(X) p Fj 3378 2608 a(\003) p Fo 3417 2644 a(,) k(and) g(let) p Fm 0 2764 a(l) p Fk 31 2728 a(2) 29 2789 y(+) p Fo 88 2764 a(\() p Fg(K) p Fo 1 w(\)) e(b) s(e) g(the) h(set) g(of) e(all) f(p) s(ositiv) m (e) p Fm 32 w(X) p Fg 37 w(2) p Fm 29 w(l) p Fk 1581 2728 a(2) p Fo 1620 2764 a(\() p Fg(K) p Fo 1 w(\).) 45 b(The) 34 b(algebra) p Fm 32 w(\031) p Fj 2443 2779 a(S) p Fo 2495 2764 a(\() p Fg(B) p Fo 3 w(\() p Fg(K) p Fo 1 w(\)\)) g(is) e(in) g(standard) i(form) e(on) 0 2885 y(the) f(Hilb) s(ert) f(space) p Fm 32 w(l) p Fk 791 2849 a(2) p Fo 830 2885 a(\() p Fg(K) p Fo 1 w(\),) i(and) f(its) f (natural) g(cone) i(and) f(mo) s(dular) e(conjugation) h(are) p Fm 30 w(l) p Fk 3217 2849 a(2) 3215 2909 y(+) p Fo 3275 2885 a(\() p Fg(K) p Fo 1 w(\)) h(and) p Fm 31 w(J) p Fj 3701 2900 a(S) p Fo 3752 2885 a(.) 146 3005 y(The) j(represen) m (tation) p Fm 33 w(\031) p Fj 1039 3020 a(S) p Fo 1124 3005 a(extends) g(to) f(a) f(represen) m(tation) p Fm 33 w(\031) p Fo 32 w(:) p Fe 27 w(M) p Fg(!B) p Fo 3 w(\() p Fm(l) p Fk 2800 2969 a(2) p Fo 2840 3005 a(\() p Fg(K) p Fo 1 w(\)) p Fg 22 w(\012) 23 b(H) p Fj 3199 3020 a(R) p Fo 3263 3005 a(\)) 33 b(b) m(y) p Fm 1377 3225 a(\031) p Fo 4 w(\() p Fm(B) p Fg 27 w(\012) p Fm 22 w(C) p Fo 7 w(\)) 28 b(=) p Fm 27 w(\031) p Fj 1975 3240 a(S) p Fo 2028 3225 a(\() p Fm(B) p Fo 5 w(\)) p Fg 22 w(\012) p Fm 22 w(C) r(:) p Fo 1176 w(\(3.7\)) 0 3445 y(The) 38 b(v) m(on) g(Neumann) g(algebra) p Fm 36 w(\031) p Fo 4 w(\() p Fe(M) p Fo(\)) f(is) g(in) f(standard) i (form) e(on) i(the) f(Hilb) s(ert) f(space) p Fm 39 w(l) p Fk 3283 3409 a(2) p Fo 3322 3445 a(\() p Fg(K) p Fo 1 w(\)) p Fg 26 w(\012) 26 b(H) p Fj 3688 3460 a(R) p Fo 3752 3445 a(.) 0 3566 y(The) 34 b(natural) d(cone) i(and) g(the) g(mo) s (dular) d(conjugation) i(are) p Fg 1093 3786 a(H) p Fk 1178 3744 a(+) p Fo 1265 3786 a(=) p Fm 28 w(l) p Fk 1400 3744 a(2) 1398 3810 y(+) p Fo 1457 3786 a(\() p Fg(K) p Fo 1 w(\)) p Fg 22 w(\012) 23 b(H) p Fk 1817 3744 a(+) p Fj 1816 3813 a(R) p Fm 1880 3786 a(;) 212 b(J) p Fo 37 w(=) p Fm 27 w(J) p Fj 2367 3801 a(S) p Fg 2441 3786 a(\012) p Fm 23 w(J) p Fj 2595 3801 a(R) p Fm 2659 3786 a(:) p Fo 146 4006 a(The) 36 b(standard) f(Liouvillean,) p Fm 33 w(L) p Fl 1357 4021 a(\025) p Fo 1403 4006 a(,) g(is) g(the) g (unique) g(selfadjoin) m(t) f(op) s(erator) g(on) p Fm 35 w(l) p Fk 3091 3969 a(2) p Fo 3131 4006 a(\() p Fg(K) p Fo 1 w(\)) p Fg 23 w(\012) 25 b(H) p Fj 3493 4021 a(R) p Fo 3592 4006 a(suc) m(h) 0 4126 y(that) p Fm 892 4246 a(\031) p Fo 4 w(\() p Fm(\034) p Fl 1042 4205 a(t) 1031 4271 y(\025) p Fo 1077 4246 a(\() p Fm(A) p Fo(\)\)) i(=) h(e) p Fk 1438 4205 a(i) p Fl(tL) p Fh 1531 4217 a(\025) p Fm 1576 4246 a(\031) p Fo 4 w(\() p Fm(A) p Fo(\)e) p Fj 1827 4205 a(\000) p Fk(i) p Fl(tL) p Fh 1975 4217 a(\025) p Fm 2021 4246 a(;) p Fo 211 w(e) p Fk 2302 4205 a(i) p Fl(tL) p Fh 2395 4217 a(\025) p Fg 2441 4246 a(H) p Fk 2526 4205 a(+) p Fo 2613 4246 a(=) p Fg 27 w(H) p Fk 2801 4205 a(+) p Fm 2860 4246 a(:) p Fo 0 4421 a(\() p Fm(L) p Fl 104 4436 a(\025) p Fo 190 4421 a(implemen) m(ts) 39 b(the) j(dynamics) e(in) g(the) h(represen) m(tation) p Fm 41 w(\031) p Fo 45 w(and) g(preserv) m(es) i(the) e(natural) f (cone\).) 0 4541 y(One) 33 b(easily) f(sho) m(ws) i(that) p Fm 1253 4661 a(L) p Fl 1319 4676 a(\025) p Fo 1392 4661 a(=) p Fm 28 w(L) p Fk 1562 4676 a(0) p Fo 1624 4661 a(+) p Fm 22 w(\025\031) p Fo 4 w(\() p Fm(V) p Fo 21 w(\)) p Fg 22 w(\000) p Fm 23 w(\025J) 9 b(\031) p Fo 4 w(\() p Fm(V) p Fo 21 w(\)) p Fm(J) n(;) p Fo 1053 w(\(3.8\)) 0 4836 y(where) p Fm 1354 4956 a(L) p Fk 1420 4971 a(0) p Fo 1487 4956 a(=) 28 b([) p Fm(K) r(;) p Fg 33 w(\001) p Fo 17 w(]) p Fg 21 w(\012) p Fo 23 w(1) 22 b(+) g(1) p Fg 22 w(\012) p Fm 22 w(L) p Fj 2361 4971 a(R) p Fo 3579 4956 a(\(3.9\)) 0 5130 y(see) 34 b(e.g.) 43 b([DJP].) p 90 rotate dyy eop %%Page: 7 7 7 6 bop Fo 3731 100 a(7) 146 407 y(Consider) 40 b(the) g(pro) 5 b(jection) p Fm 40 w(P) p Fk 1269 422 a(L) p Fo 1357 407 a(:=) 39 b(1) p Fg 27 w(\012) 28 b(j) p Fo(\012) p Fj 1778 422 a(R) p Fo 1842 407 a(\)\(\012) p Fj 1988 422 a(R) p Fg 2053 407 a(j) p Fo 39 w(on) 40 b(the) g(Hilb) s(ert) e (space) p Fm 41 w(l) p Fk 3080 371 a(2) p Fo 3119 407 a(\() p Fg(K) p Fo 1 w(\)) p Fg 27 w(\012) 28 b(H) p Fj 3488 422 a(R) p Fo 3552 407 a(.) 65 b(W) -8 b(e) 0 527 y(iden) m(tify) p Fm 32 w(l) p Fk 386 491 a(2) p Fo 426 527 a(\() p Fg(K) p Fo 1 w(\)) 32 b(with) h(Ran) p Fm -1 w(P) p Fk 1071 542 a(L) p Fo 1152 527 a(b) m(y) p Fm 33 w(l) p Fk 1318 491 a(2) p Fo 1358 527 a(\() p Fg(K) p Fo 1 w(\)) p Fg 28 w(3) p Fm 28 w(B) p Fg 33 w(!) p Fm 27 w(B) p Fg 28 w(\012) p Fo 22 w(\012) p Fj 2138 542 a(R) p Fo 2203 527 a(.) 43 b(Ob) m(viously) -8 b(,) 33 b(for) p Fm 32 w(X) r(;) 17 b(B) p Fg 33 w(2) p Fm 28 w(l) p Fk 3259 491 a(2) p Fo 3299 527 a(\() p Fg(K) p Fo 1 w(\),) 675 747 y(\() p Fm(X) p Fg 30 w(\012) p Fo 22 w(\012) p Fj 993 762 a(R) p Fg 1058 747 a(j) p Fo(e) p Fj 1129 706 a(\000) p Fk(i) p Fl(tL) p Fi 1277 715 a(0) p Fo 1316 747 a(e) p Fk 1359 706 a(i) p Fl(tL) p Fh 1452 718 a(\025) p Fm 1497 747 a(B) p Fg 28 w(\012) p Fo 22 w(\012) p Fj 1768 762 a(R) p Fo 1833 747 a(\)) 28 b(=) f(T) -8 b(r) p Fj 2102 762 a(K) p Fo 2161 747 a(\() p Fm(X) p Fj 2288 706 a(\003) p Fm 2327 747 a(P) p Fk 2390 762 a(L) p Fo 2438 747 a(e) p Fj 2481 706 a(\000) p Fk(i) p Fl(tL) p Fi 2629 715 a(0) p Fo 2668 747 a(e) p Fk 2711 706 a(i) p Fl(tL) p Fh 2804 718 a(\025) p Fm 2849 747 a(P) p Fk 2912 762 a(L) p Fm 2961 747 a(B) p Fo 5 w(\)) p Fm(:) p Fo 146 967 a(W) g(e) 33 b(again) e(p) s(ostulate) h(existence) j (of) d(the) h(V) -8 b(an) 32 b(Ho) m(v) m(e) i(limit.) p Ff 0 1171 a(Assumption) i(3.A) p Fs 49 w(Ther) -5 b(e) 34 b(exists) h(an) f(op) -5 b(er) g(ator) p Fo 35 w(\000) p Fk 1973 1186 a(L) p Fo 2049 1171 a(:) p Fm 27 w(l) p Fk 2134 1134 a(2) p Fo 2174 1171 a(\() p Fg(K) p Fo 1 w(\)) p Fg(!) p Fm(l) p Fk 2458 1134 a(2) p Fo 2498 1171 a(\() p Fg(K) p Fo 1 w(\)) p Fs 35 w(such) 34 b(that) i(for) p Fm 34 w(t) p Fg 28 w(\025) p Fo 28 w(0) p Fs(,) p Fo 1218 1400 a(lim) p Fl 1212 1463 a(\025) p Fj(!) p Fk(0) p Fm 1375 1400 a(P) p Fk 1438 1415 a(L) p Fo 1486 1400 a(e) p Fj 1529 1359 a(\000) p Fk(i) p Fl(tL) p Fi 1677 1368 a(0) p Fl 1712 1359 a(=\025) p Fi 1788 1336 a(2) p Fo 1828 1400 a(e) p Fk 1871 1359 a(i) p Fl(tL) p Fh 1964 1371 a(\025) p Fl 2005 1359 a(=\025) p Fi 2081 1336 a(2) p Fm 2120 1400 a(P) p Fk 2183 1415 a(L) p Fo 2259 1400 a(=) 27 b(e) p Fk 2405 1359 a(i) p Fl(t) p Fk(\000) p Fi 2494 1370 a(L) p Fm 2541 1400 a(:) p Fo 962 w(\(3.10\)) 0 1824 y(A) 33 b(F) -8 b(ermi) 30 b(Golden) i(Rule) g(computation) f (yields) h(that) 625 2055 y(\000) p Fk 686 2070 a(L) p Fo 762 2055 a(=) c(lim) p Fl 866 2118 a(\017) p Fj(\045) p Fk(0) p Fc 1116 1961 a(X) p Fl 1018 2177 a(e) p Fj(2) p Fk(sp\([) p Fl(K) q(;) p Fj(\001) p Fk(]\)) p Fo 1358 2055 a(1) p Fl 1407 2070 a(e) p Fo 1444 2055 a(\([) p Fm(K) r(;) p Fg 17 w(\001) p Fo 17 w(]\)\)\() p Fm(\031) p Fo 4 w(\() p Fm(V) p Fo 20 w(\)) p Fg 22 w(\000) p Fm 23 w(J) 9 b(\031) p Fo 4 w(\() p Fm(V) p Fo 21 w(\)) p Fm(J) p Fo 9 w(\)) 1358 2392 y(\() p Fm(e) p Fo 23 w(+) 22 b(i) p Fm(\017) p Fg 21 w(\000) p Fm 23 w(L) p Fk 1816 2407 a(0) p Fo 1856 2392 a(\)) p Fj 1894 2350 a(\000) p Fk(1) p Fo 1988 2392 a(\() p Fm(\031) p Fo 4 w(\() p Fm(V) p Fo 21 w(\)) p Fg 22 w(\000) p Fm 23 w(J) 9 b(\031) p Fo 4 w(\() p Fm(V) p Fo 22 w(\)) p Fm(J) p Fo 9 w(\)1) p Fl 2788 2407 a(e) p Fo 2824 2392 a(\([) p Fm(K) r(;) p Fg 17 w(\001) p Fo 17 w(]\)) p Fm(;) p Fo 3530 2215 a(\(3.11\)) 0 2610 y(and) 41 b(indeed) g(one) g(can) g(pro) m(v) m(e) h(this) e(form) m(ula) f(under) i(v) m(ery) i(general) d(conditions) f([Da1,) h(DJ3].) 68 b(The) 0 2731 y(op) s(erator) 29 b(\000) p Fk 451 2746 a(L) p Fo 529 2731 a(is) g(called) g(the) h(Lev) m(el) g(Shift) f (Op) s(erator) g(for) g(the) i(standard) f(Liouvillean.) 39 b(The) 31 b(op) s(erator) 0 2851 y([) p Fm(K) r(;) p Fg 17 w(\001) p Fo(]) c(+) p Fm 27 w(\025) p Fk 398 2815 a(2) p Fo 438 2851 a(\000) p Fk 499 2866 a(L) p Fo 588 2851 a(predicts) 41 b(lo) s(cation) d(of) i(eigen) m(v) -5 b(alues) 41 b(and) g(resonances) h(of) p Fm 41 w(L) p Fl 2851 2866 a(\025) p Fo 2937 2851 a(to) e(the) h(2nd) g(order) g(of) 0 2971 y(p) s(erturbation) h(theory) i(and) f(has) g(b) s(een) h(an) e (imp) s(ortan) m(t) f(to) s(ol) h(in) g(the) h(recen) m(t) h(w) m(orks) h(on) d(return) i(to) 0 3092 y(equilibrium) 30 b([BFS,) i(DJ1,) g(DJ2,) g(JP1,) h(JP5,) g(M].) 146 3212 y(W) -8 b(e) 34 b(are) f(in) m (terested) h(in) e(relation) f(b) s(et) m(w) m(een) 36 b(\000) p Fk 1837 3227 a(L) p Fo 1918 3212 a(and) d(the) g(Da) m(vies) g (generator) g(\000) p Fk 3085 3227 a(H) p Fo 3142 3212 a(.) 45 b(Ob) m(viously) -8 b(,) 33 b(as) 0 3333 y(algebras,) p Fg 32 w(B) p Fo 3 w(\() p Fg(K) p Fo 1 w(\)) p Fg 28 w(\021) p Fm 29 w(l) p Fk 793 3296 a(2) p Fo 832 3333 a(\() p Fg(K) p Fo 1 w(\)) 28 b(=:) p Fg 28 w(V) p Fo 8 w(.) 43 b(Let) p Fm 33 w(\015) p Fo 33 w(:) p Fg 28 w(V) 8 b(!V) p Fo 40 w(b) s(e) 33 b(the) g(linear) e(in) m(v) m (ertible) h(map) g(de\014ned) i(b) m(y) p Fm 1514 3553 a(\015) p Fo 5 w(\() p Fm(B) p Fo 5 w(\)) 28 b(:=) p Fm 27 w(B) p Fo 5 w(e) p Fj 2005 3511 a(\000) p Fl(\014) s(K) q(=) p Fk(2) p Fm 2239 3553 a(:) p Fo 1264 w(\(3.12\)) p Ff 0 3773 a(Theorem) 74 b(3.1) p Fs 49 w(Assumption) 43 b(2.A) g(holds) f(if) h(and) f(only) h(if) g(Assumption) g(3.A) g (holds.) 68 b(If) 42 b(the) h(as-) 0 3893 y(sumptions) 34 b(hold,) g(then) p Fo 1500 4013 a(\000) p Fk 1561 4028 a(H) p Fo 1645 4013 a(=) p Fm 28 w(\015) p Fj 1805 3972 a(\000) p Fk(1) p Fg 1921 4013 a(\016) p Fo 22 w(\000) p Fk 2054 4028 a(L) p Fg 2125 4013 a(\016) p Fm 22 w(\015) 5 b(:) p Fo 1250 w(\(3.13\)) p Ff 0 4391 a(Remark.) p Fo 43 w(Explicitely) -8 b(,) 31 b(for) p Fm 32 w(B) p Fg 33 w(2) d(B) p Fo 3 w(\() p Fg(K) p Fo 1 w(\)) p Fg 28 w(\021) p Fm 29 w(l) p Fk 1706 4355 a(2) p Fo 1745 4391 a(\() p Fg(K) p Fo 1 w(\),) 33 b(w) m(e) h(ha) m(v) m(e) g(\000) p Fk 2388 4406 a(H) p Fo 2445 4391 a(\() p Fm(B) p Fo 5 w(\)) 27 b(=) h(\000) p Fk 2792 4406 a(L) p Fo 2840 4391 a(\() p Fm(B) p Fo 5 w(e) p Fj 3000 4355 a(\000) p Fl(\014) s(K) q(=) p Fk(2) p Fo 3233 4391 a(\)e) p Fl 3314 4355 a(\014) s(K) q(=) p Fk(2) p Fo 3493 4391 a(.) p Ff 0 4611 a(Pro) s(of.) p Fo 43 w(The) 34 b(Araki) e(p) s(erturbation) f(theory) j([BR2) o(,) f (DJP]) g(yields) f(that) 970 4831 y(\011) p Fk 1046 4846 a(0) p Fo 1113 4831 a(:=) 27 b(e) p Fj 1286 4790 a(\000) p Fl(\014) s(K) q(=) p Fk(2) p Fg 1542 4831 a(\012) p Fo 22 w(\012) p Fj 1711 4846 a(R) p Fg 1804 4831 a(2) p Fo 28 w(Dom) o(\(e) p Fj 2184 4790 a(\000) p Fl(\014) p Fk 3 w(\() p Fl(L) p Fi 2357 4799 a(0) p Fk 2392 4790 a(+) p Fl(\025\031) p Fk 2 w(\() p Fl(V) p Fk 16 w(\)\)) p Fl(=) p Fk(2) p Fo 2745 4831 a(\)) p Fm(;) p Fo 0 5051 a(that) 32 b(the) h(v) m(ector) 1390 5172 y(\011) p Fl 1466 5187 a(\025) p Fo 1539 5172 a(:=) 28 b(e) p Fj 1713 5130 a(\000) p Fl(\014) p Fk 3 w(\() p Fl(L) p Fi 1886 5139 a(0) p Fk 1921 5130 a(+) p Fl(\025\031) p Fk 2 w(\() p Fl(V) p Fk 16 w(\)\)) p Fl(=) p Fk(2) p Fo 2274 5172 a(\011) p Fk 2350 5187 a(0) p Fo 3530 5172 a(\(3.14\)) p 90 rotate dyy eop %%Page: 8 8 8 7 bop Fo 3731 100 a(8) 0 407 y(b) s(elongs) 32 b(to) g(Ker) p Fm(L) p Fl 692 422 a(\025) p Fo 738 407 a(,) h(and) f(that) h(\011) p Fl 1275 422 a(\025) p Fo 1348 407 a(=) 27 b(\011) p Fk 1527 422 a(0) p Fo 1589 407 a(+) p Fm 22 w(O) p Fo 3 w(\() p Fm(\025) p Fo(\).) 42 b(F) -8 b(or) p Fm 32 w(X) r(;) 17 b(B) p Fg 33 w(2) 28 b(B) p Fo 3 w(\() p Fg(K) p Fo 1 w(\)) p Fg 28 w(\021) p Fm 28 w(l) p Fk 2855 371 a(2) p Fo 2895 407 a(\() p Fg(K) p Fo 1 w(\)) 33 b(w) m(e) g(ha) m(v) m(e) 60 626 y(T) -8 b(r) p Fj 160 641 a(K\012H) p Fd 329 652 a(R) p Fo 391 626 a(\() p Fm(X) p Fg 30 w(\012) 22 b(j) p Fo(\012) p Fj 737 641 a(R) p Fo 802 626 a(\)\(\012) p Fj 948 641 a(R) p Fg 1012 626 a(j) p Fm 17 w(\034) p Fj 1110 585 a(\000) p Fl(t) p Fk 1099 651 a(0) p Fm 1195 626 a(\034) p Fl 1248 585 a(t) 1237 651 y(\025) p Fo 1282 626 a(\() p Fm(B) p Fg 28 w(\012) p Fo 22 w(1\)\)) 28 b(=) f(\() p Fm(X) p Fj 1904 585 a(\003) p Fg 1965 626 a(\012) p Fo 23 w(\012) p Fj 2135 641 a(R) p Fg 2200 626 a(j) p Fo(e) p Fj 2271 585 a(\000) p Fk(i) p Fl(tL) p Fi 2419 594 a(0) p Fo 2458 626 a(e) p Fk 2501 585 a(i) p Fl(tL) p Fh 2594 597 a(\025) p Fo 2639 626 a(\() p Fm(B) p Fg 27 w(\012) p Fo 23 w(1\)e) p Fj 3008 585 a(\000) p Fk(i) p Fl(tL) p Fh 3156 597 a(\025) p Fo 3201 626 a(e) p Fk 3244 585 a(i) p Fl(tL) p Fi 3337 594 a(0) p Fo 3376 626 a(1) p Fg 22 w(\012) p Fo 23 w(\012) p Fj 3617 641 a(R) p Fo 3681 626 a(\)) 88 853 y(=) g(T) -8 b(r) p Fj 292 868 a(K) p Fc 367 772 a(\000) p Fm 412 853 a(X) 8 b(P) p Fk 564 868 a(L) p Fo 612 853 a(e) p Fj 655 812 a(\000) p Fk(i) p Fl(tL) p Fi 803 821 a(0) p Fo 842 853 a(e) p Fk 885 812 a(i) p Fl(tL) p Fh 978 824 a(\025) p Fo 1024 853 a(\() p Fm(\031) p Fj 1117 868 a(S) p Fo 1169 853 a(\() p Fm(B) p Fo 5 w(\)) p Fg 22 w(\012) p Fo 22 w(1\)e) p Fj 1575 812 a(\000) p Fk(i) p Fl(tL) p Fh 1723 824 a(\025) p Fo 1768 853 a(e) p Fk 1811 812 a(i) p Fl(tL) p Fi 1904 821 a(0) p Fo 1944 853 a(1) p Fg 21 w(\012) p Fo 23 w(\012) p Fj 2184 868 a(R) p Fc 2249 772 a(\001) p Fo 88 1084 a(=) 27 b(T) -8 b(r) p Fj 292 1099 a(K) p Fc 367 1003 a(\000) p Fo 412 1084 a(e) p Fl 455 1043 a(\014) s(K) q(=) p Fk(2) p Fm 634 1084 a(X) p Fc 739 1003 a(\002) p Fm 780 1084 a(P) p Fk 843 1099 a(L) p Fo 892 1084 a(e) p Fj 935 1043 a(\000) p Fk(i) p Fl(tL) p Fi 1083 1052 a(0) p Fo 1122 1084 a(e) p Fk 1165 1043 a(i) p Fl(tL) p Fh 1258 1055 a(\025) p Fo 1303 1084 a(\() p Fm(\031) p Fj 1396 1099 a(S) p Fo 1448 1084 a(\() p Fm(B) p Fo 5 w(\)) p Fg 22 w(\012) p Fo 23 w(1\)e) p Fj 1855 1043 a(\000) p Fk(i) p Fl(tL) p Fh 2003 1055 a(\025) p Fo 2048 1084 a(e) p Fk 2091 1043 a(i) p Fl(tL) p Fi 2184 1052 a(0) p Fo 2223 1084 a(1) p Fg 22 w(\012) p Fo 23 w(\012) p Fj 2464 1099 a(R) p Fc 2528 1003 a(\003) p Fo 2587 1084 a(e) p Fj 2630 1043 a(\000) p Fl(\014) s(K) q(=) p Fk(2) p Fc 2863 1003 a(\001) p Fo 88 1315 a(=) p Fc 191 1234 a(\000) p Fm 237 1315 a(X) p Fj 326 1274 a(\003) p Fo 365 1315 a(e) p Fl 408 1274 a(\014) s(K) q(=) p Fk(2) p Fg 609 1315 a(\012) p Fo 22 w(\012) p Fj 778 1330 a(R) p Fg 843 1315 a(j) p Fc 888 1234 a(\000) p Fm 933 1315 a(\031) p Fj 988 1330 a(S) p Fc 1057 1234 a(\000) p Fm 1103 1315 a(P) p Fk 1166 1330 a(L) p Fo 1214 1315 a(e) p Fj 1257 1274 a(\000) p Fk(i) p Fl(tL) p Fi 1405 1283 a(0) p Fo 1444 1315 a(e) p Fk 1487 1274 a(i) p Fl(tL) p Fh 1580 1286 a(\025) p Fo 1625 1315 a(\() p Fm(\031) p Fj 1718 1330 a(S) p Fo 1770 1315 a(\() p Fm(B) p Fo 5 w(\)) p Fg 22 w(\012) p Fo 23 w(1\)e) p Fj 2177 1274 a(\000) p Fk(i) p Fl(tL) p Fh 2325 1286 a(\025) p Fo 2370 1315 a(e) p Fk 2413 1274 a(i) p Fl(tL) p Fi 2506 1283 a(0) p Fo 2545 1315 a(1) p Fg 22 w(\012) p Fo 23 w(\012) p Fj 2786 1330 a(R) p Fc 2851 1234 a(\001) p Fg 2918 1315 a(\012) p Fo 23 w(1) p Fc 3067 1234 a(\001) p Fo 3129 1315 a(e) p Fj 3172 1274 a(\000) p Fl(\014) s(K) q(=) p Fk(2) p Fg 3427 1315 a(\012) p Fo 23 w(\012) p Fj 3597 1330 a(R) p Fc 3662 1234 a(\001) p Fo 88 1546 a(=) p Fc 191 1466 a(\000) p Fm 237 1546 a(X) p Fj 326 1505 a(\003) p Fo 365 1546 a(e) p Fl 408 1505 a(\014) s(K) q(=) p Fk(2) p Fg 609 1546 a(\012) p Fo 22 w(\012) p Fj 778 1561 a(R) p Fg 843 1546 a(j) p Fo(e) p Fj 914 1505 a(\000) p Fk(i) p Fl(tL) p Fi 1062 1514 a(0) p Fo 1101 1546 a(e) p Fk 1144 1505 a(i) p Fl(tL) p Fh 1237 1517 a(\025) p Fo 1282 1546 a(\() p Fm(\031) p Fj 1375 1561 a(S) p Fo 1427 1546 a(\() p Fm(B) p Fo 5 w(\)) p Fg 23 w(\012) p Fo 22 w(1\)e) p Fj 1834 1505 a(\000) p Fk(i) p Fl(tL) p Fh 1982 1517 a(\025) p Fo 2027 1546 a(e) p Fk 2070 1505 a(i) p Fl(tL) p Fi 2163 1514 a(0) p Fo 2202 1546 a(e) p Fj 2245 1505 a(\000) p Fl(\014) s(K) q(=) p Fk(2) p Fg 2501 1546 a(\012) p Fo 22 w(\012) p Fj 2670 1561 a(R) p Fc 2735 1466 a(\001) p Fo 88 1777 a(=) 27 b(\() p Fm(X) p Fj 318 1736 a(\003) p Fo 358 1777 a(e) p Fl 401 1736 a(\014) s(K) q(=) p Fk(2) p Fg 601 1777 a(\012) p Fo 23 w(\012) p Fj 771 1792 a(R) p Fg 836 1777 a(j) p Fo(e) p Fj 907 1736 a(\000) p Fk(i) p Fl(tL) p Fo 1059 1777 a(e) p Fk 1102 1736 a(i) p Fl(tL) p Fh 1195 1748 a(\025) p Fo 1240 1777 a(\() p Fm(B) p Fg 27 w(\012) p Fo 23 w(1\)e) p Fj 1609 1736 a(\000) p Fl(\014) s(K) q(=) p Fk(2) p Fg 1864 1777 a(\012) p Fo 23 w(\012) p Fj 2034 1792 a(R) p Fo 2099 1777 a(\)) 22 b(+) p Fm 22 w(O) p Fo 3 w(\() p Fm(\025) p Fo(\)) 88 2009 y(=) 27 b(\() p Fm(X) p Fj 318 1967 a(\003) p Fo 358 2009 a(e) p Fl 401 1967 a(\014) s(K) q(=) p Fk(2) p Fg 601 2009 a(\012) p Fo 23 w(\012) p Fj 771 2024 a(R) p Fg 836 2009 a(j) p Fm(P) p Fk 927 2024 a(L) p Fo 974 2009 a(e) p Fj 1017 1967 a(\000) p Fk(i) p Fl(tL) p Fo 1170 2009 a(e) p Fk 1213 1967 a(i) p Fl(tL) p Fh 1306 1979 a(\025) p Fm 1351 2009 a(P) p Fk 1414 2024 a(L) p Fm 1462 2009 a(B) p Fo 5 w(e) p Fj 1584 1967 a(\000) p Fl(\014) s(K) q(=) p Fk(2) p Fg 1840 2009 a(\012) p Fo 22 w(\012) p Fj 2009 2024 a(R) p Fo 2074 2009 a(\)) 22 b(+) p Fm 22 w(O) p Fo 3 w(\() p Fm(\025) p Fo(\)) 88 2240 y(=) 27 b(T) -8 b(r) p Fj 292 2255 a(K) p Fc 367 2159 a(\000) p Fm 412 2240 a(X) p Fc 518 2159 a(\002) p Fm 559 2240 a(P) p Fk 622 2255 a(L) p Fo 670 2240 a(e) p Fj 713 2199 a(\000) p Fk(i) p Fl(tL) p Fi 861 2208 a(0) p Fo 900 2240 a(e) p Fk 943 2199 a(i) p Fl(tL) p Fh 1036 2211 a(\025) p Fm 1082 2240 a(P) p Fk 1145 2255 a(L) p Fm 1193 2240 a(B) p Fo 5 w(e) p Fj 1315 2199 a(\000) p Fl(\014) s(K) q(=) p Fk(2) p Fc 1548 2159 a(\003) p Fo 1606 2240 a(e) p Fl 1649 2199 a(\014) s(K) q(=) p Fk(2) p Fc 1828 2159 a(\001) p Fo 1895 2240 a(+) p Fm 22 w(O) p Fo 3 w(\() p Fm(\025) p Fo(\)) 0 2463 y(uniformly) 30 b(for) p Fm 32 w(t) p Fg 28 w(\025) p Fo 28 w(0.) 44 b(Hence,) 34 b(for) p Fm 32 w(X) r(;) 17 b(B) p Fg 33 w(2) 28 b(B) p Fo 3 w(\() p Fg(K) p Fo 1 w(\)) p Fg 28 w(\021) p Fm 28 w(l) p Fk 2060 2427 a(2) p Fo 2100 2463 a(\() p Fg(K) p Fo 1 w(\),) 384 2683 y(T) -8 b(r) p Fj 485 2698 a(K) p Fo 543 2683 a(\() p Fm(X) 8 b(P) p Fk 733 2698 a(H) p Fm 789 2683 a(\034) p Fj 842 2642 a(\000) p Fl(t) p Fk 831 2707 a(0) p Fm 927 2683 a(\034) p Fl 980 2642 a(t) 969 2707 y(\025) p Fm 1015 2683 a(P) p Fk 1078 2698 a(H) p Fm 1135 2683 a(B) p Fo 5 w(\)) 28 b(=) f(T) -8 b(r) p Fj 1483 2698 a(K) p Fc 1558 2602 a(\000) p Fm 1604 2683 a(X) p Fc 1709 2602 a(\002) p Fm 1751 2683 a(P) p Fk 1814 2698 a(L) p Fo 1862 2683 a(e) p Fj 1905 2642 a(\000) p Fk(i) p Fl(tL) p Fi 2053 2651 a(0) p Fo 2092 2683 a(e) p Fk 2135 2642 a(i) p Fl(tL) p Fh 2228 2654 a(\025) p Fm 2273 2683 a(P) p Fk 2336 2698 a(L) p Fm 2385 2683 a(B) p Fo 5 w(e) p Fj 2507 2642 a(\000) p Fl(\014) s(K) q(=) p Fk(2) p Fc 2740 2602 a(\003) p Fo 2798 2683 a(e) p Fl 2841 2642 a(\014) s(K) q(=) p Fk(2) p Fc 3019 2602 a(\001) p Fo 3087 2683 a(+) p Fm 22 w(O) p Fo 3 w(\() p Fm(\025) p Fo(\)) 0 2903 y(uniformly) 30 b(for) p Fm 32 w(t) p Fg 28 w(\025) p Fo 28 w(0.) 44 b(Since) 32 b(dim) p Fg 16 w(K) p Fm 29 w(<) p Fg 27 w(1) p Fo(,) g(w) m(e) i(conclude) f(that) f(for) g(a) h(giv) m(en) p Fm 32 w(t) p Fo 33 w(the) g(limit) 1254 3123 y(lim) p Fl 1248 3185 a(\025) p Fj(!) p Fk(0) p Fm 1412 3123 a(P) p Fk 1475 3138 a(L) p Fo 1523 3123 a(e) p Fj 1566 3082 a(\000) p Fk(i) p Fl(tL) p Fi 1714 3091 a(0) p Fl 1749 3082 a(=\025) p Fi 1825 3058 a(2) p Fo 1864 3123 a(e) p Fk 1907 3082 a(i) p Fl(tL) p Fh 2000 3094 a(\025) p Fl 2041 3082 a(=\025) p Fi 2117 3058 a(2) p Fm 2156 3123 a(P) p Fk 2219 3138 a(L) p Fo 2295 3123 a(=:) p Fm 28 w(T) p Fl 2497 3082 a(t) p Fk 2483 3147 a(L) p Fo 0 3373 a(exists) g(i\013) f(the) h(limit) 1336 3493 y(lim) p Fl 1330 3556 a(\025) p Fj(!) p Fk(0) p Fm 1494 3493 a(P) p Fk 1557 3508 a(H) p Fm 1614 3493 a(\034) p Fj 1667 3442 a(\000) p Fl(t=\025) p Fi 1823 3419 a(2) p Fk 1656 3517 a(0) p Fm 1863 3493 a(\034) p Fl 1916 3442 a(t=\025) p Fi 2017 3419 a(2) p Fl 1905 3521 a(\025) p Fm 2057 3493 a(P) p Fk 2120 3508 a(H) p Fo 2204 3493 a(=:) p Fm 28 w(T) p Fl 2406 3452 a(t) p Fk 2392 3518 a(H) p Fo 0 3697 a(exists.) 44 b(Moreo) m(v) m(er,) 34 b(if) e(the) h(limits) c(exist,) k (then) p Fm 1504 3917 a(T) p Fl 1575 3876 a(t) p Fk 1561 3942 a(H) p Fo 1645 3917 a(=) p Fm 28 w(\015) p Fj 1805 3876 a(\000) p Fk(1) p Fg 1921 3917 a(\016) p Fm 22 w(T) p Fl 2064 3876 a(t) p Fk 2050 3942 a(L) p Fg 2121 3917 a(\016) p Fm 22 w(\015) 5 b(:) p Fo 0 4137 a(In) 39 b(particular,) p Fm 39 w(T) p Fl 684 4101 a(t) p Fk 670 4162 a(H) p Fo 767 4137 a(is) f(a) h(semigroup) f(i\013) p Fm 38 w(T) p Fl 1626 4101 a(t) p Fk 1612 4162 a(L) p Fo 1700 4137 a(is) g(a) h(semigroup) f(and) h(their) g(generators) g(\(\000) p Fk 3378 4152 a(H) p Fo 3474 4137 a(and) g(\000) p Fk 3731 4152 a(L) p Fo 0 4257 a(resp) s(ectiv) m(ely\)) 34 b(satisfy) e(\(3.13\).) p Fb 43 w(2) p Fn 0 4640 a(4) 161 b(Nonequilibrium) 55 b(case) p Fo 0 4859 a(W) -8 b(e) 28 b(no) m(w) g(consider) f(the) h(case) g(where) h(the) e(reserv) m(oir) h (is) f(not) g(in) g(thermal) e(equilibrium,) h(namely) g(where) 0 4979 y(the) 33 b(in) m(v) -5 b(arian) m(t) 31 b(state) p Fe 1407 5100 a(M) p Fj 1512 5115 a(R) p Fg 1604 5100 a(3) p Fm 28 w(A) p Fg 28 w(7!) p Fo 27 w(\(\012) p Fj 2034 5115 a(R) p Fg 2099 5100 a(j) p Fm(A) p Fo(\012) p Fj 2270 5115 a(R) p Fo 2334 5100 a(\)) 1158 b(\(4.15\)) p 90 rotate dyy eop %%Page: 9 9 9 8 bop Fo 3731 100 a(9) 0 407 y(is) p Fs 44 w(not) p Fo 43 w(a) 44 b(\() p Fm(\034) p Fj 463 422 a(R) p Fm 528 407 a(;) 17 b(\014) p Fo 6 w(\)-KMS) 43 b(state) h(for) g(an) m(y) p Fm 44 w(\014) p Fo 6 w(.) 78 b(A) 44 b(t) m(ypical) f(example) g(is) h (a) g(free) g(Bose) h(or) e(F) -8 b(ermi) 42 b(gas) 0 527 y(with) 32 b(quasi-free) g(initial) d(state) k(whose) g(energy) h (densit) m(y) f(is) f(di\013eren) m(t) g(from) g(Planc) m(k's) h(la) m (w.) 43 b(Another) 0 648 y(example) 32 b(is) g(a) g(m) m(ultithermal) d (reserv) m(oir) 34 b(where) p Fe 1326 859 a(M) p Fj 1431 874 a(R) p Fo 1523 859 a(=) p Fe 27 w(M) p Fj 1731 874 a(R) p Fi 1791 883 a(1) p Fg 1852 859 a(\012) 23 b(\001) 17 b(\001) g(\001) j(\012) p Fe 23 w(M) p Fj 2295 874 a(R) p Fh 2355 885 a(M) p Fm 2426 859 a(;) 1389 1004 y(\034) p Fj 1431 1019 a(R) p Fo 1523 1004 a(=) p Fm 27 w(\034) p Fj 1668 1019 a(R) p Fi 1728 1028 a(1) p Fg 1790 1004 a(\012) j(\001) 17 b(\001) g(\001) j(\012) p Fm 23 w(\034) p Fj 2170 1019 a(R) p Fh 2230 1030 a(M) p Fm 2302 1004 a(;) p Fo 1360 1150 a(\012) p Fj 1430 1165 a(R) p Fo 1523 1150 a(=) 27 b(\012) p Fj 1696 1165 a(R) p Fi 1756 1174 a(1) p Fg 1818 1150 a(\012) c(\001) 17 b(\001) g(\001) j(\012) p Fo 23 w(\012) p Fj 2226 1165 a(R) p Fh 2286 1176 a(M) p Fm 2358 1150 a(;) p Fe 0 1368 a(M) p Fj 105 1383 a(R) p Fh 165 1395 a(k) p Fg 259 1368 a(3) p Fm 52 w(A) p Fg 52 w(7!) p Fo 51 w(\(\012) p Fj 761 1383 a(R) p Fh 821 1395 a(k) p Fm 864 1368 a(A) p Fo(\012) p Fj 1007 1383 a(R) p Fh 1067 1395 a(k) p Fo 1110 1368 a(\)) 47 b(is) f(a) g(\() p Fm(\034) p Fj 1482 1383 a(R) p Fh 1542 1395 a(k) p Fm 1585 1368 a(;) 17 b(\014) p Fl 1684 1383 a(k) p Fo 1726 1368 a(\)-KMS) 47 b(state) g(for) f(some) p Fm 47 w(\014) p Fl 2793 1383 a(k) p Fm 2887 1368 a(>) p Fo 52 w(0,) k(and) d(not) f (all) p Fm 45 w(\014) p Fl 3737 1383 a(k) p Fo 0 1489 a(are) 37 b(the) g(same.) 56 b(This) 37 b(case) h(has) g(attracted) f (considerable) f(atten) m(tion) g(in) h(the) g(recen) m(t) h (literature) e(on) 0 1609 y(nonequilibrium) 30 b(quan) m(tum) j (statistical) d(mec) m(hanics.) 146 1730 y(The) 53 b(standard) g (Liouvillean) c(is) j(again) f(w) m(ell) g(de\014ned) j(and) e(giv) m (en) g(b) m(y) h(\(3.8\).) 102 b(Ho) m(w) m(ev) m(er,) 59 b(in) 0 1850 y(nonequlibrium) 40 b(case) i(and) g(for) p Fm 41 w(\025) p Fg 43 w(6) p Fo(=) g(0,) p Fm 44 w(L) p Fl 1633 1865 a(\025) p Fo 1720 1850 a(t) m(ypically) e(will) g(ha) m(v) m(e) j(no) e(p) s(oin) m(t) g(sp) s(ectra.) 71 b(In) 42 b(partic-) 0 1970 y(ular,) 32 b(zero) i(will) c(not) j(b) s(e) h(an) f (eigen) m(v) -5 b(alue) 32 b(of) p Fm 33 w(L) p Fl 1707 1985 a(\025) p Fo 1753 1970 a(.) 44 b(W) -8 b(e) 34 b(recall) e(that) g (Ker) p Fm 1 w(L) p Fl 2689 1985 a(\025) p Fo 2763 1970 a(=) p Fg 28 w(f) p Fo(0) p Fg(g) p Fo 33 w(i\013) p Fm 32 w(W) p Fj 3272 1934 a(\003) p Fo 3311 1970 a(-dynamical) 0 2091 y(system) 26 b(\() p Fe(M) p Fm(;) 17 b(\034) p Fl 545 2106 a(\025) p Fo 590 2091 a(\)) 24 b(has) h(no) g(normal,) f (in) m(v) -5 b(arian) m(t) 24 b(states.) 41 b(Hence,) 28 b(in) c(nonequilibrium) e(case) k(one) f(exp) s(ects) 0 2211 y(that) 37 b(\000) p Fk 277 2226 a(L) p Fo 363 2211 a(will) e(ha) m(v) m(e) k(no) f(real) e(eigen) m(v) -5 b(alues) 38 b(and) f(hence) i(that) f(\000) p Fk 2375 2226 a(H) p Fo 2469 2211 a(and) g(\000) p Fk 2725 2226 a(L) p Fo 2811 2211 a(are) f(not) h(isosp) s(ectral.) 57 b(In) 0 2332 y(fact,) 32 b(in) g(nonequilibrium) e(case) j(one) g(exp) s (ects) p Fs 35 w(no) p Fo 32 w(direct) f(relation) f(b) s(et) m(w) m (een) k(\000) p Fk 2950 2347 a(H) p Fo 3039 2332 a(and) e(\000) p Fk 3290 2347 a(L) p Fo 3338 2332 a(.) 146 2452 y(The) i(sp) s(ectral) f (approac) m(h) g(to) g(nonequilibrium) d(quan) m(tum) j(statistical) e (mec) m(hanics) i(has) g(b) s(een) h(re-) 0 2572 y(cen) m(tly) 44 b(prop) s(osed) g(in) f([JP4].) 76 b(The) 44 b(basic) f(ob) 5 b(ject) 45 b(is) d(a) i(non-selfadjoin) m(t) d(generator) j(of) f (dynamics) 0 2693 y(called) 31 b(C-Liouvillean.) 41 b(This) 33 b(op) s(erator) f(is) g(de\014ned) i(as) e(follo) m(ws.) 146 2813 y(Assume) 39 b(that) e(\012) p Fj 799 2828 a(R) p Fo 902 2813 a(is) g(a) g(cyclic) h(\(and) f(hence) i(separating\)) e(v) m(ector) i(for) p Fe 37 w(M) p Fj 2945 2828 a(R) p Fo 3047 2813 a(and) e(let) g(\001) h(b) s(e) g(the) 0 2933 y(corresp) s(onding) 32 b(mo) s(dular) f(op) s(erator.) 43 b(W) -8 b(e) 33 b(assume) g(that) f(the) h(op) s(erator) 1304 3153 y(\(1) p Fg 22 w(\012) p Fo 23 w(\001) p Fk 1594 3112 a(1) p Fl(=) p Fk(2) p Fo 1704 3153 a(\)) p Fm(\031) p Fo 4 w(\() p Fm(V) p Fo 22 w(\)\(1) p Fg 21 w(\012) p Fo 23 w(\001) p Fj 2245 3112 a(\000) p Fk(1) p Fl(=) p Fk(2) p Fo 2410 3153 a(\)) p Fm(;) p Fo 0 3373 a(initially) 26 b(de\014ned) 32 b(on) p Fe 31 w(M) p Fo 30 w(1) p Fg 17 w(\012) p Fo 19 w(\012) p Fj 1193 3388 a(R) p Fg 1285 3373 a(\032) p Fm 28 w(l) p Fk 1421 3337 a(2) p Fo 1461 3373 a(\() p Fg(K) p Fo 1 w(\)) p Fg(\012) q(H) p Fj 1776 3388 a(R) p Fo 1840 3373 a(,) f(extends) i(to) d(an) g(elemen) m (t) g(of) p Fe 30 w(M) p Fo(.) 43 b(W) -8 b(e) 31 b(denote) g(this) 0 3494 y(elemen) m(t) h(b) m(y) p Fm 34 w(\031) p Fo 4 w(\() p Fm(V) p Fo 21 w(\)) p Fk 709 3509 a(\001) p Fo 805 3494 a(and) g(set) p Fe 34 w(L) p Fk 1213 3509 a(0) p Fo 1280 3494 a(:=) p Fm 28 w(L) p Fk 1477 3509 a(0) p Fo 1517 3494 a(,) p Fe 1221 3714 a(L) p Fl 1287 3729 a(\025) p Fo 1361 3714 a(=) p Fe 27 w(L) p Fk 1530 3729 a(0) p Fo 1592 3714 a(+) p Fm 22 w(\025\031) p Fo 4 w(\() p Fm(V) p Fo 22 w(\)) p Fg 22 w(\000) p Fm 22 w(\025J) 9 b(\031) p Fo 4 w(\() p Fm(V) p Fo 22 w(\)) p Fk 2416 3729 a(\001) p Fm 2479 3714 a(J) n(:) p Fo 972 w(\(4.16\)) 0 3934 y(The) 40 b(op) s(erator) p Fe 39 w(L) p Fl 673 3949 a(\025) p Fo 757 3934 a(is) f(called) f(the) h(C-Liouvillean) e (of) h(the) i(system) p Fg 40 w(S) p Fo 34 w(+) p Fg 27 w(R) p Fo(.) 63 b(Note) 40 b(that) e(except) j(in) 0 4054 y(trivial) d(cases) p Fm 41 w(\031) p Fo 4 w(\() p Fm(V) p Fo 22 w(\)) p Fk 764 4069 a(\001) p Fo 867 4054 a(is) i(not) g(self-adjoin) m(t,) g(and) h(hence) p Fe 41 w(L) p Fl 2234 4069 a(\025) p Fo 2320 4054 a(is) f(also) f(not) h (self-adjoin) m(t.) 65 b(Note) 40 b(also) 0 4175 y(that) p Fe 35 w(L) p Fl 280 4190 a(\025) p Fo 361 4175 a(generates) c(a) p Fm 35 w(C) p Fk 946 4190 a(0) p Fo 985 4175 a(-semigroup) e(on) p Fm 35 w(l) p Fk 1656 4138 a(2) p Fo 1696 4175 a(\() p Fg(K) p Fo 1 w(\)) p Fg 24 w(\012) 24 b(H) p Fj 2058 4190 a(R) p Fo 2123 4175 a(,) 36 b(that) p Fe 35 w(L) p Fl 2466 4190 a(\025) p Fo 2511 4175 a(\(1) p Fg 24 w(\012) p Fo 25 w(\012) p Fj 2794 4190 a(R) p Fo 2858 4175 a(\)) c(=) h(1) p Fg 23 w(\012) p Fo 25 w(\012) p Fj 3281 4190 a(R) p Fo 3345 4175 a(,) j(and) g(that) 0 4295 y(for) c(all) p Fm 31 w(A) p Fg 27 w(2) p Fe 28 w(M) p Fo(,) p Fm 1126 4415 a(\031) p Fo 4 w(\() p Fm(\034) p Fl 1276 4374 a(t) 1265 4440 y(\025) p Fo 1310 4415 a(\() p Fm(A) p Fo(\)\)1) p Fg 22 w(\012) p Fo 23 w(\012) p Fj 1738 4430 a(R) p Fo 1830 4415 a(=) c(e) p Fk 1977 4374 a(i) p Fl(t) p Fa(L) p Fh 2068 4386 a(\025) p Fm 2113 4415 a(\031) p Fo 4 w(\() p Fm(A) p Fo(\)1) p Fg 22 w(\012) p Fo 23 w(\012) p Fj 2562 4430 a(R) p Fm 2627 4415 a(;) p Fo 0 4590 a(see) 34 b([JP4]) f(for) f(details.) p Ff 0 4818 a(Assumption) k(4.A) p Fs 49 w(Ther) -5 b(e) 34 b(exists) h(an) f(op) -5 b(er) g(ator) p Fo 35 w(\000) p Fk 1973 4833 a(C) p Fo 2055 4818 a(:) p Fm 28 w(l) p Fk 2141 4782 a(2) p Fo 2181 4818 a(\() p Fg(K) p Fo 1 w(\)) p Fg(!) p Fm(l) p Fk 2465 4782 a(2) p Fo 2504 4818 a(\() p Fg(K) p Fo 1 w(\)) p Fs 36 w(such) 34 b(that) h(for) p Fm 35 w(t) p Fg 28 w(\025) p Fo 28 w(0) p Fs(,) p Fo 1217 5048 a(lim) p Fl 1211 5110 a(\025) p Fj(!) p Fk(0) p Fm 1375 5048 a(P) p Fk 1438 5063 a(L) p Fo 1486 5048 a(e) p Fj 1529 5006 a(\000) p Fk(i) p Fl(t) p Fa(L) p Fi 1675 5015 a(0) p Fl 1709 5006 a(=\025) p Fi 1785 4983 a(2) p Fo 1825 5048 a(e) p Fk 1868 5006 a(i) p Fl(t) p Fa(L) p Fh 1959 5018 a(\025) p Fl 2000 5006 a(=\025) p Fi 2076 4983 a(2) p Fm 2115 5048 a(P) p Fk 2178 5063 a(L) p Fo 2254 5048 a(=) 27 b(e) p Fk 2400 5006 a(i) p Fl(t) p Fk(\000) p Fi 2489 5017 a(C) p Fm 2542 5048 a(:) p Fo 961 w(\(4.17\)) p 90 rotate dyy eop %%Page: 10 10 10 9 bop Fo 3682 100 a(10) 0 407 y(A) 33 b(F) -8 b(ermi) 30 b(Golden) i(Rule) g(computation) f(yields) 591 634 y(\000) p Fk 652 649 a(C) p Fo 734 634 a(=) d(lim) p Fl 838 696 a(\017) p Fj(\045) p Fk(0) p Fc 1088 539 a(X) p Fl 990 755 a(e) p Fj(2) p Fk(sp\([) p Fl(K) q(;) p Fj(\001) p Fk(]\)) p Fo 1330 634 a(1) p Fl 1379 649 a(e) p Fo 1416 634 a(\([) p Fm(K) r(;) p Fg 17 w(\001) p Fo 17 w(]\)\() p Fm(\031) p Fo 4 w(\() p Fm(V) p Fo 20 w(\)) p Fg 22 w(\000) p Fm 23 w(J) 9 b(\031) p Fo 4 w(\() p Fm(V) p Fo 21 w(\)) p Fk 2368 649 a(\001) p Fm 2431 634 a(J) p Fo 9 w(\)) 1330 970 y(\() p Fm(e) p Fo 23 w(+) 22 b(i) p Fm(\017) p Fg 21 w(\000) p Fe 23 w(L) p Fk 1788 985 a(0) p Fo 1828 970 a(\)) p Fj 1866 929 a(\000) p Fk(1) p Fo 1960 970 a(\() p Fm(\031) p Fo 4 w(\() p Fm(V) p Fo 21 w(\)) p Fg 22 w(\000) p Fm 23 w(J) 9 b(\031) p Fo 4 w(\() p Fm(V) p Fo 21 w(\)) p Fk 2609 985 a(\001) p Fm 2672 970 a(J) p Fo 9 w(\)1) p Fl 2822 985 a(e) p Fo 2859 970 a(\([) p Fm(K) r(;) p Fg 17 w(\001) p Fo 17 w(]\)) p Fm(;) p Fo 3530 793 a(\(4.18\)) 0 1184 y(and) 40 b(one) h(can) f(pro) m(v) m(e) i (this) e(form) m(ula) e(under) j(v) m(ery) h(general) e(conditions) f ([DJ3) o(].) 67 b(As) 41 b(exp) s(ected,) j(the) 0 1304 y(op) s(erator) c([) p Fm(K) r(;) p Fg 17 w(\001) p Fo(]) 27 b(+) p Fm 28 w(\025) p Fk 800 1268 a(2) p Fo 839 1304 a(\000) p Fk 900 1319 a(C) p Fo 996 1304 a(predicts) 41 b(the) g(lo) s(cation) e(of) h(resonances) j(of) p Fe 40 w(L) p Fl 2725 1319 a(\025) p Fo 2811 1304 a(to) e(the) g(second) h (order) f(of) 0 1424 y(p) s(erturbation) f(theory) i([JP4) q(].) 69 b(The) 43 b(op) s(erator) d(\000) p Fk 1883 1439 a(C) p Fo 1980 1424 a(is) g(called) h(the) g(Lev) m(el) h(Shift) f(Op) s (erator) f(for) h(the) 0 1545 y(C-Liouvillean.) p Ff 0 1767 a(Theorem) 74 b(4.1) p Fs 49 w(Assumption) 43 b(2.A) g(holds) f(if) h(and) f(only) h(if) g(Assumption) g(4.A) g (holds.) 68 b(If) 42 b(the) h(as-) 0 1888 y(sumptions) 34 b(hold,) g(then) p Fo 1694 2008 a(\000) p Fk 1755 2023 a(H) p Fo 1839 2008 a(=) 28 b(\000) p Fk 2004 2023 a(C) p Fm 2059 2008 a(:) p Ff 0 2403 a(Pro) s(of:) p Fo 43 w(The) 34 b(iden) m(tities) 584 2591 y(T) -8 b(r) p Fj 684 2606 a(K) p Fo 742 2591 a(\() p Fm(X) 8 b(P) p Fk 932 2606 a(L) p Fm 980 2591 a(\034) p Fj 1033 2550 a(\000) p Fl(t) p Fk 1022 2615 a(0) p Fm 1118 2591 a(\034) p Fl 1171 2550 a(t) 1160 2615 y(\025) p Fm 1206 2591 a(P) p Fk 1269 2606 a(L) p Fo 1317 2591 a(\)) 28 b(=) f(T) -8 b(r) p Fj 1586 2606 a(K\012H) p Fd 1755 2617 a(R) p Fc 1833 2510 a(\000) p Fm 1879 2591 a(X) p Fg 30 w(\012) 22 b(j) p Fo(\012) p Fj 2187 2606 a(R) p Fo 2252 2591 a(\)\(\012) p Fj 2398 2606 a(R) p Fg 2463 2591 a(j) p Fm 17 w(\034) p Fj 2561 2550 a(\000) p Fl(t) p Fk 2550 2615 a(0) p Fm 2645 2591 a(\034) p Fl 2698 2550 a(t) 2687 2615 y(\025) p Fo 2733 2591 a(\() p Fm(B) p Fg 27 w(\012) p Fo 22 w(1\)) p Fc 3058 2510 a(\001) p Fo 1383 2818 a(=) 27 b(\() p Fm(X) p Fj 1613 2777 a(\003) p Fg 1674 2818 a(\012) p Fo 23 w(\012) p Fj 1844 2833 a(R) p Fg 1909 2818 a(j) p Fo(e) p Fj 1980 2777 a(\000) p Fk(i) p Fl(t) p Fa(L) p Fi 2126 2786 a(0) p Fo 2165 2818 a(e) p Fk 2208 2777 a(i) p Fl(t) p Fa(L) p Fh 2299 2789 a(\025) p Fo 2344 2818 a(\() p Fm(\031) p Fj 2437 2833 a(S) p Fo 2489 2818 a(\() p Fm(B) p Fo 5 w(\)) p Fg 22 w(\012) p Fo 23 w(1\)1) p Fg 21 w(\012) p Fo 23 w(\012) p Fj 3093 2833 a(R) p Fo 3158 2818 a(\)) 1383 3046 y(=) g(\() p Fm(X) p Fj 1613 3004 a(\003) p Fg 1674 3046 a(\012) p Fo 23 w(\012) p Fj 1844 3061 a(R) p Fg 1909 3046 a(j) p Fo(e) p Fj 1980 3004 a(\000) p Fk(i) p Fl(t) p Fa(L) p Fi 2126 3013 a(0) p Fo 2165 3046 a(e) p Fk 2208 3004 a(i) p Fl(t) p Fa(L) p Fh 2299 3016 a(\025) p Fm 2344 3046 a(B) p Fg 27 w(\012) p Fo 23 w(\012) p Fj 2615 3061 a(R) p Fo 2680 3046 a(\)) 1383 3273 y(=) g(T) -8 b(r) p Fj 1586 3288 a(K) p Fo 1645 3273 a(\() p Fm(X) 8 b(P) p Fk 1835 3288 a(L) p Fo 1882 3273 a(e) p Fj 1925 3232 a(\000) p Fk(i) p Fl(t) p Fa(L) p Fi 2071 3241 a(0) p Fo 2110 3273 a(e) p Fk 2153 3232 a(i) p Fl(t) p Fa(L) p Fh 2244 3244 a(\025) p Fm 2290 3273 a(P) p Fk 2353 3288 a(L) p Fo 2401 3273 a(\)) p Fm(;) p Fo 0 3481 a(yield) 32 b(that) p Fm 1265 3602 a(P) p Fk 1328 3617 a(H) p Fm 1385 3602 a(\034) p Fj 1438 3560 a(\000) p Fl(t) p Fk 1427 3626 a(0) p Fm 1523 3602 a(\034) p Fl 1576 3560 a(t) p Fm 1606 3602 a(P) p Fk 1669 3617 a(H) p Fo 1754 3602 a(=) p Fm 28 w(P) p Fk 1921 3617 a(L) p Fo 1969 3602 a(e) p Fj 2012 3560 a(\000) p Fk(i) p Fl(t) p Fa(L) p Fi 2158 3569 a(0) p Fo 2197 3602 a(e) p Fk 2240 3560 a(i) p Fl(t) p Fa(L) p Fh 2331 3572 a(\025) p Fm 2376 3602 a(P) p Fk 2439 3617 a(L) p Fm 2487 3602 a(;) p Fo 1016 w(\(4.19\)) 0 3774 y(and) h(the) g(result) f(is) g(immediate.) p Fb 41 w(2) p Fo 146 3943 a(W) -8 b(e) 30 b(remark) e(that) h (C-Liouvillean) d(is) i(also) h(w) m(ell-de\014ned) g(in) f(the) h (thermal) f(equilibrium) e(and) j(that) 0 4063 y(in) j(this) g(case) p Fe 1122 4183 a(L) p Fl 1188 4198 a(\025) p Fo 1262 4183 a(=) 27 b(e) p Fl 1408 4142 a(\014) p Fk 3 w(\() p Fl(L) p Fi 1526 4151 a(0) p Fk 1562 4142 a(+) p Fl(\025\031) p Fk 2 w(\() p Fl(V) p Fk 16 w(\)\)) p Fl(=) p Fk(2) p Fm 1914 4183 a(L) p Fl 1980 4198 a(\025) p Fo 2026 4183 a(e) p Fj 2069 4142 a(\000) p Fl(\014) p Fk 3 w(\() p Fl(L) p Fi 2242 4151 a(0) p Fk 2277 4142 a(+) p Fl(\025\031) p Fk 2 w(\() p Fl(V) p Fk 17 w(\)\)) p Fl(=) p Fk(2) p Fm 2630 4183 a(;) p Fo 873 w(\(4.20\)) 0 4356 y(see) 47 b([JP4].) 83 b(Theorem) 46 b(3.1) f(can) h(b) s(e) g(also) e(pro) m(v) m (en) j(using) f(relations) e(\(4.19\)) h(and) g(\(4.20\)) g(and) h(the) 0 4476 y(argumen) m(t) 32 b(of) g(Section) h(5.6) f(in) g([DJP].) p Fn 0 4808 a(5) 161 b(Some) 52 b(remarks) p Fo 0 5027 a(Theorem) 34 b(3.1) f(extends) j(to) d(a) h(large) e(class) i(of) g (un) m(b) s(ounded) h(p) s(erturbations) p Fm 33 w(V) p Fo 22 w(.) 47 b(All) 32 b(what) i(is) f(needed) 0 5147 y(is) h(that) g(\() p Fe(M) p Fm(;) 17 b(\034) p Fl 542 5162 a(\025) p Fo 587 5147 a(\)) 34 b(and) p Fm 34 w(L) p Fl 916 5162 a(\025) p Fo 996 5147 a(are) g(w) m(ell-de\014ned) h(and) f (that) g(the) h(basic) f(results) g(of) g(Araki's) g(p) s(erurbation) 0 5268 y(theory) 43 b(hold.) 73 b(The) 44 b(recen) m(t) g(result) e ([DJP]) h(giv) m(es) g(a) f(set) i(of) e(su\016cien) m(t) i (conditions.) 72 b(Consider) 44 b(an) p 90 rotate dyy eop %%Page: 11 11 11 10 bop Fo 3682 100 a(11) 0 407 y(un) m(b) s(ounded) 34 b(self-adjoin) m(t) d(op) s(erator) p Fm 32 w(V) p Fo 54 w(on) p Fg 32 w(K) 24 b(\012) e(H) p Fj 1928 422 a(R) p Fo 2025 407 a(and) 33 b(assume:) 0 648 y(\(1\)) p Fm 32 w(V) p Fo 54 w(is) f(a\016liated) f(with) p Fe 33 w(M) p Fo(.) 0 888 y(\(2\)) p Fm 32 w(L) p Fk 223 852 a(semi) p Fl 223 914 a(\025) p Fo 398 888 a(is) h(essen) m(tially) g (self-adjoin) m(t) f(Dom) n(\() p Fm(L) p Fk 1773 852 a(semi) 1773 913 y(0) p Fo 1916 888 a(\)) p Fg 22 w(\\) p Fo 22 w(Dom) o(\() p Fm(V) p Fo 21 w(\)) i(for) p Fg 32 w(j) p Fm(\025) p Fg(j) p Fm 27 w(<) p Fo 27 w(1.) 0 1129 y(\(3\)) p Fm 32 w(L) p Fl 223 1144 a(\025) p Fo 301 1129 a(is) f(essen) m(tially) h(self-adjoin) m(t) d(on) j(Dom) o (\() p Fm(L) p Fk 1813 1144 a(0) p Fo 1852 1129 a(\)) p Fg 22 w(\\) p Fo 23 w(Dom) n(\() p Fm(\031) p Fo 4 w(\() p Fm(V) p Fo 22 w(\)\)) p Fg 22 w(\\) p Fo 22 w(Dom) o(\() p Fm(J) 9 b(\031) p Fo 4 w(\() p Fm(V) p Fo 21 w(\)) p Fm(J) p Fo 9 w(\)) 33 b(for) p Fg 32 w(j) p Fm(\025) p Fg(j) p Fm 27 w(<) p Fo 27 w(1.) 0 1370 y(\(4\)) p Fg 32 w(k) p Fo(e) p Fj 250 1334 a(\000) p Fl(\014) s(\025\031) p Fk 2 w(\() p Fl(V) p Fk 17 w(\)) p Fl(=) p Fk(2) p Fo 619 1370 a(\011) p Fk 695 1385 a(0) p Fg 734 1370 a(k) p Fm 27 w(<) p Fg 28 w(1) p Fo 32 w(for) p Fg 32 w(j) p Fm(\025) p Fg(j) p Fm 27 w(<) p Fo 28 w(1.) 0 1611 y(Then) 48 b(the) f(results) g(of) g([DJP]) f(yield) g(that) h(Theorem) g(3.1) f (holds) h(with) f(the) h(same) g(pro) s(of) f(for) g(the) 0 1731 y(un) m(b) s(ounded) g(p) s(erturbation) p Fm 45 w(V) p Fo 21 w(.) 81 b(In) 45 b(particular,) h(Theorem) g(3.1) e(holds) h(for) f(P) m(auli-Fierz) f(systems) 0 1851 y(with) 32 b(b) s(osonic) g(reserv) m(oirs.) 146 1972 y(The) k(pro) s(of) d(of) h (Theorem) g(4.1) g(requires) h(no) f(estimates) g(and) g(follo) m(ws) f (from) g(the) i(iden) m(tit) m(y) f(\(4.19\).) 0 2092 y(Ob) m(viously) -8 b(,) 40 b(this) e(theorem) h(holds) f(whenev) m(er) j(C-Liouvillean) 36 b(can) j(b) s(e) g(meaningfully) d(de\014ned,) 41 b(see) 0 2213 y([DJ3) o(].) 146 2333 y(The) 30 b(results) f(of) g(this) f(note) h(bridge) f(the) i(gap) e(b) s(et) m(w) m(een) j(the) e(large) f (b) s(o) s(dy) h(of) f(literature) g(on) g(Mark) m(o-) 0 2453 y(vian) 37 b(semigroups) f(for) h(op) s(en) g(quan) m(tum) h (systems) g(and) g(the) f(recen) m(t) i(in) m(v) m(estigations) d(of) h (op) s(en) g(quan-) 0 2574 y(tum) i(systems) j(based) f(on) f (algebraic) e(and) i(sp) s(ectral) g(tec) m(hniques.) 68 b(The) 41 b(main) d(ob) 5 b(jects) 42 b(of) d(the) i(t) m(w) m(o) 0 2694 y(approac) m(hes|the) 35 b(Da) m(vies) e(generator) h(and) f(the) h (Lev) m(el) g(Shift) f(Op) s(erator) g(for) g(the) h(standard) g(and) g (C-) 0 2814 y(Liouvillean|) 28 b(determine) j(eac) m(h) i(other) f(and) f(hence) i(the) f(results) g(of) f(one) h(approac) m(h) g(can) g(b) s (e) g(used) g(in) 0 2935 y(the) h(con) m(text) g(of) f(the) h(other.) 43 b(This) 33 b(link) e(is) h(exploited) f(in) h(detail) f(in) g(the) i (forthcoming) d(article) h([DJ3) o(].) 146 3055 y(Finally) -8 b(,) 38 b(w) m(e) i(men) m(tion) d(an) i(early) f(w) m(ork) i([JP3]) f (where) h(the) f(relation) e(b) s(et) m(w) m(een) k(\000) p Fk 3199 3070 a(H) p Fo 3294 3055 a(and) e(\000) p Fk 3551 3070 a(L) p Fo 3638 3055 a(has) 0 3176 y(b) s(een) d(studied.) 53 b(In) 36 b(this) f(w) m(ork) i(one) e(can) h(\014nd) g(an) g(algorithm) c(ho) m(w) k(to) f(construct) i(\000) p Fk 3150 3191 a(H) p Fl(=) p Fk(L) p Fo 3322 3176 a(from) d(\000) p Fk 3616 3191 a(L) p Fl(=) p Fk(H) p Fo 3752 3176 a(.) 0 3296 y(This) e(algorithm) d(can) j(b) s(e) g(also) f(used) i(to) f(pro) m(v) m(e) h(Theorem) f(4.1) g([Pi].) 43 b(The) 33 b(direct) f(pro) s (of) f(of) g(Theorem) 0 3416 y(3.1) h(giv) m(en) h(in) f(this) g(note) h (is) f(ho) m(w) m(ev) m(er) j(considerably) d(simpler.) p Fn 0 3749 a(References) p Fo 0 3968 a([BR1]) 103 b(Brattelli,) 33 b(O.,) j(Robinson,) f(D.) g(W.:) p Fs 48 w(Op) -5 b(er) g(ator) 37 b(A) n(lgebr) -5 b(as) 37 b(and) f(Quantum) h(Statistic) -5 b(al) 37 b(Me-) 347 4089 y(chanics,) c(V) -7 b(olume) 35 b(1) p Fo(.) d(Springer-V) -8 b(erlag,) 31 b(Berlin,) g(second) j (edition) d(1987.) 0 4292 y([BR2]) 103 b(Brattelli,) 33 b(O.,) j(Robinson,) f(D.) g(W.:) p Fs 48 w(Op) -5 b(er) g(ator) 37 b(A) n(lgebr) -5 b(as) 37 b(and) f(Quantum) h(Statistic) -5 b(al) 37 b(Me-) 347 4412 y(chanics,) c(V) -7 b(olume) 35 b(2) p Fo(.) d(Springer-V) -8 b(erlag,) 31 b(Berlin,) g(second) j (edition) d(1996.) 0 4616 y([BFS]) 106 b(Bac) m(h,) 36 b(V.,) f(F) -8 b(r\177) -49 b(ohlic) m(h,) 33 b(J.,) j(Sigal,) d(I.:) 48 b(Return) 34 b(to) h(equilibrium.) c(J.) k(Math.) g(Ph) m(ys.) p Ff 36 w(41) p Fo(,) g(3985) 347 4736 y(\(2000\).) 0 4940 y([Da1]) 120 b(Da) m(vies,) 27 b(E.) e(B.:) 40 b(Mark) m(o) m(vian) 26 b(master) f(equations.) g(Comm) m(un.) g(Math.) h(Ph) m(ys.) p Ff 27 w(39) p Fo(,) h(91) d(\(1974\).) 0 5143 y([Da2]) 120 b(Da) m(vies,) 32 b(E.) h(B.:) 44 b(Mark) m(o) m(vian) 33 b(master) f(equations) h(I) s(I.) g(Math.) g(Ann.) p Ff 33 w(219) p Fo(,) g(147) e(\(1976\).) p 90 rotate dyy eop %%Page: 12 12 12 11 bop Fo 3682 100 a(12) 0 407 y([Da3]) 120 b(Da) m(vies,) 32 b(E.) h(B.:) 44 b(One) 33 b(parameter) f(semigroups,) g(Academic) g (Press) i(1980) 0 605 y([DJ1]) 119 b(Derezi) s(\023) -51 b(nski,) 26 b(J.,) i(Jak) -5 b(\024) -44 b(s) q(i) m(\023) e(c,) 27 b(V.:) 41 b(Sp) s(ectral) 26 b(theory) h(of) e(P) m(auli-Fierz) g(op) s (erators,) i(J.) g(F) -8 b(unc.) 26 b(Anal.) p Ff 347 726 a(180) p Fo(,) 33 b(241) e(\(2001\).) 0 924 y([DJ2]) 119 b(Derezi) s(\023) -51 b(nski,) 51 b(J.,) i(Jak) -5 b(\024) -44 b(s) q(i) m(\023) e(c,) 52 b(V.:) 75 b(Return) 49 b(to) f(equilibrium) d (for) j(P) m(auli-Fierz) e(systems.) k(T) -8 b(o) 347 1045 y(app) s(ear) 32 b(in) g(Ann.) h(Henri) f(P) m(oincar) m(\023) -46 b(e.) 0 1243 y([DJ3]) 119 b(Derezi) s(\023) -51 b(nski,) 31 b(J.,) i(Jak) -5 b(\024) -44 b(s) q(i) m(\023) e(c,) 32 b(V.:) 44 b(In) 33 b(preparation.) 0 1441 y([DJP]) 102 b(Derezi) s(\023) -51 b(nski,) 44 b(J.,) i(Jak) -5 b(\024) -44 b(s) q(i) m(\023) e(c,) 45 b(V.,) h(Pillet,) e(C.) f(A.:) 65 b(P) m(erturbation) 43 b(theory) g(of) p Fm 43 w(W) p Fj 3282 1405 a(\003) p Fo 3321 1441 a(-dynamics,) 347 1562 y(Liouvilleans) 30 b(and) j(KMS-states.) g(T) -8 b(o) 33 b(app) s(ear) f(in) g(Rev.) h(Math.) g(Ph) m(ys.) 0 1760 y([F]) 229 b(F) -8 b(ermi,) 48 b(E.:) p Fs 71 w(Nucle) -5 b(ar) 48 b(physics.) p Fo 46 w(Notes) f(compiled) e(b) m(y) i(Orear) f (J.,) k(Rosenfeld) d(A.H.) f(and) 347 1881 y(Sc) m(hluter) 33 b(R.A.) g(The) g(Univ) m(ersit) m(y) g(of) f(Chicago) g(Press,) j (Chicago,) d(1950.) 0 2079 y([GFVKS]) 48 b(Gorini,) 29 b(V.,) j(F) -8 b(rigerio,) 29 b(A.,) i(V) -8 b(erri,) 31 b(M.,) h(Kossak) m(o) m(wski,) h(A.,) e(Sudarshan,) h(E,C.G.:) 44 b(Prop-) 347 2200 y(erties) 28 b(of) g(quan) m(tum) g(mark) m(o) m (vian) g(master) g(equations.) h(Rep.) f(Math.) h(Ph) m(ys.) p Ff 30 w(13) p Fo(,) g(149) e(\(1978\).) 0 2398 y([Ha]) 171 b(Haag,) 32 b(R.:) p Fs 43 w(L) -5 b(o) g(c) g(al) 35 b(quantum) g(physics.) p Fo 31 w(Springer-V) -8 b(erlag,) 31 b(Berlin,) h(1993.) 0 2597 y([Haa]) 122 b(Haak) m(e,) 37 b(F.:) p Fs 49 w(Statistic) -5 b(al) 38 b(tr) -5 b(e) g(atment) 37 b(of) h(op) -5 b(en) 37 b(systems) g(by) h(gener) -5 b(alize) g(d) 36 b(master) h(e) -5 b(quation.) p Fo 347 2717 a(Springer) 32 b(T) -8 b(racts) 33 b(in) f(Mo) s(dern) h(Ph) m (ysics) p Ff 35 w(66) p Fo(,) f(Springer-V) -8 b(erlag,) 31 b(Berlin,) g(1973.) 0 2915 y([JP1]) 128 b(Jak) -5 b(\024) -44 b(s) q(i) m(\023) e(c,) 30 b(V.,) h(Pillet,) d(C.-A.:) 43 b(On) 30 b(a) g(mo) s(del) e(for) i(quan) m(tum) g(friction) e(I) s(I) s (I.) i(Ergo) s(dic) g(prop) s(erties) 347 3036 y(of) i(the) h(spin-b) s (oson) f(system.) i(Comm) m(un.) e(Math.) h(Ph) m(ys.) p Ff 34 w(178) p Fo(,) g(627) f(\(1996\).) 0 3234 y([JP2]) 128 b(Jak) -5 b(\024) -44 b(s) q(i) m(\023) e(c,) 40 b(V.,) h(Pillet,) d (C.-A.:) 57 b(Sp) s(ectral) 39 b(theory) g(of) g(thermal) f (relaxation.) f(J.) i(Math.) h(Ph) m(ys.) p Ff 347 3355 a(38) p Fo(,) 33 b(1757) e(\(1997\).) 0 3553 y([JP3]) 128 b(Jak) -5 b(\024) -44 b(s) q(i) m(\023) e(c,) 34 b(V.,) g(Pillet,) f (C.-A.:) 47 b(F) -8 b(rom) 32 b(resonances) k(to) e(master) g (equations.) g(Ann.) h(Inst.) g(Henri) 347 3674 y(P) m(oincar) m(\023) -46 b(e) p Ff 33 w(67) p Fo(,) 32 b(425) g(\(1997\).) 0 3872 y([JP4]) 128 b(Jak) -5 b(\024) -44 b(s) q(i) m(\023) e(c,) 24 b(V.,) h(Pillet,) e(C.-A.:) 39 b(Non-equilibrium) 20 b(steady) k(states) g(for) e(\014nite) h(quan) m(tum) g(systems) 347 3992 y(coupled) 33 b(to) f(thermal) f(reserv) m(oirs.) j(Comm) m(un.) e (Math.) h(Ph) m(ys.) p Ff 34 w(226) p Fo(,) g(131) f(\(2002\).) 0 4191 y([JP5]) 128 b(Jak) -5 b(\024) -44 b(s) q(i) m(\023) e(c,) 38 b(V.,) g(Pillet,) f(C.-A.:) 53 b(Mathematical) 35 b(theory) j(of) e (non-equilibrium) e(quan) m(tum) k(sta-) 347 4311 y(tistical) 30 b(mec) m(hanics.) j(J.) g(Stat.) f(Ph) m(ys.) p Ff 35 w(108) p Fo(,) g(787) g(\(2002\).) 0 4510 y([KTH]) 74 b(Kub) s(o,) 32 b(R.,) h(T) -8 b(o) s(da,) 33 b(M.,) g(Hashitsume,) g (N.:) p Fs 44 w(Statistic) -5 b(al) 35 b(Physics) g(II.) f(None) -5 b(quilibrium) 35 b(Sta-) 347 4630 y(tistic) -5 b(al) 35 b(Me) -5 b(chanics.) p Fo 31 w(Springer-V) d(erlag,) 31 b(Berlin,) g(1985.) 0 4829 y([M]) 204 b(Merkli,) 48 b(M.:) 69 b(P) m(ositiv) m(e) 46 b(comm) m(utators) e(in) h(non-equilibrium) d (quan) m(tum) j(statistical) e(me-) 347 4949 y(c) m(hanics.) 33 b(Comm) m(un.) f(Math.) h(Ph) m(ys.) p Ff 35 w(223) p Fo(,) f(327) g(\(2001\).) 0 5147 y([LeSp]) 81 b(Leb) s(o) m(witz,) 60 b(J.,) h(Sp) s(ohn,) f(H.:) 89 b(Irrev) m(ersible) 55 b(thermo) s(dynamics) f(for) g(quan) m(tum) h(systems) 347 5268 y(w) m(eakly) 33 b(coupled) g(to) f(thermal) f(reserv) m(oirs.) j (Adv.) f(Chem.) g(Ph) m(ys.) p Ff 35 w(39) p Fo(,) g(109) e(\(1978\).) p 90 rotate dyy eop %%Page: 13 13 13 12 bop Fo 3682 100 a(13) 0 407 y([P) m(a1]) 132 b(P) m(auli,) 37 b(W.:) 52 b(F) -8 b(esrshrift) 36 b(zum) h(60.) f(Gerburstage) h(A.) g (Sommerfeld,) f(S.30,) i(Leipzig,) f(Hirzel) 347 527 y(1928.) 0 731 y([P) m(a2]) 132 b(P) m(auli,) 40 b(W.:) p Fs 58 w(Pauli) h(L) -5 b(e) g(ctur) g(es) 41 b(on) g(Physics:) 57 b(V) -7 b(olume) 41 b(4.) g(Statistic) -5 b(al) 41 b(Me) -5 b(chanics.) p Fo 38 w(Edited) 347 851 y(b) m(y) 33 b(C.P) -8 b(.) 34 b(Enz,) f(The) h(MIT) f(Press,) i(Cam) m(bridge,) d(1973.) 0 1054 y([Pi]) 199 b(Pillet,) 30 b(C.-A.:) 44 b(Priv) -5 b(ate) 32 b(comm) m(unication.) 0 1258 y([Ru]) 167 b(Ruelle,) 41 b(D.:) 59 b(Natural) 40 b(nonequilibrium) e(states) j(in) f(quan) m (tum) h(statistical) d(mec) m(hanics.) j(J.) 347 1378 y(Stat.) 32 b(Ph) m(ys.) p Ff 35 w(98) p Fo(,) g(57) g(\(2000\).) 0 1582 y([St]) 201 b(Stratila,) 24 b(S.:) p Fs 40 w(Mo) -5 b(dular) 28 b(the) -5 b(ory) 28 b(in) g(op) -5 b(er) g(ator) 27 b(algebr) -5 b(as.) p Fo 24 w(Abacus) 26 b(Press,) i(T) -8 b(urn) m(bridge) 25 b(W) -8 b(ells,) 347 1702 y(1981.) 0 1905 y([StZs]) 103 b(Stratila,) 35 b(S.,) k(Zsido,) e(L.:) p Fs 51 w(L) -5 b(e) g(ctur) g(es) 39 b(on) g(von) f(Neumann) g(algebr) -5 b(as) p Fo(.) 36 b(Abacus) i(Press,) h(T) -8 b(urn-) 347 2026 y(bridge) 32 b(W) -8 b(ells,) 32 b(1979.) 0 2229 y([VH]) 147 b(V) -8 b(an) 24 b(Ho) m(v) m(e,) j(L.:) 39 b(Master) 25 b(equation) f(and) g(approac) m(h) g(to) g(equilibrium) d (for) j(quan) m(tum) g(systems.) 347 2350 y(In) p Fs 41 w(F) -7 b(undamental) 40 b(pr) -5 b(oblems) 42 b(in) g(statistic) -5 b(al) 42 b(me) -5 b(chanics) p Fo(,) 41 b(com) m(bined) g(b) m(y) g (E.G.D.) f(Cohen,) 347 2470 y(North-Holand,) 31 b(Amsterdam) h(1962.) 0 2673 y([W]) 193 b(W) -8 b(eissk) m(opf,) 31 b(V.,) g(Wigner,) f(E.P) -8 b(.:) 43 b(Berec) m(hn) m(ung) 32 b(der) e(nat) s(\177) -51 b(urlic) m(hen) 29 b(Linien) m(breite) g(auf) g(Grund) 347 2794 y(der) k(Diracsc) m(hen) g(Lic) m(h) m(ttheorie.) f(Z.) h(Ph) m (ys.) p Ff 34 w(63) p Fo(,) g(54) f(\(1930\).) p 90 rotate dyy eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF