 07113 Vladimir S. Buslaev and Catherine Sulem
 Linear adiabatic dynamics generated by operators with continuous spectrum.I
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May 4, 07

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Abstract. We are interested in the asymptotic behavior of the solution to the
Cauchy problem for the linear evolution equation
$$ i\varepsilon \partial_t \psi = A(t) \psi, \quad A(t) =A_0 +V(t),\quad
\psi(0) = \psi_0, $$
in the limit $\varepsilon \to 0$. A case of special interest is when
the operator $A(t)$ has continuous spectrum and the initial data
$\psi_0$ is, in particular, an improper eigenfunction of the
continuous spectrum of $A(0)$. Under suitable assumptions on $A(t)$,
we derive a formal asymptotic solution of the problem whose leading
order has an explicit representation.
A key ingredient is a reduction of the original Cauchy problem to the
study of the semiclassical pseudodifferential operator ${\lM}=
M(t, i\varepsilon\partial_t)$ with compact operatorvalued
symbol $M(t,E) = V_1(t)(A_0EI)^{1} V_2(t)$ , $V(t) =V_2(t)V_1(t),$
and an asymptotic analysis of its spectral
properties. We illustrate our approach with
a detailed presentation of the example of the Schr\"odinger equation
on the axis with the $\delta$function potential: $A(t)
=\partial_{xx} + \alpha(t) \delta(x).$
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