 03476 Ira Herbst, Erik Skibsted
 Absence of quantum states corresponding to unstable classical channels:
homogeneous potentials of degree zero
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Oct 28, 03

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Abstract. We develop a general theory of absence of quantum states
corresponding to unstable classical channels. A principal example
treated in detail is the following: Consider a realvalued potential
$ V$ on $ \mathbf{R}^{n}$, $ n\geq2$,
which is smooth outside zero and homogeneous of degree zero. Suppose
that the restriction of $ V$ to the
unit sphere $S^{n1}$ is a Morse function. We prove that there
are no $L^{2}$solutions to the Schr\"odinger equation
$i\partial_t \phi=(2^{1}\Delta +V)\phi$
which asymptotically
in time are concentrated near local maxima or saddle points of $
V_{S^{n1}}$. Consequently all states concentrate asymptotically
in time near the local minima. Shortrange perturbations are
included.
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