03-476 Ira Herbst, Erik Skibsted

Absence of quantum states corresponding to unstable classical channels: homogeneous potentials of degree zero (621K, Postscript) 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 real-valued 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 $S^{n-1}$ 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^{n-1}}$. Consequently all states concentrate asymptotically in time near the local minima. Short-range perturbations are included.

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