Ira Herbst, Erik Skibsted
Absence of quantum states corresponding to unstable classical channels: 
 homogeneous potentials of degree zero
(621K, Postscript)

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.