Singularities of solutions of some Schrödinger equations on ${\text{R}}^n $

Abstract

The fundamental solution (or propagator) k0(t, x, y) of this equation, with initial condition fco(0, x, y) = U=zl S(Xj jy), has singularities which can easily be determined by using a well-known explicit formula for k0 [2, 6] for t ^ mn/u)j and taking distribution limits as t —• m7r/co;.. The result is: (i) if L = G)jt/n is not an integer for any ƒ, then k(tt x, y) is smooth in (ii) if L € Z for ƒ = j , . . . , jr then k(t, x, y) is a smooth 8 -function supported on the (n r)-dimensional plane {XJS = (-1)%/^}. In quantum mechanics, k(t, x, y) is the (probability) amplitude that a particle certainly at y at time 0 has arrived at x at time t. Semi-classically, k(t> x, y) describes a swarm of classical particles with initial positions all at y and with initial momenta uniformly distributed through R. The occurrence of singularities in k(t, x, y) coincides with the appearance of an infinite density of classical particles at a given point. In this announcement, we show how this description of singularities can be extended to the case where the Hamiltonian is H0 + V, V being (multiplication by) a potential function on R which is assumed to belong to the symbol class ^ ( R ) ; i.e. \d$V = 0( 0, where = (1 + \x\) Semi-classically, the swarm is now additionally influenced by the force -grad V(x); since this force approaches 0 at °°, the higher energy particles in the swarm are little influenced by the perturbation, and so they will tend to re-accumulate at the same points as before. It turns out that these points also form the singular locus for the perturbed wave functions, demonstrating the stability of the picture described in (i), (ii) above.