To Olga Alexandrovna Ladyzhenskaya, with respect and admiration Introduction In the Schrödinger picture, the change in time of the wave function of a quantum system is governed by the Schrödinger equation where H is the Hamiltonian of the system (we choose the units so that = 1). If the operator H does not change in time, then, given ψ(0) = ψ 0 , we have ψ(t) = e −itH ψ 0 . The fundamental solution of the time-dependent Schrödinger equation is the distributional kernel of the solution operator

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... −itH . For a non-relativistic quantum particle of mass 1 moving in the space of n dimensions in a potential field V (x), the operator H has the form (1) H = − 1 2