Abstract
A quantum system with a self-adjoint Hamiltonian has a well-defined unitary time evolution operator even if there is no ground state. However, if the spectrum of the Hamiltonian is unbounded from below, the Brownian (or imaginary time) functional integral may diverge for all times. Working in one-dimensional non-relativistic quantum mechanics with potentials which are unbounded from below at infinity and at the origin, we show how to regulate the Brownian functional integral, analytically continue to quantum mechanical time, and then remove the cutoff to obtain the quantum Green's function associated with any self-adjoint Hamiltonian. Our constructions illustrate the equivalence of the operator and functional integral approaches to quantum mechanics.
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