Product formula for resolvents of normal operators and the modified Feynman integral

Abstract

We extend the theory of the "modified Feynman integal" developed by the second author by extending his product formula for the imaginary resolvents of selfadjoint (unbounded) operators to those of normal operators. This enables us to establish the convergence of the "modified Feynman integral" for Hamiltonians with highly singular complex (instead of real) potentials. Such Hamiltonians arise naturally in the study of the Schrödinger equation associated with dissipative quantum mechanical systems. By slightly altering the proof of our results, we also give a very general (operator-theoretic) interpretation of Nelson’s "Feynman integral by analytic continuation in the mass parameter" that is valid for singular potentials with an arbitrary sign. An interesting aspect of our "product formula for the imaginary resolvents of normal operators" is that it extends, and in some sense unifies, the above two approaches to the Feynman integral.