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

Authors: António de Bivar-Weinholtz and Michel L. Lapidus
Journal: Proc. Amer. Math. Soc. 110 (1990), 449-460
MSC: Primary 47D03; Secondary 28C20, 47A60, 47B25, 81S40
MathSciNet review: 1013964
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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.

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Additional Information

Keywords: Semigroups of operators, product formula for normal operators, modified Feynman integral, Schrödinger equation, Hamiltonians with singular complex potentials, dissipative quantum mechanical systems, magnetic vector potentials, analytic continuation in mass, Feynman path integrals, potentials with arbitrary signed singularity
Article copyright: © Copyright 1990 American Mathematical Society