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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Product formula for resolvents of normal operators and the modified Feynman integral
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by António de Bivar-Weinholtz and Michel L. Lapidus PDF
Proc. Amer. Math. Soc. 110 (1990), 449-460 Request permission

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
  • © Copyright 1990 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 110 (1990), 449-460
  • MSC: Primary 47D03; Secondary 28C20, 47A60, 47B25, 81S40
  • DOI: https://doi.org/10.1090/S0002-9939-1990-1013964-6
  • MathSciNet review: 1013964