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Miloslav Znojil
Reviewer number:
Item's zbl-Number:
DE 017 497 140
Casahorrán, J.
Quantum mechanical tunneling: Differential operators, zeta-functions and determinants.
Forthschr. Phys. 50, No. 3 - 4, 405 - 424 (2002).
81Q30Feynman integrals and graphs; applications of algebraic topology and algebraic geometry
Primary Classification:
81Q20Semiclassical techniques including WKB and Maslov methods
Secondary Classification:
81S40Path integrals
path-integral method; double well potential; Wick rotation; instanton solutions; corrections; semiclassical technique; shape invariance; closed solutions; multiinstanton corrections.

A characteristic difficulty with quantization is that the related
practical calculations (e.g., of Feynmann integrals) may often
become prohibitively complicated (and still, mathematically, not
too well founded) in the majority of the realistic
phenomenological models in physics. This is the reason why the
author in question picked up an oversimplified model, viz., a
particle in a one-dimensional double well, and applied to it a
complicated universal method, viz., the evaluation of corrections
to an instanton solution based on a Wick rotation in the complex
plane of time. In this way one arrives at the auxiliary
differential operators (of the second order), the eigenvalues and
eigenfunctions of which are obtainable in closed form
(construction based, by the way, on the so called shape invariance
recollected in Appendix A). Then one evaluates the determinants by
means of the so called zeta function method summarized briefly in
Appendix B. In this way, with a patient emphasis on the key
technicalities (first of all, on the zero-mode removal and on the
multi-instanton corrections in the dilute-gas approximation), the
author equips the current method with a well enhanced credit of a
well deserved mathematical rigor. All done with care and
pedagogical routine.
Remarks to the editors:

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