Valley views: instantons, large order behaviors, and supersymmetry

Abstract

The elucidation of the properties of the instantons in the topologically trivial sector has been a long-standing puzzle. Here we claim that the properties can be summarized in terms of the geometrical structure in the configuration space, the valley. The evidence for this claim is presented in various ways. The conventional perturbation theory and the non-perturbative calculation are unified, and the ambiguity of the Borel transform of the perturbation series is removed. A 'proof' of Bogomolny's "trick" is presented, which enables us to go beyond the dilute-gas approximation. The prediction of the large order behavior of the perturbation theory is confirmed by explicit calculations, in some cases to the 478-th order. A new type of supersymmetry is found as a by-product, and our result is shown to be consistent with the non-renormalization theorem. The prediction of the energy levels is confirmed with numerical solutions of the Schrödinger equation.

-fold supersymmetry in quantum systems with position-dependent mass

“…In contrast to the ordinary constant-mass case, not only the form of potential but also the position dependence of mass affect various aspects of PDM systems. In particular, dynamical N -fold supersymmetry breaking can take place through the nonperturbative effect due to quantum tunneling [39,40]. Hence, it is quite interesting if we can experimentally observe such a phenomenon in realistic systems such as semiconductors, quantum dots, and so on.…”

Corrigendum to “sl(M+1) construction of quasi-solvable quantum M-body systems” [Ann. Phys. 309 (2004) 239–280]

“…In this case, the operator H is said to be quasi-exactly solvable (on S). Otherwise, the solvable spectra and the corresponding vectors of V N only give local solutions of the characteristic equation and have, at most, restrictive meanings in the perturbation theory defined on the physical space L 2 (S) [9,16,19]. A quasi-solvable operator H of several variables is said to be solvable if it preserves an infinite flag of finite dimensional functional spaces V N ,…”

“…In order that the total gauged Hamiltonian (8.7) can be cast in the Schrödinger form by a gauge transformation, we must have, 9) in addition to Eq. (5.15).…”

On Disagreement About Nonperturbative Corrections in Triple-Well Potential

“…Although the perturbation series is generally divergent, as we have already observed in the previous section (cf., Eqs. (17)), the asymptotic property of the perturbation series nevertheless ensures that for a sufficiently small value of the expansion parameter a partial sum of the first finite terms in the perturbation series gives an asymptotic value of the perturbative correction 5 . As a consequence, however small the value of the expansion parameter is, there exists a critical order m c at which the absolute value of the perturbative correction |g 2m c…”

“…We would like to thank the organizers of the international conference "New Frontiers in Quantum Mechanics" (July [5][6][7][8] 2004, Shizuoka University, Japan) where the present work started. This work was partially supported by the Grand-in-Aid for Scientific Research No.14740158 (M. S.) and by a Spanish Ministry of Education, Culture and Sports research fellowship (T. T.).…”