Hideaki Aoyama

N fold supersymmetry in quantum mechanics: General formalism

Abstract We report general properties of N -fold supersymmetry in one-dimensional quantum mechanics. N -fold supersymmetry is characterized by supercharges which are N th polynomials of momentum. Relations between the anti-commutator of the supercharges and the Hamiltonian, the spectra, the Witten index, the non-renormalization theorems and the quasi-solvability are examined. We also present further investigation about a particular class of N -fold supersymmetric models which we dubbed typexa0A. Algebraic equations which determine a part of spectra of typexa0A models are presented, and the non-renormalization theorem are generalized. Finally, we present a possible generalization of N -fold supersymmetry in multi-dimensional quantum mechanics.

General Forms of a N-fold Supersymmetric Family

Abstract We report general forms of one family of the N -fold supersymmetry in one-dimensional quantum mechanics. The N -fold supersymmetry is characterized by the supercharges which are N th order in differential operators. The family reported here is defined as a particular form of the supercharges and is referred to as “type A”. We show that a quartic and a periodic potentials, which were previously found to be N -fold supersymmetric by the authors, are realized as special cases of this type A family.

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. Evidence for this claim is presented in various ways. The conventional perturbation theory and the non-perturbatioe calculation are unified, and the ambiguity of the Borel transform of the perturbation series is removed. A ‘proof’ of Bogomolnys “trick” is presented, which enables us to go beyond the dilute-gas approximation. The prediction of the large order behavior of perturbation theory is confirmed by explicit calculations, in some cases to the 478th 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 Schrodinger equation.

Multi-instanton calculus in N = 2 supersymmetric QCD

Microscopic tests of the exact results are performed in N = 2 SU(2) supersymmetric QCD. We construct the multi-instanton solution in N = 2 supersymmetric QCD and calculate the two-instanton contribution F2 to the prepotential F explicitly. For Nf = 1,2, instanton calculus agrees with the prediction of the exact results, however, for Nf = 3, we find a discrepancy between them.

N fold supersymmetry for a periodic potential

Abstract We report a new type of supersymmetry, “ N -fold supersymmetry”, in one-dimensional quantum mechanics. Its supercharges are N th order polynomials of momentum: it reduces to ordinary supersymmetry for N =1 , but for other values of N the anticommutator of the supercharges is not the ordinary Hamiltonian, but is a polynomial of the Hamiltonian. (For this reason, the original Hamiltonian is referred to as the “Mother Hamiltonian”.) This supersymmetry shares some features with the ordinary variety, the most notable of which is the non-renormalization theorem. An N -fold supersymmetry was earlier found for a quartic potential whose supersymmetry is spontaneously broken. Here we report that it also holds for a periodic potential, albeit with somewhat different supercharges, whose supersymmetry is not broken.

sl(2) construction of type A N-fold supersymmetry

N-fold supersymmetry is an extension of the ordinary supersymmetry in one-dimensional quantum mechanics. One of its major property is quasi-solvability, which means that energy eigenvalues can be obtained for a portion of the spectra. We show that recently found type A N-fold supersymmetry can be constructed by using sl(2) algebra, which provides a basis for the quasi-solvability. By this construction we find a condition for the type A N-fold supersymmetry which is less restrictive than the condition known previously. Several explicitly known models are also examined in the light of this construction.

Recent Developments of the Theory of Tunneling

Path-integral approach in imaginary and complex time has been proven successful in treating the tunneling phenomena in quantum mechanics and quantum field theories. Latest developments in this field, the proper valley method in imaginary time, its application to various quantum systems, complex time formalism, asympton theory for the large order analysis of the perturbation theory, are reviewed in a self-contained manner.

Classification of type A N-fold supersymmetry

Abstract Type A N -fold supersymmetry of one-dimensional quantum mechanics can be constructed by using sl(2) generators represented on a finite-dimensional functional space. Using this sl(2) formalism we show a general method of constructing typexa0A N -fold supersymmetric models. We also present systematic generation of known models and several new models using this method.

Valleys in quantum mechanics

Abstract Conventionally, perturbative and non-perturbative calcula-tions are performed independently. In this paper, valleys in the configuration space in quantum mechanics are investigated as a way to treat them in a unified manner. All the known results of the interplay of them are reproduced naturally. The prescription for separating the non-perturbative contribution from the perturbative is given in terms of the analytic continuation of the valley parameter. Our method is illustrated on a new series of examples with the asymmetric double-well potential. We obtain the non-perturbative part explicitly, which leads to the prediction of the large order behavior of the perturbative series. We calculate the first 200 perturbative coefficients for a wide range of parameters and confirm the agreement with the prediction of the valley method.

FAKE INSTABILITY IN THE EUCLIDEAN FORMALISM OF QUANTUM TUNNELING

The Euclidean path-integral formalism is studied for the system without any unstable state. While the usual bounce calculation yields a fake complex energy spectrum, proper account of the global structure of the functional space is shown to lead to the real eigenvalues. For this purpose, the valley instanton is constructed by the proper valley method and the dilute-gas approximation is carried out. This establishes a general imaginary-time formalism, unifying the instanton and the bounce method. {copyright} {ital 1997} {ital The American Physical Society}