Witten index, axial anomaly, and Krein's spectral shift function in supersymmetric quantum mechanics

A new method is presented to study supersymmetric quantum mechanics. Using relative scattering techniques, basic relations are derived between Krein’s spectral shift function, the Witten index, and the anomaly. The topological invariance of the spectral shift function is discussed. The power of this method is illustrated by treating various models and calculating explicitly the spectral shift function, the Witten index, and the anomaly. In particular, a complete treatment of the two‐dimensional magnetic field problem is given, without assuming that the magnetic flux is quantized.

[1]  Barry Simon,et al.  Topological invariance of the Witten index , 1988 .

[2]  D. Bollé,et al.  Krein's spectral shift function and fredholm determinants as efficient methods to study supersymmetric quantum mechanics , 1987 .

[3]  Hans L. Cycon,et al.  Schrodinger Operators: With Application to Quantum Mechanics and Global Geometry , 1987 .

[4]  A. Niemi,et al.  Index theorems on open infinite manifolds , 1986 .

[5]  A. Niemi,et al.  Fermion number fractionization in quantum field theory , 1986 .

[6]  Hirayama.  Topological invariants for Dirac operators on open spaces. , 1986, Physical review. D, Particles and fields.

[7]  Jackiw.  Erratum: Fractional charge and zero modes for planar systems in a magnetic field , 1984, Physical review. D, Particles and fields.

[8]  F. Gesztesy.  Scattering theory for one-dimensional systems with nontrivial spatial asymptotics , 1986 .

[9]  R. L. Elsenbaumer,et al.  Handbook of conducting polymers , 1986 .

[10]  Su,et al.  Vacuum charge: Another study in 1+1 dimensions. , 1985, Physical review. D, Particles and fields.

[11]  Sathiapalan.  Generalized supersymmetric quantum mechanics. , 1985, Physical review letters.

[12]  B. Skagerstam,et al.  Witten's index and supersymmetric quantum mechanics , 1985 .

[13]  Blankenbecler,et al.  Axial and parity anomalies and vacuum charge: A direct approach. , 1985, Physical review. D, Particles and fields.

[14]  Sukhatme,et al.  Supersymmetric quantum mechanics and large-N expansions. , 1985, Physical review letters.

[15]  Blankenbecler,et al.  Fractional charge and spectral asymmetry in one dimension: A closer look. , 1985, Physical review. D, Particles and fields.

[16]  Brown,et al.  Topological invariance of the Witten index and related quantities. , 1985, Physical review. D, Particles and fields.

[17]  J. Lott.  Vacuum charge and the eta function , 1984 .

[18]  A. Comtet,et al.  Anomalous behaviour of the witten index-exactly soluble models , 1984 .

[19]  D. Boyanovsky,et al.  Fractional indices in supersymmetric theories , 1984 .

[20]  A. Niemi,et al.  SPECTRAL ASYMMETRY ON AN OPEN SPACE , 1984 .

[21]  C. Imbimbo,et al.  Topological invariance in supersymmetric theories with a continuous spectrum , 1984 .

[22]  M. Stone.  Zero modes, boundary conditions and anomalies on the lattice and in the continuum , 1984 .

[23]  M. Cheney.  Two‐dimensional scattering: The number of bound states from scattering data , 1984 .

[24]  L. S. Koplienko.  Trace formula for nontrace-class perturbations , 1984 .

[25]  M. Hirayama.  Supersymmetric Quantum Mechanics and Index Theorem , 1983 .

[26]  C. Callias.  The heat equation with singular coefficients , 1983 .

[27]  Israel Gohberg,et al.  Topics in differential and integral equations and operator theory , 1983 .

[28]  Edward Witten,et al.  Constraints on Supersymmetry Breaking , 1982 .

[29]  J. Schrieffer,et al.  Solitons with Fermion Number 1/2 in Condensed Matter and Relativistic Field Theories , 1981 .

[30]  Andrew J. Hanson,et al.  Gravitation, Gauge Theories and Differential Geometry , 1980 .

[31]  A. Heeger,et al.  Soliton excitations in polyacetylene , 1980 .

[32]  F. Gesztesy,et al.  On the high-energy behaviour of scattering phase shifts for Coulomb-like potentials , 1980 .

[33]  Roger G. Newton,et al.  Inverse scattering. I. One dimension , 1980 .

[34]  E. Weinberg.  Parameter counting for multimonopole solutions , 1979 .

[35]  Y. Aharonov,et al.  Ground state of a spin-½ charged particle in a two-dimensional magnetic field , 1979 .

[36]  B. Simon.  Trace ideals and their applications , 1979 .

[37]  Barry Simon,et al.  Methods of modern mathematical physics. III. Scattering theory , 1979 .

[38]  H. Vega.  Fermions and vortex solutions in Abelian and non-Abelian gauge theories , 1978 .

[39]  C. Callias.  Axial anomalies and index theorems on open spaces , 1978 .

[40]  P. Deift.  Applications of a commutation formula , 1978 .

[41]  T. Dreyfus.  The determinant of the scattering matrix and its relation to the number of eigenvalues , 1978 .

[42]  U. Schmincke.  On Schrödinger's factorization method for Sturm-Liouville operators , 1978, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[43]  Tosio Kato,et al.  Asymptotic behalrior of the scattering phase for exterior domains , 1978 .

[44]  R. Newton.  Noncentral potentials: The generalized Levinson theorem and the structure of the spectrum , 1977 .

[45]  J. Kiskis.  Fermions in a Pseudoparticle Field , 1977 .

[46]  LEVINSONtheorem für SCHRÖDINGERoperatoren , 1977 .

[47]  Claudio Rebbi,et al.  Solitons with Fermion Number 1/2 , 1976 .

[48]  Barry Simon,et al.  Quantum Mechanics for Hamiltonians Defined as Quadratic Forms , 1971 .

[49]  M. Kreĭn,et al.  Introduction to the theory of linear nonselfadjoint operators , 1969 .

[50]  Tosio Kato.  Perturbation theory for linear operators , 1966 .

[51]  R. Newton.  Scattering theory of waves and particles , 1966 .

[52]  H. Bremermann.  Distributions, complex variables, and Fourier transforms , 1968 .

[53]  R. Kronig,et al.  The electronic structure of metals , 1949 .

[54]  D. F. Hays,et al.  Table of Integrals, Series, and Products , 1966 .