4,570 research outputs found
Double Shape Invariance of Two-Dimensional Singular Morse Model
A second shape invariance property of the two-dimensional generalized Morse
potential is discovered. Though the potential is not amenable to conventional
separation of variables, the above property allows to build purely
algebraically part of the spectrum and corresponding wave functions, starting
from {\it one} definite state, which can be obtained by the method of
-separation of variables, proposed recently.Comment: 9 page
Non-linear Supersymmetry for non-Hermitian, non-diagonalizable Hamiltonians: II. Rigorous results
We continue our investigation of the nonlinear SUSY for complex potentials
started in the Part I (math-ph/0610024) and prove the theorems characterizing
its structure in the case of non-diagonalizable Hamiltonians. This part
provides the mathematical basis of previous studies. The classes of potentials
invariant under SUSY transformations for non-diagonalizable Hamiltonians are
specified and the asymptotics of formal eigenfunctions and associated functions
are derived. Several results on the normalizability of associated functions at
infinities are rigorously proved. Finally the Index Theorem on relation between
Jordan structures of intertwined Hamiltonians depending of the behavior of
elements of canonical basis of supercharge kernel at infinity is proven.Comment: 31 pp., comments on PT symmetry and few relevant refs are adde
SUSY Intertwining Relations of Third Order in Derivatives
The general solution of the intertwining relations between a pair of
Schr\"odinger Hamiltonians by the supercharges of third order in derivatives is
obtained. The solution is expressed in terms of one arbitrary function. Some
properties of the spectrum of the Hamiltonian are derived, and wave functions
for three energy levels are constructed. This construction can be interpreted
as addition of three new levels to the spectrum of partner potential: a ground
state and a pair of levels between successive excited states. Possible types of
factorization of the third order supercharges are analysed, the connection with
earlier known results is discussed.Comment: 17
Systems with Higher-Order Shape Invariance: Spectral and Algebraic Properties
We study a complex intertwining relation of second order for Schroedinger
operators and construct third order symmetry operators for them. A modification
of this approach leads to a higher order shape invariance. We analyze with
particular attention irreducible second order Darboux transformations which
together with the first order act as building blocks. For the third order
shape-invariance irreducible Darboux transformations entail only one sequence
of equidistant levels while for the reducible case the structure consists of up
to three infinite sequences of equidistant levels and, in some cases, singlets
or doublets of isolated levels.Comment: 18 pages, LaTeX, editorial page is remove
Two-Dimensional Supersymmetry: From SUSY Quantum Mechanics to Integrable Classical Models
Two known 2-dim SUSY quantum mechanical constructions - the direct
generalization of SUSY with first-order supercharges and Higher order SUSY with
second order supercharges - are combined for a class of 2-dim quantum models,
which {\it are not amenable} to separation of variables. The appropriate
classical limit of quantum systems allows us to construct SUSY-extensions of
original classical scalar Hamiltonians. Special emphasis is placed on the
symmetry properties of the models thus obtained - the explicit expressions of
quantum symmetry operators and of classical integrals of motion are given for
all (scalar and matrix) components of SUSY-extensions. Using Grassmanian
variables, the symmetry operators and classical integrals of motion are written
in a unique form for the whole Superhamiltonian. The links of the approach to
the classical Hamilton-Jacobi method for related "flipped" potentials are
established.Comment: 19 page
The Extended Chiral Quark Model confronts QCD
We discuss the truncation of low energy effective action of QCD below the
chiral symmetry breaking (CSB) scale, including all operators of dimensionality
less or equal to 6 which can be built with quark and chiral fields. We perform
its bosonization in the scalar, pseudoscalar, vector and axial-vector channels
in the large-N_c and leading-log approximation. Constraints on the coefficients
of the effective lagrangian are derived from the requirement of Chiral Symmetry
Restoration (CSR) at energies above the CSB scale in the scalar-pseudoscalar
and vector-axial-vector channels, from matching to QCD at intermediate scales,
and by fitting some hadronic observables. In this truncation two types of
pseudoscalar states (massless pions and massive Pi'-mesons), as well as a
scalar, vector and axial-vector one arise as a consequence of dynamical chiral
symmetry breaking. Their masses and coupling constants as well as a number of
chiral structural constants are derived. A reasonable fit of all parameters
supports a relatively heavy scalar meson (quarkonium) with the mass \sim 1 GeV
and a small value of axial pion-quark coupling constant g_A \simeq 0.55.Comment: Talk at QCD99, Montpellier, July 1999, 7 pages, Late
Polynomial SUSY in Quantum Mechanics and Second Derivative Darboux Transformation
We give the classification of second-order polynomial SUSY Quantum Mechanics
in one and two dimensions. The particular attention is paid to the irreducible
supercharges which cannot be built by repetition of ordinary Darboux
transformations. In two dimensions it is found that the binomial superalgebra
leads to the dynamic symmetry generated by a central charge operator.Comment: 10 pages, LaTeX, preprint SPbU-IP-94-0
- …