974 research outputs found
Intertwining relations of non-stationary Schr\"odinger operators
General first- and higher-order intertwining relations between non-stationary
one-dimensional Schr\"odinger operators are introduced. For the first-order
case it is shown that the intertwining relations imply some hidden symmetry
which in turn results in a -separation of variables. The Fokker-Planck and
diffusion equation are briefly considered. Second-order intertwining operators
are also discussed within a general approach. However, due to its complicated
structure only particular solutions are given in some detail.Comment: 18 pages, LaTeX20
Classical Integrable 2-dim Models Inspired by SUSY Quantum Mechanics
A class of integrable 2-dim classical systems with integrals of motion of
fourth order in momenta is obtained from the quantum analogues with the help of
deformed SUSY algebra. With similar technique a new class of potentials
connected with Lax method is found which provides the integrability of
corresponding 2-dim hamiltonian systems. In addition, some integrable 2-dim
systems with potentials expressed in elliptic functions are explored.Comment: 19 pages, LaTeX, final version to be published in J.Phys.
Matrix Hamiltonians: SUSY approach to hidden symmetries
A new supersymmetric approach to the analysis of dynamical symmetries for
matrix quantum systems is presented. Contrary to standard one dimensional
quantum mechanics where there is no role for an additional symmetry due to
nondegeneracy, matrix hamiltonians allow for non-trivial residual symmetries.
This approach is based on a generalization of the intertwining relations
familiar in SUSY Quantum Mechanics. The corresponding matrix supercharges, of
first or of second order in derivatives, lead to an algebra which incorporates
an additional block diagonal differential matrix operator (referred to as a
"hidden" symmetry operator) found to commute with the superhamiltonian. We
discuss some physical interpretations of such dynamical systems in terms of
spin 1/2 particle in a magnetic field or in terms of coupled channel problem.
Particular attention is paid to the case of transparent matrix potentials.Comment: 20 pages, LaTe
Factorization of non-linear supersymmetry in one-dimensional Quantum Mechanics. II: proofs of theorems on reducibility
In this paper, we continue to study factorization of supersymmetric (SUSY)
transformations in one-dimensional Quantum Mechanics into chains of elementary
Darboux transformations with nonsingular coefficients. We define the class of
potentials that are invariant under the Darboux - Crum transformations and
prove a number of lemmas and theorems substantiating the formulated formerly
conjectures on reducibility of differential operators for spectral equivalence
transformations. Analysis of the general case is performed with all the
necessary proofs.Comment: 13 page
New Two-Dimensional Quantum Models Partially Solvable by Supersymmetrical Approach
New solutions for second-order intertwining relations in two-dimensional SUSY
QM are found via the repeated use of the first order supersymmetrical
transformations with intermediate constant unitary rotation. Potentials
obtained by this method - two-dimensional generalized P\"oschl-Teller
potentials - appear to be shape-invariant. The recently proposed method of
separation of variables is implemented to obtain a part of their
spectra, including the ground state. Explicit expressions for energy
eigenvalues and corresponding normalizable eigenfunctions are given in analytic
form. Intertwining relations of higher orders are discussed.Comment: 21 pages. Some typos corrected; imrovements added in Subsect.4.2;
some references adde
Factorization of nonlinear supersymmetry in one-dimensional Quantum Mechanics. I: general classification of reducibility and analysis of the third-order algebra
We study possible factorizations of supersymmetric (SUSY) transformations in
the one-dimensional quantum mechanics into chains of elementary Darboux
transformations with nonsingular coefficients. A classification of irreducible
(almost) isospectral transformations and of related SUSY algebras is presented.
The detailed analysis of SUSY algebras and isospectral operators is performed
for the third-order case.Comment: 16 page
On two-dimensional superpotentials: from classical Hamilton-Jacobi theory to 2D supersymmetric quantum mechanics
Superpotentials in supersymmetric classical mechanics are no
more than the Hamilton characteristic function of the Hamilton-Jacobi theory
for the associated purely bosonic dynamical system. Modulo a global sign, there
are several superpotentials ruling Hamilton-Jacobi separable supersymmetric
systems, with a number of degrees of freedom greater than one. Here, we explore
how supersymmetry and separability are entangled in the quantum version of this
kind of system. We also show that the planar anisotropic harmonic oscillator
and the two-Newtonian centers of force problem admit two non-equivalent
supersymmetric extensions with different ground states and Yukawa couplings.Comment: 14 pages, 2 figures, version to appear in J. Phys. A: Math. Ge
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