60 research outputs found
Shape Invariance and Its Connection to Potential Algebra
Exactly solvable potentials of nonrelativistic quantum mechanics are known to
be shape invariant. For these potentials, eigenvalues and eigenvectors can be
derived using well known methods of supersymmetric quantum mechanics. The
majority of these potentials have also been shown to possess a potential
algebra, and hence are also solvable by group theoretical techniques. In this
paper, for a subset of solvable problems, we establish a connection between the
two methods and show that they are indeed equivalent.Comment: Latex File, 10 pages, One figure available on request. Appeared in
the proceedings of the workshop on "Supersymmetric Quantum Mechanics and
Integrable Models" held at University of Illinois, June 12-14, 1997; Ed. H.
Aratyn et a
Generating Complex Potentials with Real Eigenvalues in Supersymmetric Quantum Mechanics
In the framework of SUSYQM extended to deal with non-Hermitian Hamiltonians,
we analyze three sets of complex potentials with real spectra, recently derived
by a potential algebraic approach based upon the complex Lie algebra sl(2, C).
This extends to the complex domain the well-known relationship between SUSYQM
and potential algebras for Hermitian Hamiltonians, resulting from their common
link with the factorization method and Darboux transformations. In the same
framework, we also generate for the first time a pair of elliptic partner
potentials of Weierstrass type, one of them being real and the other
imaginary and PT symmetric. The latter turns out to be quasiexactly solvable
with one known eigenvalue corresponding to a bound state. When the Weierstrass
function degenerates to a hyperbolic one, the imaginary potential becomes PT
non-symmetric and its known eigenvalue corresponds to an unbound state.Comment: 20 pages, Latex 2e + amssym + graphics, 2 figures, accepted in Int.
J. Mod. Phys.
Semi-fermionic representation for spin systems under equilibrium and non-equilibrium conditions
We present a general derivation of semi-fermionic representation for spin
operators in terms of a bilinear combination of fermions in real and imaginary
time formalisms. The constraint on fermionic occupation numbers is fulfilled by
means of imaginary Lagrange multipliers resulting in special shape of
quasiparticle distribution functions. We show how Schwinger-Keldysh technique
for spin operators is constructed with the help of semi-fermions. We
demonstrate how the idea of semi-fermionic representation might be extended to
the groups possessing dynamic symmetries (e.g. singlet/triplet transitions in
quantum dots). We illustrate the application of semi-fermionic representations
for various problems of strongly correlated and mesoscopic physics.Comment: Review article, 40 pages, 11 figure
The su(1,1) dynamical algebra from the Schr\"odinger ladder operators for N-dimensional systems: hydrogen atom, Mie-type potential, harmonic oscillator and pseudo-harmonic oscillator
We apply the Schr\"odinger factorization to construct the ladder operators
for hydrogen atom, Mie-type potential, harmonic oscillator and pseudo-harmonic
oscillator in arbitrary dimensions. By generalizing these operators we show
that the dynamical algebra for these problems is the Lie algebra.Comment: 10 page
Levinson's Theorem for Non-local Interactions in Two Dimensions
In the light of the Sturm-Liouville theorem, the Levinson theorem for the
Schr\"{o}dinger equation with both local and non-local cylindrically symmetric
potentials is studied. It is proved that the two-dimensional Levinson theorem
holds for the case with both local and non-local cylindrically symmetric cutoff
potentials, which is not necessarily separable. In addition, the problems
related to the positive-energy bound states and the physically redundant state
are also discussed in this paper.Comment: Latex 11 pages, no figure, submitted to J. Phys. A Email:
[email protected], [email protected]
Exact and quasiexact solvability of second-order superintegrable quantum systems: I. Euclidean space preliminaries
We show that second-order superintegrable systems in two-dimensional and three-dimensional Euclidean space generate both exactly solvable (ES) and quasiexactly solvable (QES) problems in quantum mechanics via separation of variables, and demonstrate the increased insight into the structure of such problems provided by superintegrability. A principal advantage of our analysis using nondegenerate superintegrable systems is that they are multiseparable. Most past separation of variables treatments of QES problems via partial differential equations have only incorporated separability, not multiseparability. Also, we propose another definition of ES and QES. The quantum mechanical problem is called ES if the solution of Schrödinger equation can be expressed in terms of hypergeometric functions mFn and is QES if the Schrödinger equation admits polynomial solutions with coefficients necessarily satisfying a three-term or higher order of recurrence relations. In three dimensions we give an example of a system that is QES in one set of separable coordinates, but is not ES in any other separable coordinates. This example encompasses Ushveridze's tenth-order polynomial QES problem in one set of separable coordinates and also leads to a fourth-order polynomial QES problem in another separable coordinate set
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