Georg Junker

Schrodinger operators with complex potential but real spectrum

Abstract Several aspects of complex-valued potentials generating a real and positive spectrum are discussed. In particular, we construct complex-valued potentials whose corresponding Schrodinger eigenvalue problem can be solved analytically.

Conditionally Exactly Solvable Potentials: A Supersymmetric Construction Method☆

Abstract We present in this paper a rather general method for the construction of so-called conditionally exactly solvable potentials. This method is based on algebraic tools known from supersymmetric quantum mechanics. Various families of one-dimensional potentials are constructed whose corresponding Schrodinger eigenvalue problem can be solved exactly under certain conditions of the potential parameters. Examples of quantum systems on the real line and the half line as well as on some finite interval are studied in detail.

Conditionally exactly solvable problems and non-linear algebras

Abstract Using ideas of supersymmetric quantum mechanics we construct a class of conditionally exactly solvable potentials which are supersymmetric partners of the linear and radial harmonic oscillator. Furthermore we show that this class of problems possesses some symmetry structures which belong to non-linear algebras.

Non-linear coherent states associated with conditionally exactly solvable problems

Abstract Recently, based on a supersymmetric approach, new classes of conditionally exactly solvable problems have been found. These systems exhibit a symmetry structure characterized by non-linear algebras. In this paper the associated “non-linear” coherent states are constructed and some of their properties are discussed in detail.

Quasiclassical path-integral approach to supersymmetric quantum mechanics

{}From Feynmans path integral, we derive quasi-classical quantization rules in supersymmetric quantum mechanics (SUSY-QM). First, we derive a SUSY counterpart of Gutzwillers formula, from which we obtain the quantization rule of Comtet, Bandrauk and Campbell when SUSY is good. When SUSY is broken, we arrive at a new quantization formula, which is found as good as and even sometime better than the WKB formula in evaluating energy spectra for certain one-dimensional bound state problems. The wave functions in the stationary phase approximation are also derived for SUSY and broken SUSY cases. Insofar as a broken SUSY case is concerned, there are strong indications that the new quasi-classical approximation formula always overestimates the energy eigenvalues while WKB always underestimates.

Intertwining relations of non-stationary Schrödinger operators

General first- and higher-order intertwining relations between non-stationary one-dimensional Schrodinger operators are introduced. For the first-order case it is shown that the intertwining relations imply some hidden symmetry which in turn results in an R-separation of variables. The Fokker-Planck and diffusion equations 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.

Supersymmetric classical mechanics

We study the classical properties of a supersymmetric system which is often used as a model for supersymmetric quantum mechanics. It is found that the classical dynamics of the bosonic as well as the fermionic degrees of freedom is fully described by a so--called quasi--classical solution. We also comment on the importance of this quasi--classical solution in the semi--classical treatment of the supersymmetric quantum model.

Solvable potentials, non-linear algebras, and associated coherent states

Using the Darboux method and its relation with supersymmetric quantum mechanics we construct all SUSY partners of the harmonic oscillator. With the help of the SUSY transformation we introduce ladder operators for these partner Hamiltonians and shown that they close a quadratic algebra. The associated coherent states are constructed and discussed in some detail.

Remarks on the quasi-classical propagator of area-preserving maps

Abstract The path-integral-like expression for the quantum propagator of discrete-time area-preserving maps is evaluated approximately by neglecting higher than second-order terms in an expansion of the action about the classical paths. In the resulting quasi-classical approximation for the propagator special attention is paid to the recursive nature of its amplitude and the possible appearance of Maslov-like phases. Using a further approximation for the amplitude we arrive at explicit expressions which clearly show the differences between the contributions of stable and unstable classical paths. An estimate for the range of validity for the quasi-classical approximation is also given.

Coherent states for a polynomial su(1,1) algebra and a conditionally solvable system

In a previous paper (2007 J. Phys. A: Math. Theor. 40 11105), we constructed a class of coherent states for a polynomially deformed su(2) algebra. In this paper, we first prepare the discrete representations of the nonlinearly deformed su(1, 1) algebra. Then we extend the previous procedure to construct a discrete class of coherent states for a polynomial su(1, 1) algebra which contains the Barut–Girardello set and the Perelomov set of the SU(1, 1) coherent states as special cases. We also construct coherent states for the cubic algebra related to the conditionally solvable radial oscillator problem.