Families Of Isospectral Potentials Research Articles

An introduction to analysis of Rényi complexity ratio of quantum states for central potential

AbstractRényi complexity ratio of two density functions is introduced for three and multidimensional quantum systems. Localization property of several density functions are defined and five theorems about near continuous property of Rényi complexity ratio are proved by Lebesgue measure. Some properties of Rényi complexity ratio are demonstrated and investigated for different quantum systems. Exact analytical forms of Rényi entropy, Rényi complexity ratio, statistical complexities based on Rényi entropy for integral order have been presented for solutions of pseudoharmonic and a family of isospectral potentials. Some properties of Rényi complexity ratio are verified for six diatomic molecules (CO, NO, N2, CH, H2, and ScH) and for other quantum systems.

Generalized quantum similarity index: An application to pseudoharmonic oscillator with isospectral potentials in 3D

AbstractExact solutions of a pseudoharmonic oscillator and a family of isospectral potentials are investigated in spherical coordinates. The entropic moment, generalized quantum similarity index, and some quantum information measures are investigated analytically and numerically for two density functions of two quantum systems with same energy and one quantum system with different energies. Analytical results are compared for 19 selected molecules and verified by some physical and artificial values of the spectroscopic parameters.

Isospectral Trigonometric Pöschl-Teller Potentials with Position Dependent Mass Generated by Supersymmetry

In this work a position dependent mass Hamiltonian with the same spectrum of the trigonometric Pöschl-Teller one was constructed by means of the underlying potential algebra. The corresponding wave functions are determined by using the factorization method. A new family of isospectral potentials are constructed by applying a Darboux transformation. An example is presented in order to illustrate the formalism.

PHASE RIGIDITY OF POINT INTERACTIONS

In this paper, we investigate phase rigidity of one-dimensional point interactions. With the aid of supersymmetric quantum mechanics (SUSYQM) we generate family of isospectral potentials describing point interactions. We than demonstrate that for SUSYQM generated bound states in the continuum (BIC) phase rigidity is always zero, while for bound states from discrete part of spectrum phase rigidity may vary from zero to unity, depending on the strength of point interaction.

Darboux transformation of the Schrödinger equation

The recent developments in the theory of the generation of potentials for which the Schrodinger equation has an exact solution are discussed. The generalization of the Darboux transformation to the nonstationary Schrodinger equation is studied in detail. The supersymmetric generalization of the nonstationary Schrodinger equation is formulated. Versions corresponding to exact and spontaneously broken supersymmetry are discussed. New, exactly solvable nonstationary potentials are obtained as examples. The stationary Darboux transformation is viewed as a special case of the new transformation. Families of isospectral potentials with the spectra of the harmonic oscillator and the hydrogen-like atom are obtained. The effectiveness of these methods for describing the coherent states of the transformed Hamiltonians is demonstrated.

Isospectral deformation of quantum potentials and the Liouville equation

A quantum problem on an isospectral deformation of one-dimensional potentials (and of corresponding wave functions) is considered. The isospectral deformation defined in the form of a phase flow is shown to obey a system of coupled Liouville equations. In a simple case of an individual flow the well-known integrable Liouville equation arises; its solution provides known families of isospectral potentials. Operators performing this deformation are studied; their unitary property is proved. An evolution of spectral shift operators is determined using those unitary operators. An asymptotical behavior of both a potential and wave functions under this isospectral deformation is studied. It is shown, in particular, that the deformation of the Rosen-Morse potential and that of the harmonic oscillator's potential have common analytical properties. The approach used in the paper can be extended to the case of a deformation leading to a shift of one selected energy level. In the case of the simplest individual flow we get a generalization of the integrable Liouville equation.

SOLITONS FROM SUPERSYMMETRY

Supersymmetry transformations constitute a new, powerful technique of generating families of isospectral potentials. We show that isospectral families of reflectionless potentials provide surprisingly simple expressions for the pure multi-soliton solutions of the KdV and other nonlinear evolution equations.