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We show that the radial harmonic oscillator problem in the position-dependent mass background of the type $m(\alpha;r) = (1+\alpha r^2)^{-2}$, $\alpha>0$, can be solved by using a point canonical transformation mapping the corresponding…
The classical Schr\"odinger equation with a harmonic trap potential $V(x)=|x|^2$, describing the quantum harmonic oscillator, has been studied quite extensively in the last twenty years. Its ground states are bell-shaped and unique, among…
We study the generalized quantum isotonic oscillator Hamiltonian given by H=-d^2/dr^2+l(l+1)/r^2+w^2r^2+2g(r^2-a^2)/(r^2+a^2)^2, g>0. Two approaches are explored. A method for finding the quasi-polynomial solutions is presented, and…
We use a power-series expansion to calculate the eigenvalues of anharmonic oscillators bounded by two infinite walls. We show that for large finite values of the separation of the walls, the calculated eigenvalues are of the same high…
We consider the numerical integration of the matrix Hill's equation. Parametric resonances can appear and this property is of great interest in many different physical applications. Usually, the Hill's equations originate from a Hamiltonian…
In this work we study a class of anharmonic oscillators on $\mathbb{R}^n$ corresponding to Hamiltonians of the form $A(D)+V(x)$, where $A(\xi)$ and $V(x)$ are $C^{\infty}$ functions enjoying some regularity conditions. Our class includes…
A simple and algorithmic description of matrix shape invariant potentials is presented. The complete lists of generic matrix superpotentials of dimension $2\times2$ and of special superpotentials of dimension $3\times3$ are given…
This paper is a natural continuation of the previous paper \cite{TyuVo13} where generalized oscillator representations for Calogero Hamiltonians with potential $V(x)=\alpha/x^2$, $\alpha\geq-1/4$, were constructed. In this paper, we present…
We discuss the maximum kinematical invariance group of the quantum harmonic oscillator from a view point of the Ermakov-type system. A six parameter family of the square integrable oscillator wave functions, which seems cannot be obtained…
Spectral components of one-dimensional Schr\"odinger operator with complex potential are investigated. An effective upper bound for the total number of eigenvalues and spectral singularities is established. For dissipative Schr\"odinger…
We test the analytical expressions for the first two eigenvalues of the harmonic oscillator with a Gaussian perturbation proposed recently. Our numerical eigenvalues show that those expressions are valid in an interval of the coupling…
The dynamical law obeyed by the one-dimensional physical systems in the scale relativity approach is reduced to a Riccati nonlinear differential equation. Applied to the harmonic oscillator potential, we show that such an approach permits…
In this paper, we generalize several results of the article "Analytic continuation of eigenvalues of a quartic oscillator" of A. Eremenko and A. Gabrielov. We consider a family of eigenvalue problems for a Schr\"odinger equation with even…
We describe a method for the calculation of accurate energy eigenvalues and expectation values of observables of separable quantum-mechanical models. We discuss the application of the approach to one-dimensional anharmonic oscillators with…
In this paper, closed formulas for the eigenvectors of a particular class of matrices generated by generalized permutation matrices, named generalized circulant matrices, are presented.
The large-N expansion technique is tested via an anomalous, soft-core potential which admits the tunneling through its central barrier. The precision of the approximation is found sensitive to the asymptotic component of the interaction.…
Summation by parts is used to find the sum of a finite series of generalized harmonic numbers involving a specific polynomial or rational function. The Euler-Maclaurin formula for sums of powers is used to find the sums of some finite…
A symmetric characteristic singular integral equation with two fixed singularities at the endpoints in the class of functions bounded at the ends is analyzed. It reduces to a vector Hilbert problem for a half-disc and then to a vector…
We investigate the combinatorics of the general formulas for the powers of the operator $h \partial^k$, where $h$ is a central element of a ring and $\partial$ is a differential operator. This generalizes previous work on the powers of…
$C_{\lambda}$-extended oscillator algebras are realized as generalized deformed oscillator algebras. For $\lambda = 3$, the spectrum of the corresponding bosonic oscillator Hamiltonian is shown to strongly depend on the algebra parameters.…