Related papers: Quasi-exactly solvable potentials in Wigner-Dunkl …
We propose a new analytical method to solve for the nonexactly solvable Schrodinger equation. Successfully, it is applied to a class of spiked harmonic oscillators and truncated Coulomb potentials. The utility of this method could be…
We propose a Wigner quasiprobability distribution function for Hamiltonian systems in spaces of constant curvature --in this paper on hyperboloids--, which returns the correct marginals and has the covariance of the Shapiro functions under…
We prove that a linear d-dimensional Schr{\"o}dinger equation on $\mathbb{R}^d$ with harmonic potential $|x|^2$ and small t-quasiperiodic potential $i\partial\_t u -- \Delta u + |x|^2 u + \epsilon V (t\omega, x)u = 0, x \in \mathbb{R}^d$…
We study fractional quantum Hall states in the cylinder geometry with open boundaries. By truncating the Coulomb interactions between electrons we show that it is possible to construct infinitely many exact eigenstates including the ground…
Recently, a class of PT-invariant quantum mechanical models described by the non-Hermitian Hamiltonian $H=p^2+x^2(ix)^\epsilon$ was studied. It was found that the energy levels for this theory are real for all $\epsilon\geq0$. Here, the…
We investigate the one-dimensional Coulomb potential with application to a class of quasirelativistic systems, so-called Dirac-Weyl materials, described by matrix Hamiltonians. We obtain the exact solution of the shifted and truncated…
We consider supersymmetric quantum mechanical models with both local and nonlocal potentials. We present a nonlocal deformation of exactly solvable local models. Its energy eigenfunctions and eigenvalues are determined exactly. We observe…
In this work we introduce two different generalizations of the Fokker-Planck equation in (1+1) dimensions by replacing the spatial derivatives in terms of generalized Dunkl-type derivatives involving reflection operators. As applications of…
In this work, we show that the completeness relation for the eigenvectors, which is an essential assumption of quantum mechanics, remains true if the Hamiltonian, having a discrete spectrum, is modified by a delta potential (to be made…
We study the Schr\"{o}dinger operators on a non-compact star graph with the Coulomb-type potentials having singularities at the vertex. The convergence of regularized Hamiltonians $H_\varepsilon$ with cut-off Coulomb potentials coupled with…
The two-dimensional Dirac Hamiltonian with equal scalar and vector potentials has been proved commuting with the deformed orbital angular momentum $L$. When the potential takes the Coulomb form, the system has an SO(3) symmetry, and…
We deform the real potential of Poeschl and Teller by a shift of its coordinate in imaginary direction. We show that the new model remains exactly solvable. Its bound states are constructed in closed form. Wave functions are complex and…
Non-relativistic potential models are considered of the pure power V(r) = sgn(q) r^q and logarithmic V(r) = ln(r) types. Envelope representations and kinetic potentials are employed to show that these potentials are actually in a single…
The synchronization properties of two self-sustained quantum oscillators are studied in the Wigner representation. Instead of considering the quantum limit of the quantum van-der-Pol master equation we derive the quantum master equation…
The relationship between the quasi-exactly solvable problems and W-algebras is revealed. This relationship enabled one to formulate a new general method for building multi-dimensional and multi-channel exactly and quasi-exactly solvable…
We propose a more direct approach to constructing differential operators that preserve polynomial subspaces than the one based on considering elements of the enveloping algebra of sl(2). This approach is used here to construct new exactly…
There exists an exact relationship between the quasi-exactly solvable problems of quantum mechanics and models of square and rectangular random complex matrices. This relationship enables one to reduce the problem of constructing…
Using supersymmetric quantum mechanics we develop a new method for constructing quasi-exactly solvable (QES) potentials with two known eigenstates. This method is extended for constructing conditionally-exactly solvable potentials (CES).…
A general procedure is presented to construct conditionally solvable (CES) potentials using the techniques of supersymmetric quantum mechanics.The method is illustrated with potentials related to the harmonic oscillator problem.Besides…
Several kinds of q-orthogonal polynomials with |q|=1 are constructed as the main parts of the eigenfunctions of new solvable discrete quantum mechanical systems. Their orthogonality weight functions consist of quantum dilogarithm functions,…