Related papers: Quasi-exactly solvable Fokker-Planck equations
The Schr\" odinger equations which are exactly solvable in terms of associated special functions are directly related to some self-adjoint operators defined in the theory of hypergeometric type equations. The fundamental formulae occurring…
We extend the notion of Dirac oscillator in two dimensions, to construct a set of potentials. These potentials becomes exactly and quasi-exactly solvable potentials of non-relativistic quantum mechanics when they are transformed into a…
We present a mathematically rigorous quantum-mechanical treatment of a one-dimensional nonrelativistic quantum dual theories (with oscillator and Coulomb-like potentials) and compare their spectra and the sets of eigenfunctions. We…
Solutions of stationary Fokker-Planck equations in the narrow beam regime are commonly approximated by either ballistic linear transport or by a Fermi pencil-beam equation. We present a rigorous approximation analysis of these three models…
It is demonstrated that quasi-exactly solvable models of quantum mechanics admit an interesting duality transformation which changes the form of their potentials and inverts the sign of all the exactly calculable energy levels. This…
By considering a unified treatment, we present quasi exact polynomial solutions to both the Klein-Gordon and Dirac equations with the family of soft-core Coulomb potentials $V_q(r)=-Z/\left(r^q+\beta^q\right)^{1/q}$, $Z>0$, $\beta>0$,…
We describe the real quasi-exactly solvable spectral locus of the PT-symmetric quartic using the Nevanlinna parametrization.
The Schrodinger equations which are exactly solvable in terms of associated special functions are directly related to some self-adjoint operators defined in the theory of hypergeometric type equations. The fundamental formulae occurring in…
How to accurately solve time-dependent Schr\"odinger equation is an interesting and important problem. Here, we propose a novel method to obtain the exact Floquet solutions of the Schr\"odinger equation for periodically driven systems by…
We have generated, using an sl(2,R) formalism, several new classes of quasi-solvable elliptic potentials, which in the appropriate limit go over to the exactly solvable forms. We have obtained exact solutions of the corresponding spectral…
A new class of completely integrable models is constructed. These models are deformations of the famous integrable and exactly solvable Gaudin models. In contrast with the latter, they are quasi-exactly solvable, i.e. admit the algebraic…
After a short outline of the factorization and Grassmann picture of the one-dimensional (1D) Fokker-Planck (FP) equation, we consider a class of spatially-inhomogeneous solutions of the 2D FP equation with symmetric 2D (super)potentials. We…
We present a mathematically rigorous quantum-mechanical treatment of a two-dimensional nonrelativistic quantum dual theories (with oscillator and Coulomb like potentials) on a plane and compare their spectra and the sets of eigenfunctions.…
PT symmetric complex potential V(r) = - r^4 + i a r^3 + b r^2 + i c r + i d/r + e/r^2 is studied. Arbitrarily large multiplets of its closed bound-state solutions with real energies are shown obtainable quasi-exactly (i.e., with a certain…
In this paper, by studying a class of 1-D Sturm-Liouville problems with periodic coefficients, we show and classify the solutions of periodic Schrodinger equations in a multidimensional case, which tells that not all the solutions are Bloch…
We determine the Lie point symmetries of the Fokker-Planck equation and provide examples of solutions of this equation. The Fokker-Planck equation admits a conserved form, hence there is an auxiliary system associated to this equation and…
We classify (1+3)-dimensional Fokker-Planck equations with a constant diagonal diffusion matrix that are solvable by the method of separation of variables. As a result, we get possible forms of the drift coefficients $B_1(\vec x),B_2(\vec…
A new class of quasi exactly solvable potentials with a variable mass in the Schroedinger equation is presented. We have derived a general expression for the potentials also including Natanzon confluent potentials. The general solution of…
Some multidimensional generalizations of the Fokker-Planck equation used by R. Friedrich and J. Peinke for the description of a turbulent cascade as a stochastic process of Markovian type, are considered. The exact solutions of the Cauchy…
We construct a family of self-similar solutions with fat tails to a quadratic kinetic equation. This equation describes the long time behaviour of weak solutions with finite mass to the weak turbulence equation associated to the nonlinear…