Related papers: One-Dimensional Quasi-Exactly Solvable Schr\"oding…
Quantum superintegrable systems are solvable eigenvalue problems. Their solvability is due to symmetry, but the symmetry is often "hidden". The symmetry generators of 2nd order superintegrable systems in 2 dimensions close under commutation…
We present a geometric formulation of existence of time quasi-periodic solutions. As an application, we prove the existence of quasi-periodic solutions of $b$ frequencies, $b\leq d+2$, in arbitrary dimension $d$ and for arbitrary non…
Algebraic structure of a class of differential equations including Heun is shown to be related with the deformations of sl(2) algebra. These include both quadratic and cubic ones. The finite dimensional representation of cubic algebra is…
A general method based on the polynomial deformations of the Lie algebra sl(2,R) is proposed in order to exhibit the quasi-exactly solvability of specific Hamiltonians implied by quantum physical models. This method using the…
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…
It is shown that the Dunkl harmonic oscillator on the line can be generalized to a quasi-exactly solvable one, which is an anharmonic oscillator with $n+1$ known eigenstates for any $n\in \N$. It is also proved that the Hamiltonian of the…
The Dirac equation for an electron in two spatial dimensions in the Coulomb and homogeneous magnetic fields is an example of the so-called quasi-exactly solvable models. The solvable parts of its spectrum was previously solved from the…
We present what we believe to be the first known example of an exact quasiperiodic localized stable solution with spatially symmetric large-amplitude oscillations in a non-integrable Hamiltonian lattice model. The model is a one-dimensional…
The quantum version of a non-linear oscillator, previouly analyzed at the classical level, is studied. This is a problem of quantization of a system with position-dependent mass of the form $m={(1+\lambda x^2)}^{-1}$ and with a…
We build the coherent states for a family of solvable singular Schr\"odinger Hamiltonians obtained through supersymmetric quantum mechanics from the truncated oscillator. The main feature of such systems is the fact that their…
Representations of the quantum q-oscillator algebra are studied with particular attention to local Hamiltonian representations of the Schroedinger type. In contrast to the standard harmonic oscillators such systems exhibit a continuous…
From the algebraic treatment of the quasi-solvable systems, and a q-deformation of the associated $su(2)$ algebra, we obtain exact solutions for the q-deformed Schrodinger equation with a 3-dimensional q-deformed harmonic oscillator…
Two families of quasi exactly solvable 2*2 matrix Schroedinger operators are constructed. The first one is based on a polynomial matrix potential and depends on three parameters. The second is a one-parameter generalisation of the scalar…
We systematically describe and classify 1-dimensional Schr\"odinger equations that can be solved in terms of hypergeometric type functions. Beside the well-known families, we explicitly describe 2 new classes of exactly solvable…
Let group generators having finite-dimensional representation be realized as Hermitian linear differential operators without nhomogeneous terms as takes place, for example, for the SO(n) group. Then orresponding group Hamiltonians…
The stationary 1D Schr\"odinger equation with a polynomial potential $V(q)$ of degree N is reduced to a system of exact quantization conditions of Bohr-Sommerfeld form. They arise from bilinear (Wronskian) functional relations pairing…
We propose the notion of $E_{2}$-quasi-exact solvability and apply this idea to find explicit solutions to the eigenvalue problem for a non-Hermitian Hamiltonian system depending on two parameters. The model considered reduces to the…
It is shown that all PDM Schroedinger equations admitting more than five dimensional Lie symmetry algebras (whose completed list can be found in paper~[{\it J.~Math. Phys.} {\bf 58}, , 083508 (2017)] are exactly solvable. The corresponding…
We associate with each simple Lie algebra a system of second-order differential equations invariant under a non-compact real form of the corresponding Lie group. In the limit of a contraction to a Schr\"odinger algebra, these equations…
We construct one soliton solutions for the nonlinear Schroedinger equation with variable quadratic Hamiltonians in a unified form by taking advantage of a complete (super) integrability of generalized harmonic oscillators. The soliton wave…