Related papers: Quasi Exactly Solvable 2$\times$2 Matrix Equations
Matrix quasi exactly solvable operators are considered and new conditions are determined to test whether a matrix differential operator possesses one or several finite dimensional invariant vector spaces. New examples of $2\times 2$-matrix…
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…
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…
We discuss a universal algebraic approach to quasi-exactly solvable models which allows us to interpret them as constrained Hamiltonian systems with a finite number of physical states. Using this approach we reproduce well-known…
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…
We reconsider the quasi exactly solvable matrix models constructed recently by R. Zhdanov. The 2$\times$2 matrix operators representing the algebra sl(2) are generalized to matrices of arbitrary dimension and a similar construction is…
Several explicit examples of quasi exactly solvable `discrete' quantum mechanical Hamiltonians are derived by deforming the well-known exactly solvable Hamiltonians of one degree of freedom. These are difference analogues of the well-known…
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…
A correspondence between the sextic anharmonic oscillator and a pair of third-order ordinary differential equations is used to investigate the phenomenon of quasi-exact solvability for eigenvalue problems involving differential operators…
Exactly integrable systems connected to semisimple algebras of second rank with an arbitrary choice of grading are presented in explicit form. General solutions of these systems are expressed in terms of matrix elements of two fundamental…
A new non-Hermitian E2-quasi-exactly solvable model is constructed containing two previously known models of this type as limits in one of its three parameters. We identify the optimal finite approximation to the double scaling limit to the…
We present a general procedure for determining quasi-exact solvability of the Dirac and the Pauli equation with an underlying $sl(2)$ symmetry. This procedure makes full use of the close connection between quasi-exactly solvable systems and…
The O(N) invariant quartic anharmonic oscillator is shown to be exactly solvable if the interaction parameter satisfies special conditions. The problem is directly related to that of a quantum double well anharmonic oscillator in an…
The algebraic structures underlying quasi-exact solvability for spin 1/2 Hamiltonians in one dimension are studied in detail. Necessary and sufficient conditions for a matrix second-order differential operator preserving a space of wave…
In this paper we discuss constraints on two-dimensional quantum-mechanical systems living in domains with boundaries. The constrains result from the requirement of hermicity of corresponding Hamiltonians. We construct new two-dimensional…
We introduce a new concept of infinite quasi-exactly solvable models which are constructable through multi-parameter deformations of known exactly solvable ones. The spectral problem for these models admits exact solutions for infinitely…
A second-order differential (q-difference) eigenvalue equation is constructed whose solutions are generating functions of the dual (q-)Hahn polynomials. The fact is noticed that these generating functions are reduced to the (little…
We review three examples of quasi exactly solvable (QES) Hamitonians which possess multiple algebraisations. This includes the most prominent example, the Lame equation, as well as recently studied many-body Hamiltonians with Weierstrass…
New examples of matrix quasi exactly solvable Schroedinger operators are constructed. One of them constitutes a matrix generalization of the quasi exactly solvable anharmonic oscillator, the corresponding invariant vector space is…
First examples of quasi-exactly solvable models describing spin-orbital interaction are constructed. In contrast with other examples of matrix quasi-exactly solvable models discussed in the literature up to now, our models admit (but still…