Quasi-exactly solvable problems and random matrix theory

Abstract

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 topological (1/N ) expansions in random matrix models to the problem of constructing semiclassical expansions for observables in quasi-exactly solvable problems. Lie algebraic aspects of this relationship are also discussed.

Exact analytical solutions of the Schrödinger equation for a two dimensional purely sextic double-well potential

“…[1][2][3] There are some one-dimensional model potentials for which the DWAO has analytical exact solutions: sextic potential combining with quartic terms, [4][5][6][7][8][9][10][11][12][13] purely sextic double-well potential, 14 or even higher order (up to ten) of polynomial potential. 15 Interestingly, for the studies, we have known the lowest order of polynomial potentials allowed to have exact analytical solutions is six.…”

On Time-Dependent Quasi-Exactly Solvable Problems

“…In turns out that equations of the type (1) are typical not only for QES models but appear in many branches of mathematical physics and this fact enables one to establish a deep relationship between QES models and many other seemingly unrelated models. For example, QES models turn out to be equivalent to completely integrable Gaudin spin chains [1,10] (for which system (1) plays the role of the Bethe ansatz equations for the coordinates of elementary spin excitations), to random matrix models [4,5] (for which system (1) determines the distribution of eigenvalues of large random matrices), and also to purely classical models of 2dimensional elctrostatics [1,2,6] respectively hydrostatics of point vortices [3]. For both the latter models system (1) determines the equilibrium positions of pointwise classical objects (the charged Coulomb particles or resp.…”

On time-dependent quasi-exactly solvable problems

“…In turns out that equations of the type (1) are typical not only for QES models but appear in many branches of mathematical physics and this fact enables one to establish a deep relationship between QES models and many other seemingly unrelated models. For example, QES models turn out to be equivalent to completely integrable Gaudin spin chains [1,10] (for which system (1) plays the role of the Bethe ansatz equations for the coordinates of elementary spin excitations), to random matrix models [4,5] (for which system (1) determines the distribution of eigenvalues of large random matrices), and also to purely classical models of 2-dimensional elctrostatics [1,2,6] respectively hydrostatics of point vortices [3]. For both the latter models system (1) determines the equilibrium positions of pointwise classical objects (the charged Coulomb particles or resp.…”