New Findings in Quantum Mechanics (Partial Algebraization of the Spectral Problem)

Abstract

We discuss a new class of spectral problems discovered recently which occupies an intermediate position between the exactly-solvable problems (like the famous harmonic oscillator) and all others. The problems belonging to this class are distinguished by the fact that an (arbitrary) part of the eigenvalues and eigenfunctions can be found algebraically, but not the whole spectrum. The reason explaining the existence of the quasi-exactly-solvable problems is a hidden dynamical symmetry present in the Hamiltonian. For one-dimensional motion, this hidden symmetry is SU(2). The simplest one-dimensional system admitting algebraization for a part of the spectrum is the anharmonic oscillator with the x6 anharmonicity and a relation between the coefficients in front of x2 and x6. We review also more complicated cases with the emphasis on pedagogical aspects. The groups SU (2)× SU (2), SO(3) and SU(3) generate two-dimensional problems with the partial algebraization of the spectrum. Typically we get Schrödinger-type equations in curved space. An intriguing relation between the algebraic structure of the Hamiltonian and the geometry of the space emerges. Another interesting development is the use of the graded algebras which allow one to construct multi-component quasi-exactly-solvable Hamiltonians.

Exact solutions of a new elliptic Calogero–Sutherland model

“…It follows that H N has (at most) (6), (7), with χ k (z) a symmetric polynomial of degree at most Nm, as claimed. The algebra sl(N + 1), which plays a fundamental role in the partial integrability of the Hamiltonian (2), (3), is sometimes called a hidden symmetry algebra [26], since in this approach the Hamiltonian need not be a Casimir element. Note that the Hamiltonian (2), (3) must certainly have eigenfunctions which do not belong to the algebraic sector (6), (7), since its discrete spectrum is infinite.…”

Periodic Quasi-Exactly Solvable Models

“…Quasi-exactly solvable (QES) Hamiltonians, interconnecting a diverse array of physical problems, have been the subject of extensive study in recent times [1,2,3,4,5,6]. These systems, containing a finite number of exactly obtainable eigenstates, have been linked with classical electrostatic problems, as also to the finite dimensional irreducible representations of certain algebras.…”

“…These systems, containing a finite number of exactly obtainable eigenstates, have been linked with classical electrostatic problems, as also to the finite dimensional irreducible representations of certain algebras. Some of these studies employ group theoretical methods, others are based on the symmetry of the relevant differential equations [2,3,4]. The key to the existence of the finite number of identifiable states is the quasi-exact solvability condition, relating certain potential parameters of these dynamical systems.…”

Dynamical algebra of quasi-exactly-solvable potentials

“…Independently, other authors, particularly the Soviet ones, have also been working on such types of problems, whose solutions are closely related to dynamical groups. An excellent review of this approach appeared in a recent work by Shiffman [7].…”

“…It is also clear from Eqs. (8) - (10) that the ansatz (7) and especially the polynomial a(x) determine the desired potential V(x), so we must take care with the "gauged potential" b, V(x). Once we choose a (x) as a polynomial of maximum power n, then the highest power of EV(x) will be 2(n -1), as can be seen from (10 )a, …”