New exact solutions for polynomial oscillators in large dimensions

Abstract

A new type of exact solvability is reported. Schrödinger equation is considered in a very large spatial dimension D ≫ 1 and its central polynomial potential is allowed to depend on "many" (= 2q) coupling constants. In a search for its bound states possessing an exact and elementary wave functions ψ (proportional to a harmonicoscillator-like polynomial of a freely varying, i.e., not just small, degree N), the "solvability conditions" are known to form a complicated nonlinear set which requires a purely numerical treatment at a generic choice of D, q and N. Assuming that D is large we discovered and demonstrate that this problem may be completely factorized and acquires an amazingly simple exact solution at all N and up to q = 5 at least.

Quasi-exact minus-quartic oscillators in strong-core regime

“…During our study we felt particularly motivated by the technical difficulties arising in connection with the explicit construction of the quartic QES charges. Although, implicitly, they are defined by the coupled pair of the Magyari's polynomial algebraic equations for two unknowns, their practical determination must usually rely upon the computerized, Gröbner-basis-based algebraic manipulation techniques and numerical root-searching [26]. In addition, it is quite unpleasant that the complexity of the latter algorithm grows fairly quickly with the growth of the degree N of the polynomial wave functions as well as with the growth of the angular momentum ℓ.…”

-symmetric quantum toboggans

“…is only exceptionally tractable non-numerically at q > 2 [28]. We were still able to recollect a few explicit formulae for q = 2 [5].…”

Generalization of quasi-exactly solvable and isospectral potentials

“…These kinds of potentials have drawn attention to the study of P T -symmetry and pseudo-harmonicity of the Hamiltonian [21,22].…”

“…In a recent work, Gönöul et al [20] have shown that two kinds of potentials as mentioned above can be linked together in N -dimensional space. Nonsingular (polynomial) potentials are also very much fascinating nowadays because of their mathematical beauty like P T symmetry [21] and their application in different branches of physics [22]. Therefore, it will be very much interesting to develop a unified approach for constructing QES potentials of singular and non-singular type.…”