A moments' analysis of quasi-exactly solvable systems: a new perspective on the sextic potential $gx^6+bx^4+mx^2 +{\beta \over {x^2}}$

Abstract

Abstract. There continues to be great interest in understanding quasi-exactly solvable (QES) systems.

Proof that Half-Harmonic Oscillators become Full-Harmonic Oscillators after the "Wall Slides Away"

“…This will generally produce faster converging bounds for the ground state, since the resulting missing moment order, m s , is reduced. If R(χ) = Ψ gr (χ), and the quantum system is of the type exactly solvable or quasi-exactly solvable, the MER analysis for the corresponding F representation will usually be able to identify the exact eigenergies of the system [16,17]. As noted, this is the case for the b = 0 parameter value for Eq.…”

“…Through the Herglotz extension of the probability density, F (χ) = Ψ(χ)Ψ * (χ), the corresponding F (χ) configuration will satisfy a fourth order linear differential equation, amenable to EMM. This was used to correctly predict the onset of PT symmetry breaking for the quantum system V (x) = ix 3 + iax [15], as well as bound complexvalued Regge poles used in atomic scattering [17,18].…”

Pointwise Reconstruction of Wave Functions from Their Moments through Weighted Polynomial Expansions: An Alternative Global-Local Quantization Procedure

“…The ground state is of QES form for m = −3, with lowest energy state E gr = 0 [20,36]. The HD analysis will recover the QES states; however, if we slightly perturb things, m → m + δm, the resulting HD energies do not suggest the existence of QES states.…”

“…The same applies to the case of quasi-exactly solvable (QES) systems [33][34][35][36][37]. In this case, only a subset of the discrete states can be calculated in closed form.…”

The Hill determinant method revisited

“…For a quartic potential such solutions had earlier been found in [27] to be possible for a set of specific V 4 values in the potential x 2 + V 4 x 4 . Leach [28] cited some early work on this problem of quasi-exact solubility for the sextic potential but a modern detailed treatment of the problem was given by Handy and coworkers using their highly effective new development of the Hill determinant method, the orthogonal polynomial projection quantization method (OPPQ) [29,30]. In the present work we shall use only the particular case of infinite X 0 in our Hill-series method [21] and thus treat only the standard Hill determinant approach which has been subjected to criticism in the literature.…”