Quasi-exactly solvable quartic: elementary integrals and asymptotics

Abstract

We study elementary eigenfunctions y=p exp(h) of operators L(y)=y"+Py, where p, h and P are polynomials in one variable. For the case when h is an odd cubic polynomial, we found an interesting identity which is used to describe the spectral locus. We also establish some asymptotic properties of the QES spectral locus.

20 pages, 1 figure. Added Introduction and several references, corrected misprints

References in corpus (7)

• The ODE/IM Correspondence
• Eigenvalues of PT-symmetric oscillators with polynomial potentials
• On the rational monodromy-free potentials with sextic growth
• Remarks on PT-pseudo-norm in PT-symmetric quantum mechanics
• On eigenvalues of the Schrödinger operator with a complex-valued polynomial potential
• Singular perturbation of polynomial potentials in the complex domain with applications to PT-symmetric families
• Irreducibility of some spectral determinants

Cited by in corpus (5)

• On Conjectures of A. Eremenko and A. Gabrielov
• Quasi-exactly solvable quartic: real algebraic spectral locus
• Trajectories of a quadratic differential related to a quasi-exactly solvable sextic oscillator
• On spectral asymptotic of quasi-exactly solvable quartic
• Two-parametric PT-symmetric quartic family