Explicit energy expansion for general odd-degree polynomial potentials

  title={Explicit energy expansion for general odd-degree polynomial potentials},
  author={Asiri Nanayakkara and Thilagarajah Mathanaranjan},
  journal={Physica Scripta},
In this paper we derive an almost explicit analytic formula for asymptotic eigenenergy expansion of arbitrary odd-degree polynomial potentials of the form V (x) = (ix)2N+1 + β1x2N + β2x2N−1 + ··· + β2Nx, where β′k are real or complex for 1 ⩽ k ⩽ 2N. The formula can be used to find semiclassical analytic expressions for eigenenergies up to any order, very efficiently. Each term of the expansion is given explicitly as a multinomial of the parameters β1,β2… and β2N of the potential. Unlike in the… 
1 Citations



Asymptotic behavior of eigen energies of non-Hermitian cubicpolynomial systems

The asymptotic behavior of the eigenvalues of a non-Hermitian cubic polynomial system H = (P2/2) + µx3 + ax2 + bx, where µ, a, and b are constant parameters that can be either real or complex, is

On Eigenvalues of the Schrödinger Operator with an Even Complex-Valued Polynomial Potential Per Alexandersson

In this paper, we generalize several results in the article “Analytic continuation of eigenvalues of a quartic oscillator” of A. Eremenko and A. Gabrielov [4].We consider a family of eigenvalue

Exact resolution method for general 1D polynomial Schrödinger equation

The stationary 1D Schrodinger equation with a polynomial potential V(q) of degree N is reduced to a system of (complex) exact quantization conditions of Bohr-Sommerfeld form. They arise from bilinear

Schrödinger type eigenvalue problems with polynomial potentials: Asymptotics of eigenvalues

For integers m 3 and 1 ' m 1, we study the eigenvalue problem u 00 (z) + (( 1) ' (iz) m P (iz))u(z) = u (z) with the boundary conditions that u(z) decays to zero as z tends to innit y along the rays

On the Semiclassical Expansion for 1-Dim x^N Potentials

We obtain a simple formula for the semiclassical series for potentials V (x) = x N (N even) and derive almost explicit formulae for the WKB approximation of the energy eigenvalues of such potentials.

Exact semiclassical expansions for one-dimensional quantum oscillators

A set of rules is given for dealing with WKB expansions in the one-dimensional analytic case, whereby such expansions are not considered as approximations but as exact encodings of wave functions,

The high energy semiclassical asymptotics of loci of roots of fundamental solutions for polynomial potentials

In the case of polynomial potentials all solutions to 1D Schrödinger equations are entire functions totally determined by loci of their roots and their behaviour at infinity. In this paper a

Some properties of WKB series

We investigate some properties of the WKB series for arbitrary analytic potentials and then specifically for potentials xN (N even), where more explicit formulae for the WKB terms are derived. Our

Eigenvalues of PT-symmetric oscillators with polynomial potentials

We study the eigenvalue problem −u''(z) − [(iz)m + Pm−1(iz)]u(z) = λu(z) with the boundary condition that u(z) decays to zero as z tends to infinity along the rays in the complex plane, where Pm−1(z)