Exact resolution method for general 1D polynomial Schrödinger equation

Abstract

The stationary 1D Schrödinger equation with a polynomial potential $V(q)$ of degree N is reduced to a system of exact quantization conditions of Bohr-Sommerfeld form. They arise from bilinear (Wronskian) functional relations pairing spectral determinants of (N+2) generically distinct operators, all the transforms of one quantum Hamiltonian under a cyclic group of complex scalings. The determinants' zeros define (N+2) semi-infinite chains of points in the complex spectral plane, and they encode the original quantum problem. Each chain can now be described by an exact quantization condition which constrains it in terms of its neighbors, resulting in closed equilibrium conditions for the global chain system; these are supplemented by the standard (Bohr-Sommerfeld) quantization conditions, which bind the infinite tail of each chain asymptotically. This reduced problem is then probed numerically for effective solvability upon test cases (mostly, symmetric quartic oscillators): we find that the iterative enforcement of all the quantization conditions generates discrete chain dynamics which appear to converge geometrically towards the correct eigenvalues/eigenfunctions. We conjecture that the exact quantization then acts by specifying reduced chain dynamics which can be stable (contractive) and thus determine the exact quantum data as their fixed point. (To date, this statement is verified only empirically and in a vicinity of purely quartic or sextic potentials $V(q)$.)

flatex text.tex, 4 files Submitted to: J. Phys. A: Math. Gen

Cited by in corpus (9)

• MHD alpha^2-dynamo, Squire equation and PT-symmetric interpolation between square well and harmonic oscillator
• Pseudo-differential equations, and the Bethe Ansatz for the classical Lie algebras
• Robustness of adiabatic passage trough a quantum phase transition
• Functional relations in Stokes multipliers - Fun with $x^6+\alpha x^2$ potential-
• A Solvable Deformation of Quantum Mechanics
• Stokes Multipliers, Spectral Determinants and T-Q relations
• ODE/IQFT correspondence for the generalized affine $\mathfrak{sl}(2)$ Gaudin model
• Half-Line non-self-adjoint Schrödinger operators with polynomial potentials: Asymptotics of eigenvalues
• Heat Kernel Asymptotic Expansion on Unbounded Domains with Polynomially Confining Potentials