Exact resolution method for general 1D polynomial Schrödinger equation

Abstract

The stationary 1D Schr\"odinger 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 $V(q)$.)