On Oscillation Theorem for Two-Component Schrodinger Equation

TL;DR

This paper investigates the oscillation theorem for two-component Schrödinger equations, revealing that while the conventional theorem fails in general, specific valid statements hold for certain coupled two-component systems.

Contribution

It establishes conditions under which the oscillation theorem applies to two-component Schrödinger equations, clarifying when classical node counting rules remain valid.

Findings

Ground state ($v=0$) is non-degenerate.

The average number of nodes in two components does not exceed the eigenstate index.

Conventional oscillation theorem is violated in general for multi-component systems.

Abstract

Conventional one-dimensional oscillation theorem is found to be violated for multi-component Schr\"{o}dinger equations in a general case while for two-component eigenstates coupled by the sign-constant potential operator the following statements are valid: (1) the ground state ($v = 0$) is not degenerate; and (2) the arithmetic mean of nodes $n_1$, $n_2$ for the two-component wavefunction never exceeds the ordering number $v$ of eigenstate: $(n_1 + n_2)/2\leq v$.