Absence of Quantum States Corresponding to Unstable Classical Channels

TL;DR

This paper proves that quantum states corresponding to certain unstable classical trajectories do not exist, specifically for Hamiltonian systems with orbits diverging to infinity and certain Schrödinger operators.

Contribution

It establishes a new theorem linking the absence of quantum states to classical instability in specific Hamiltonian systems.

Findings

Quantum states do not exist for systems with unstable orbits

The theorem applies to Schrödinger operators with Morse potentials on the sphere

Provides insight into quantum-classical correspondence in unstable regimes

Abstract

We consider Hamiltonian systems of a certain class with unstable orbits moving to infinity. We prove a theorem showing that analogous quantum states do not exist. This theorem is applied to Schrodinger operators with potentials of degree zero which are Morse when restricted to the unit sphere.