On the Bohr formula for the one-dimensional Schrödinger operator with increasing potential

Abstract

This article contains the justification of the quasi-classical asymptotic formula (usually attributed to N. Bohr) for the counting function N(λ) of the 1-D Schrödinger operator with potential V increasing at infinity. Results were known under strong regularity conditions on V. Using direct methods based on the phase formalism, we give much wider sufficient conditions on V for the applicability of the Bohr formula. Several counterexamples show that these conditions cannot be significantly improved.

Keywords:

• Schrödinger operator
• Bohr formula
• Discrete spectrum
• Sturm oscillation theorem
• AMS: 34L20
• 58C40