Discrete vs. Continuous in the Semiclassical Limit: Bottom of the Spectrum for Periodic Potentials
Abstract
Abstract. We compare the bottom of the spectrum of discrete and continuous Schrödinger operators with periodic potentials with barriers at the boundaries of their fundamental domains. Our results show that these energy levels coincide in the semiclassical limit, and we provide an explicit rate of convergence. We demonstrate the optimality of our results by using Bohr–Sommerfeld quantization conditions for potentials exhibiting nondegenerate wells and by numerical experiments for more general potentials. We also investigate the dependence of the spectrum of the discrete semiclassical Schrödinger operator on the semiclassical parameter [Formula: see text] and show that it can be discontinuous.
Fields of Study & Subjects
Spectral Theory in Mathematical Physics
Quantum Mechanics and Non-Hermitian Physics
Quantum chaos and dynamical systems
Publication details
- Journal / source
- SIAM Journal on Mathematical Analysis
- Volume
- 58
- Issue
- 1
- Pages
- 162–181
- Publication date
- 8 January 2026
- Language
- EN