On Explicit Estimations for the Bloch Eigenvalues of the One-dimensional Schrödinger Operator and the Kronig-Penney Model
Cemile Nur, Oktay Veliev
TL;DR
The paper addresses spectral characterization of the one-dimensional Schrödinger operator $L(q)$ with a $1$-periodic real potential, providing explicit large-eigenvalue asymptotics and reliable small-eigenvalue estimates with Kronig-Penney as a benchmark. Using perturbative expansions with Fourier-type coefficients and auxiliary quantities $a_k(\lambda)$, $b_k(\lambda)$, and $D(k)$, it derives the large-eigenvalue formula $\lambda_{n,j}=(2\pi n)^2+D(2n)+(-1)^j|q_{2n}-S_{2n}+2Q_0Q_{2n}|+o(n^{-2})$ and related higher-order terms. For small eigenvalues, the work establishes iteration-based equations with convergence criteria (e.g., $M\le \frac{4\pi^{2}(2n-1)}{3}$) and provides fixed-point computations with explicit error bounds, plus a Kronig-Penney-specific treatment. Numerical illustrations include a detailed Kronig-Penney example and first-gap estimates, highlighting practical applicability to band-structure analysis in solid-state systems.
Abstract
In this paper, we consider the small and large eigenvalues of the one-dimensional Schrödinger operator L(q) with a periodic, real and locally integrable potential q. First we explicitly write out the first and second terms of the asymptotic formulas for the large periodic and antiperiodic eigenvalues and illustrate these formulas for the Kronig-Penney model. Then we give estimates for the small periodic and antiperiodic eigenvalues and for the length of the first gaps in the case of the Kronig-Penney model. Moreover, we give error estimations and present a numerical example.