A remark on two dimensional periodic potentials. (English) Zbl 0539.35059

This paper is concerned with estimates on the spectrum for the two dimensional operator \(-\Delta +q\) when q is a periodic potential. Let A be the lattice generated by the two linearly independent vectors \(a_ 1,a_ 2\) in \(R^ 2\), and suppose that q is a bounded, real valued, periodic function on A. Let the bands \(B_ n(q)\) be defined as the images in \(R^ 2\) of the function \(\lambda_ n(\cdot,q)\), where \(\lambda_ n(k,q)\), \(k\in R^ 2\) is the n-th eigenvalue for the boundary value problem \((-\Delta +q)u=\lambda u, u(x+a_ j)=u(x) e^{ika_ j}, j=1,2\). The main result is that there exist constants c(q) and C(q) such that \[ [(| B| /\pi)n-cn^{1/4},(| B| /\pi)n+cn^{1/4}]\subset B_ n(q)\subset [(| B| /\pi)n- Cn^{1/3},(| B| /\pi)n+Cn^{1/3}] \] for n sufficiently large and \(| B|\) in the area of a fundamental cell of the lattice dual to A. An important corollary to this theorem is that the spectrum of \(- \Delta +q\) contains a ray \([\lambda^*,\infty)\) and in particular there is at most a finite number of gaps. This is in contrast with the one dimensional case.
Reviewer: W.Rundell


35P15 Estimates of eigenvalues in context of PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
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