## 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

### MSC:

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

### Keywords:

gaps of eigenvalues; periodic potential
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