Spectral asymptotics for Hill’s equation near the potential maximum.(English)Zbl 0786.34080

Let $$P(h)$$ be the one-dimensional periodic Schrödinger operator $P(h)= -{h^ 2\over 2}{d^ 2\over dx^ 2}+ V(x).$ The spectrum of $$P$$ is composed by a union of closed intervals, called “bands”. It plays an important role in many applications. The interval between two consecutive “bands” is called a “gap”. This article gives very elaborate asymptotic formulae for the sizes of the bands and gaps of the spectrum of $$P(h)$$ as $$h$$ goes to zero. This is done within the framework of turning point theory under the assumption that $$V: \mathbb{R}\to\mathbb{R}$$ is real analytic, $$2\pi$$-periodic, $$V(x)\leq 0$$ (with equality exactly at the points $$2\pi k$$, $$k\in \mathbb{Z}$$), and $$V''(0)<0$$. The paper makes a very interesting use of methods from microlocal analysis that have been developed recently by Helffer and Sjöstrand.

MSC:

 34L20 Asymptotic distribution of eigenvalues, asymptotic theory of eigenfunctions for ordinary differential operators 34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.) 34E20 Singular perturbations, turning point theory, WKB methods for ordinary differential equations 47E05 General theory of ordinary differential operators 34B30 Special ordinary differential equations (Mathieu, Hill, Bessel, etc.)