zbMATH — the first resource for mathematics

Discrete spectrum of a two-dimensional periodic elliptic second order operator perturbed by a decaying potential. I: Semibounded gap. (English. Russian original) Zbl 1070.47041
St. Petersbg. Math. J. 12, No. 4, 535-567 (2001); translation from Algebra Anal. 12, No. 4, 36-78 (2001).
The authors consider a periodic elliptic differential operator \(A\) of second order in two dimensions perturbed by a decaying potential \(V(x)\). Assuming that \(\inf\text{spec}\,A=0\), they investigate the number of negative eigenvalues of \(A-\alpha V\) as \(\alpha \to \infty \). It turns out that the asymptotics of the number of negative eigenvalues contains two competing components. The first component is a Weyl (quasiclassical) term, while the second one is defined by a “threshold” effect on the lower bound of the spectrum of \(A\). It is described in terms of effective masses on the lower bound of the spectrum. The authors prove that for some \(A\) and \(V\) (e.g., sufficiently fast decaying potentials) the Weyl component dominates the threshold one, while in other situations (e.g., slowly decaying potentials) the threshold term dominates the Weyl term. The formulations of results in the two-dimensional situation are essentially different from those for higher dimensions.
[Part 2 appeared ibid. 15, No. 2, 249–287 (2004); translation from Algebra Anal. 15, No. 2, 128–189 (2003; Zbl 1070.47042), reviewed below.]

47F05 General theory of partial differential operators
35P15 Estimates of eigenvalues in context of PDEs
37A10 Dynamical systems involving one-parameter continuous families of measure-preserving transformations
PDF BibTeX Cite