A discreteness criterion for the spectrum of a quasielliptic operator

Abstract

For the spectrum of the operator

$$u = \sum\nolimits_{j = 1}^n {( - 1)^{m_j } D_j^{2m_j } u + q(x)u,} $$

to be discrete, where the m_j are arbitrary positive integers such that$\sum\nolimits_{j = 1}^n {\tfrac{1}{{2m_j }}< 1}$, and q(x) ≥ 1, it is necessary and sufficient that$\int\limits_K {q (x) dx \to \infty }$, when the cube K tends to infinity while preserving its dimensions.