Spectrum of multidimensional operators with periodic coefficients
Consider in \(L_ 2(R^ v)\) a selfadjoint operator H of the form \(H^ 0+Q\). Here \(H^ 0=J(i\partial /\partial x)\) is a differential or pseudo- differential operator with constant coefficients, J is a real-valued function, smooth except possibly at the point 0, satisfying \(J(tx)=| t|^ rJ(x),\) (t\(\in R\), \(r>0)\), \(J(x)>0\) for \(x\neq 0\) and an additional convexity condition. The operator Q is assumed to be bounded, selfadjoint and commuting with shifts by certain vectors. Assuming further the continuity of a certain mapping, the spectrum of H is continuous, and consists of a union of real closed intervals with lacunae between them. The paper presents conditions involving v and r that ensure that the number of the lacunae in the spectrum is finite.
pseudo-differential operator with constant coefficients
scientific article; zbMATH DE number 3903370