Discrete spectrum of a dynamic matrix (Q1086800)

The author investigates the spectrum of the operator \(H=H_ 0+V\) on \(\oplus^{\nu}_{1}L^ 2(T)\), where \(T=[0,2\pi]^{\mu}\), \(\nu\geq 1\), \(\mu\geq 3\), \[ (H_ 0f)(q)=K^{1/2}(q)M^{-1}K^{1/2}(q)f(q) \] \[ (Vf)(q)=(2\pi)^{-\mu}K^{1/2}(q)\sum_{n}\Lambda_ ne^{inq}\int_{T}K^{1/2}(q')e^{-inq'}f(q')dq'. \] Here \(q=(q^{(1)},q^{(2)},...,q^{(\mu)})\), \(nq=\sum n^{(\alpha)}q^{(\alpha)}\), \(K(q)\in C^{\infty}(T)\) is a non- negative \(\nu\times \nu\)-matrix, \(M_ n,M,\Lambda_ n=M_ n^{-1}- M^{-1}\) are \(\nu\times \nu\)-matrices satisfying certain conditions.