Reduction of the Bose quadratic operator to the normal form (Q1385862)

The paper concerns the Bose quadratic operator \[ H_2(\psi^+,\psi)=\frac{1}{2}\iint_{M^2}\{B(x,y)\psi^+(x)\psi^+(y) +2A(x,y)\psi^+(x)\psi(y)+B^*(x,y)\psi(x)\psi(y)\}dxdy, \] where \(B(x,y)=B(y,x); A^*(y,x)=A(x,y).\) The operator \(H_2\) which has previously been used in \textit{V. V. Kucherenko} and \textit{V. P. Maslov} [Dokl. Akad. Nauk 348, No. 2, 169-172 (1996)] and \textit{V. P. Maslov} and \textit{O. Yu. Shvedov} [Teor. Mat. Fiz. 98, No. 2, 266-288 (1994; Zbl 0832.58036)] is considered. It is proved that a proper Bogolyubov's \textbf{UV}-transformation reduces \(H_2\) to the direct sum of special kind. Each typical term \(H^j(a_j^+,a_j)\) in the sum can be reduced to one of the canonical forms which correspond to the canonical forms of quadratic Hamiltonians in classical mechanics. A new theorem concerning the spectrum of the operator \(H^j(a_j^+,a_j)\) has been announced.