Spectral properties of the generalized Friedrichs model
The author studies the operator \(A\) acting in the Hilbert space \[ \mathbb{C}\oplus L_2(M,d\mu) \] given by \[ (AF)_0=e_0f_0+\alpha\int_M\bar{v}(q)f_1(q)\,d\mu(q), \] \[ (AF)_1(q)=\alpha v(q)f_0+a(q)f_1(q)+\alpha\int_M D(q,q')f_1(q') \,d\mu(q'), \] where \(F=(f_0,f_1)\). This operator is called a generalized Friedrichs operator. The main results of this paper extend to this class of operators results previously known for the class of Friedrichs operators. The results are: {\parindent=5mm \begin{itemize}\item[1.] For sufficiently small values of \(\alpha\), the spectrum of \(A\) consists of an absolutely continuous part equal to the range of the function \(a\), and a possible single eigenvalue, smaller than the minimum of \(a\). \item[2.] The existence and completeness of wave operators related to \(A\). \end{itemize}} Suitable conditions on \(a\), \(v\), \(D\) and the constant \(\alpha>0\) are imposed to this effect. The author mentions that similar results (with similar conditions) where stated in [\textit{N.\,Angelescu, R.\,A.\thinspace Minlos} and \textit{V.\,A.\thinspace Zagrebnov}, Rev.\ Math.\ Phys.\ 17, No.\,10, 1111--1142 (2005; Zbl 1086.81040)], though complete proofs are given here.
scientific article; zbMATH DE number 5790865
scientific article