A nonself-adjoint 1D singular Hamiltonian system with an eigenparameter in the boundary condition
The author considers the one-dimensional Hamiltonian system \[ \ell_1(y)=J\frac{dy}{dx}+By=\lambda Ay \text{ on } I=[a,b), \] where \(\lambda \in \mathbb C\), \[ J=\begin{pmatrix} 0&-1\\1&0 \end{pmatrix}, \;A=\begin{pmatrix} a&c\\c&b \end{pmatrix}, \;B=\begin{pmatrix} p&q\\q&r \end{pmatrix}, \] all entries of \(A\) and \(B\) are real-valued locally integrable functions on \(I\), and \(A(x)>0\) for almost all \(x\in I\). \(L_{\max}\) denotes the corresponding maximal operator on the weighted Lebesgue space \(L_A^2(I,\mathbb C)\). It is assumed that the system is in the limit circle case, i.\,e., the minimal operator has deficiency indices \((2,2)\). Let \(\beta ,\gamma \in {\mathbb R}^2\) such that \(\alpha =\gamma ^*J\beta >0\) and define \(R_0(y)=\beta ^*y(a)\), \(R_0'(y)=\gamma ^*y(a)\) for \(y\in D(L_{\max})\). Let \([y,z]_X=z^*(x)Jy^*(x)\), and let \(u\) and \(v\) be the solutions of \(\ell_1(y)=0\) satisfying \(u^*(a)=(1,0)\), \(v^*(a)=(0,1)\). Define \(B_1^+(y)=[y,u]_b\), \(B_2^+(y)=[y,v]_b\), \(R_+(y)=B_1^+(y)-hB_2^+(y)\), where \(h\in \mathbb C\) with Im\,\(h>0\). The boundary value problem for \(\ell_1\) with boundary conditions \(R_0(y)=\lambda R_1(y)\), \(R_+(y)=0\) has the operator realization \(A_h\). Let \(H=L_A^2(I,\mathbb C^2)\oplus \mathbb C\), where the inner product on \(\mathbb C\) carries the weight \(\alpha ^{-1}\). The operator \(A_h\) on \(H\) has domain consisting of all \(\hat y=(y_1,y_2)^T\) with \(y_1\in D(L_{\max})\), \(R_+(y_1)=0\), \(y_2=R_0'(y_1)\) and acts as \[ A_h\hat y=\begin{pmatrix} A^{-1}\ell_1(y)& R_0(y_1)\end{pmatrix}. \] It is shown that \(A_h\) has the following properties: \(A_h\) is maximal dissipative; \(A_h\) has a self-adjoint dilation, which is explicitly constructed; \(A_h\) is completely nonself-adjoint; for all \(h\) with \(\operatorname{Im}h>0\), with the possible exception of one number, the spectrum of \(A_h\) is purely discrete and belongs to the open upper half-plane, each eigenvalue has finite multiplicity, and the system of eigenvectors and associated vectors of \(A_h\) is complete in \(H\).
eigenvalue parameter in the boundary condition
completeness of systems of eigenvectors and associated vectors
scientific article; zbMATH DE number 6162453
scientific article