Regular spectral problems of hyperbolic type for a system of first-order ordinary differential equations
There are considered spectral problems generated by the equation \[ y-A(x)y=\lambda B(x)y, \quad x\in [0,1] \] in the space \(L_{2}(C_{n};[0,1])\) of vector functions and the boundary conditions \[ U(y)=U_{0}y(0)+U_{1}y(1)=0. \] Here \(\lambda\) is a spectral parameter, \(U_{0}\) and \(U_{1}\) are complex \(n \times n\) matrices, \(A(x)=\{{{a}_{jk}}(x)\}_{j,k=1}^{n}\) and \(B(x) = \{{{b}_{jk}}(x)\}_{j,k=1}^{n}\) are matrix functions, and \(y(x) = (y_{1}(x),y_{2}(x), \dots,y_{n}(x))^{T}\). It is assumed that the \({{a}_{jk}}(x)\) are complex-valued functions in the space \(L_{1}[0,1]\), the functions \(b_{jk}(x)\) are absolutely continuous, and the matrix \(B(x)\) has \(n\) distinct nonzero real eigenvalues for each \(x \in [0,1]\).
spectral problems for ordinary differential equations
completeness and basis property of eigenfunctions of boundary value problems
scientific article; zbMATH DE number 7455560
scientific article