On the completeness of elementary solutions of a class of second-order differential equations with operator coefficients. (English. Russian original) Zbl 1201.34133
Math. Notes 86, No. 5, 747-750 (2009); translation from Mat. Zametki 86, No. 5, 797-800 (2009).
Consider the boundary value problem \(d^2u(t)/dt^2+(pA+A_1)(du/dt)+(qA^2+A_2)u(t)=0\), \(t\in\mathbb R_+\), \(u(0)=\varphi\), where \(u:\mathbb R_+\to H\), \(\varphi\in H\), \(H\) is a given Hilbert space, and the coefficients satisfy:
(1) \(A\) is a positive definite self-adjoint operator with completely continuous inverse;
(2) \(p,q\in\mathbb R\), \(q<0\);
(3) \(B_1=A_1A^{-1}\) and \(B_2=A_2A^{-2}\) are linear bounded operators on \(H\).
Derivatives are regarded in the sense of distributions. The authors indicate conditions on the coefficients of the given problem which ensure its regular solvability, the completeness in the trace space \(H_{3/2}\) of the system of eigenvectors and associated vectors corresponding to the eigenvalues from the left half-plane, and the completeness of the decreasing elementary solutions in the space of regular solutions of the given problem. No proofs are given.
(1) \(A\) is a positive definite self-adjoint operator with completely continuous inverse;
(2) \(p,q\in\mathbb R\), \(q<0\);
(3) \(B_1=A_1A^{-1}\) and \(B_2=A_2A^{-2}\) are linear bounded operators on \(H\).
Derivatives are regarded in the sense of distributions. The authors indicate conditions on the coefficients of the given problem which ensure its regular solvability, the completeness in the trace space \(H_{3/2}\) of the system of eigenvectors and associated vectors corresponding to the eigenvalues from the left half-plane, and the completeness of the decreasing elementary solutions in the space of regular solutions of the given problem. No proofs are given.
Reviewer: Zoran Kadelburg (Beograd)
MSC:
| 34L10 | Eigenfunctions, eigenfunction expansions, completeness of eigenfunctions of ordinary differential operators |
| 34G10 | Linear differential equations in abstract spaces |
| 47E05 | General theory of ordinary differential operators |
Keywords:
second-order differential equation; self-adjoint operator; operator pencil; entire functionReferences:
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