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A completeness theorem for a dissipative Schrödinger problem with the spectral parameter in the boundary condition. (English) Zbl 1145.34054
The authors consider a $\lambda$-linear boundary value problem for a singular second order differential expression $\ell$ on the half-line $(0,\infty)$, where it is assumed that both endpoints $0$ and $\infty$ are in the limit circle case. A dissipative boundary condition is imposed at $0$ and the spectral parameter appears linearly in the boundary condition at $\infty$. The $\lambda$-dependent boundary condition at $\infty$ is transformed into a constant boundary condition for a suitable operator $A$ in $L^2((0,\infty))\oplus\mathbb C$ which is a maximal dissipative extension of the minimal operator associated to $\ell$ in $L^2((0,\infty))$. By means of the minimal selfadjoint dilation of the maximal dissipative operator $A$ it is shown that the eigenfunctions of the original boundary value problem are complete in $L^2((0,\infty))$. Furthermore, the characteristic function of the maximal dissipative operator $A$ is identified with the Lax-Phillips scattering matrix of the selfadjoint dilation.

34L10Eigenfunctions, eigenfunction expansions, completeness of eigenfunctions (ODE)
34B05Linear boundary value problems for ODE
34B40Boundary value problems for ODE on infinite intervals
47A20Dilations, extensions and compressions of linear operators
47A40Scattering theory of linear operators
47A45Canonical models for contractions and nonselfadjoint operators
47B44Accretive operators, dissipative operators, etc. (linear)
34L40Particular ordinary differential operators
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