Uniform convergence of biorthogonal series for the Schrödinger operator with multipoint boundary conditions
The author considers the problem of convergence of the biorthogonal expansion in root functions for the one-dimensional Schrödinger operator in \(L^2[0,1]\) with complex-valued potential and multipoint boundary conditions. The multipoint boundary forms may be supported on an infinite set of points. A set of sufficient conditions for the absolute and uniform convergence of the biorthogonal expansion is given. A part of these conditions is explicit ( the sequence of the coefficients of the boundary form belongs to \(l^1\)), another part of them is implicit and is expressed as constraints on the root functions and on the spectral parameter. It is also assumed that the operator defined by the multipoint boundary conditions is densely defined. The paper relies on a previous paper of the author [Differ. Equations 27, No. 1, 64--75 (1991; Zbl 0745.34072)] and on \textit{V. A. Il'in}'s book, [Spectral theory of differential operators. Selfadjoint operators (Russian) Moskva: Nauka (1991; Zbl 0759.47026)].
biorthogonal expansions in root functions
scientific article; zbMATH DE number 1916122