Indefinite elliptic spectral problems
The author considers the spectral problem of the form \[ L u = \lambda B u, \qquad x \in G \subset\mathbb{R}^n, \tag{1} \] \[ B_j u |_{\Gamma} = 0, \quad j= \overline{1,m}, \tag{2} \] where \(L\) is a selfadjoint uniformly elliptic operator of order \(2m\); \(\{B_j\}\) is a normal system of boundary operators covering \(L\) on \(\Gamma\); \(Bu\) is an operator \(g(x)u\); and \(g(x)\) is a real-valued measurable function assuming positive and negative values on \(G\). Let the sets \(G^+=\{x\in G:g(x) >0\}\) and \(G^-=\{x\in G:g(x) <0\}\) be open and let \(L_{2,g}(G^+ \cup G^-)\) be the space of measurable functions \(u(x)\) on \(G^+ \cup G^-\) such that \(u |g|^{1/2} \in L_{2}(G^+ \cup G^-)\). The author exhibits conditions which ensure that the eigenfunctions and associated functions to problem (1), (2) are dense in the space \(L_{2,g}(G^+ \cup G^-)\) and constitute an unconditional basis for this space. Similar conditions in terms of interpolation equalities were stated in the earlier article by the author [Sib. Adv. Math. 4, No. 2, 87-121 (1994; Zbl 0851.35100)]. Here, sufficient conditions on the function \(g(x) \) and the domain \(G\) are given ensuring the basis property.
basis property for the eigenelements and associated functions
scientific article; zbMATH DE number 1431677
scientific article