On basis property of eigenfunctions and associated functions for nonlocal boundary value problems
The author considers the spectral problem \[ -D_x^2 u(x,y) + (-1)^m D_y^{2m} u(x,y) + \sum\limits_{j=0}^{m-1} a_j(x,y) D_xD_y^j u(x,y) + \] \[ +\sum\limits_{j=0}^{2m-1} b_j(x,y) D_y^j u(x,y) + q(y)u(x,y) = \lambda u(x,y) ,\;(x,y) \in Q=(0,1)\times \mathbb{R}, \eqno{(1)} \] \[ L_{\nu}u = \alpha_{\nu}u^{(m_{\nu})}_{x=0} + \beta_{\nu} u^{(m_{\nu})}_{x=1} + \alpha_{\nu 0} u{}_{x=0} + \beta_{\nu 0} u{}_{x=1} =0,\;\;\nu =1,2, \eqno{(2)} \] where \(\alpha_{\nu 0} = \beta_{\nu 0} =0 \) for \(m_{\nu} =0\). The problem is reduced to the following problem in the space \(L_2(0,1;H)\) (\(H\) is a Hilbert space; to problem (1), (2) there corresponds \(H=L_2(\mathbb{R})\)): \[ -u''(x) + A u(x) + A_1(x) u'(x) + A_2(x) u(x) = \lambda u(x),\;\;L_{\nu}u=0. \eqno{(3)} \] Here \(A\) is a positive operator whose resolvent set contains a set of the form \(\{\lambda:\;{}\lambda{} \geq r,\;{}{\operatorname{arg}}\lambda{} \geq \delta \}\) and such that \(A^{-1} \in \sigma_p\) for some \(p\). The operators \(A_1\) and \(A_2\) are assumed to be subordinate to the operator \(-\frac{d^2}{dx^2} + A\). Under natural conditions on the boundary operators \(L_{\nu}\) it is shown that the spectrum of problem (3) consists of isolated eigenvalues of finite algebraic multiplicity, and the eigenfunctions and the associated functions form a basis in \(L_2(0,1;H)\) for the Abel summation method of order \(\gamma \in ((2p+1)/2,\pi/2\delta)\).
scientific article; zbMATH DE number 1037973