Regularized sum for eigenfunctions of multi-point problem in the commensurable case (Q1392703)

Summary: Consider the eigenvalue problem which is given in the interval \([0,\pi]\) by the differential equation \[ -y''(x)+ q(x)y(x)= \lambda y(x);\quad 0\leq x\leq\pi\tag{1} \] and multi-point conditions \[ \begin{aligned} U_1(y) & = \alpha_1y(0)+ \alpha_2 y(\pi)+ \sum^n_{k=3} \alpha_ky(x_k\pi)= 0,\\ U_2(y) & = \beta_1y(0)+ \beta_2y(\pi)+ \sum^n_{k= 3} \beta_ky(x_k\pi)= 0,\end{aligned}\tag{2} \] where \(q(x)\) is sufficiently smooth function defined in the interval \([0,\pi]\). We assume that the points \(x_3,x_4,\dots, x_n\) divide the interval \([0,1]\) to commensurable parts and \(\alpha_1\beta_2- \alpha_2\beta_1\neq 0\). Let \(\lambda_{k,s}= \rho^2_{k,s}\) be the eigenvalues of the problem (1)--(2) for which we shall assume that they are simple, where \(k\), \(s\), are positive integers and suppose that \(H_{k,s}(x,\xi)\) are the residue of Green's function \(G(x,\xi,\rho)\) for the problem (1)--(2) at the points \(\rho_{k,s}\). The aim of this work is to calculate the regularized sum which is given by the form: \[ \sum_{(k)} \sum_{(s)} [\rho^{-\sigma}_{k,s} H_{k,s}(x,\xi)- R_{k,s}(\sigma, x,\xi,\rho)]= S_\sigma(x,\xi).\tag{3} \] The above summation can be represented by the coefficients of the asymptotic expansion of the function \(G(x,\xi,\rho)\) in negative powers of \(k\). In series (3) \(\sigma\) is an integer, while \(R_{k,s}(\sigma,x,\xi,\rho)\) is a function of variables \(x\), \(\xi\) and defined in the square \([0,\pi]\times [0,\pi]\) which ensure the convergence of the series (3).