Convergence of Spectral Decompositions for a Singular Differential Operator with General Boundary Conditions

Abstract

We investigate the general boundary-value problem for the operator lu = −u′′ + q(x)u , 0 < x < 1, If the complex-valued coefficients q(x) is summable on (0,1), the integral $$ {\int}_0^1x\left(1-x\right)\left|q(x)\right| dx $$ converges. The asymptotic solutions of the equation lu = μ2u derived in this article are used to construct the asymptotic spectrum of the problem, to classify the boundary conditions, and to prove theorems asserting that the system of root functions is complete and forms an unconditional basis in L2 (0,1).