A first-order boundary value problem with boundary condition on a countable set of points

Abstract

LetE={E n } be the family of subspaces spanning the eigenfunctions and adjoint functions of the boundary-value problem $$ - i\frac{{dy}}{{dx}} = \lambda y, - \alpha \leqslant x \leqslant \alpha , U(y) \equiv \int_{ - a}^a {y(t)} d\sigma (t) = 0$$ that correspond to “close” eigenvalues (in the sense of the distance defined as the maximal of the Euclidean and the hyperbolic metrics). For a purely discrete measuredσ, it is shown that the systemE does not form an unconditional basis of subspaces inL2(−a, a) if at least one of the end points ±a is mass-free.