Minimal spectral functions of an ordinary differential operator

Abstract

AbstractLetl[y]be a formally self-adjoint differential expression of an even order on the interval [0,b〉(b ≤ ∞) and letL0be the corresponding minimal operator. By using the concept of a decomposing boundary triplet, we consider the boundary problem formed by the equationl[y] − λy = f,f ∈ L2[0, b〉, and the Nevanlinna λ-dependent boundary conditions with constant values at the regular endpoint 0. For such a problem we introduce the concept of them-function, which in the case of self-adjoint separated boundary conditions coincides with the classical characteristic (Titchmarsh–Weyl) function. Our method allows one to describe all minimal spectral functions of the boundary problem, i.e. all spectral functions of the minimally possible dimension. We also improve (in the case of intermediate deficiency indicesn±(L0)and non-separated boundary conditions) the known estimate of the spectral multiplicity of the (exit space) self-adjoint extensionà ⊃ L0. Results are obtained for expressionsl[y]with operator-valued coefficients and arbitrary (equal or unequal) deficiency indicesn±(L0).