On subordinacy and spectral multiplicity for a class of singular differential operators

Abstract

The spectral multiplicity of self-adjoint operators H associated with singular differential expressions of the formis investigated. Based on earlier work of I. S. Kac and recent results on subordinacy, complete sets of necessary and sufficient conditions for the spectral multiplicity to be one or two are established in terms of: (i) the boundary behaviour of Titchmarsh–Weyl m-functions, and (ii) the asymptotic properties of solutions of Lu = λu, λ∈ℝ, at the endpoints a and b. In particular, it is shown that H has multiplicity two if and only if L is in the limit point case at both a and b and the set of all λ for which no solution of Lu = λu is subordinate at either a or b has positive Lebesgue measure. The results are completely general, subject only to minimal restrictions on the coefficients p(r), q(r)and w(r), and the assumption of separated boundary conditions when L is in the limit circle case at both endpoints.

Equality of the Spectral and Dynamical Definitions of Reflection

“…Let (2) ac be the set of λ ∈ R, where J has multiplicity 2, so automatically a.c. spectrum (see [21,23,24,39]). P ± ,r commute with J , so they take Ran(P (2) ac (J )) to itself.…”

Spectral Multiplicity of Selfadjoint Schrödinger Operators on Star-Graphs with Standard Interface Conditions

“…The Weyl function ofΓ is equal to M 0 , and Now we come to the promised way to compute the spectral multiplicity function. For the case "n = 2" this fact is proved and used in [Kac62], see also [Gil98]. It is of course not hard to believe that it holds for arbitrary n ≥ 2, however, we are not aware of an explicit reference, and therefore provide a complete proof.…”

Eigenfunction Expansions Associated with the One-dimensional Schrödinger Operator

“…It also implies that the degenerate spectrum is supported on S := {λ ∈ R : no solution of Lu = λu is subordinate at either − ∞ or at ∞}, where µ τ (S d S ) = 0 (see [8] for further details).…”

“…It turns out that in the process of reformulating the part of the expansion where the spectral multiplicity is one, the contribution of the half-line operators H −∞ and H ∞ is reflected through their respective Titchmarsh-Weyl m-functions and corresponding spectral densities. This information enables the asymptotic behaviour of the eigenfunctions at ±∞ to be determined in terms of the theory of subordinacy [8], [9], and details of this process will be demonstrated through worked examples.…”