Weyl-Titchmarsh Theory for Schrodinger Operators with Strongly Singular Potentials

Abstract

We develop Weyl-Titchmarsh theory for Schrödinger operators with strongly singular potentials such as perturbed spherical Schrödinger operators (also known as Bessel operators). It is known that in such situations one can still define a corresponding singular Weyl m-function and it was recently shown that there is also an associated spectral transformation. Here we will give a general criterion when the singular Weyl function can be analytically extended to the upper half plane. We will derive an integral representation for this singular Weyl function and give a criterion when it is a generalized Nevanlinna function. Moreover, we will show how essential supports for the Lebesgue decomposition of the spectral measure can be obtained from the boundary behavior of the singular Weyl function. Finally, we will prove a local Borg-Marchenko type uniqueness result. Our criteria will in particular cover the aforementioned case of perturbed spherical Schrödinger operators.2000 Mathematics Subject Classification. Primary 34B20, 34L05; Secondary 34B24, 47A10.

Quadratic operator pencils associated with the conservative Camassa-Holm flow

“…With the Borel measure µ defined by 27) for λ 1 , λ 2 ∈ R with λ 1 < λ 2 , the claim follows along the lines of [40,Lemma 3.3].…”

“…We are now going to introduce a singular Weyl-Titchmarsh function associated with T, following essentially [40] (see also [25,26,28,35,39]). To this end, we say that some function φ : C×R → C is a real entire solution of the homogeneous differential equation (2.5) if the following two properties hold:…”

“…These self-adjoint realizations are mostly of an auxiliary nature, whereas the more convenient objects seem to be associated quadratic operator pencils in H 1 (J) which will be introduced next. We will also introduce (singular) Weyl-Titchmarsh functions, which are basic objects of spectral theory for Sturm-Liouville problems (for further information on singular WeylTitchmarsh functions we refer to [25,26,28,35,39,40]). All this will be done for the cases of bounded intervals (in Section 3), semi-axes (in Section 4) and the whole line (in Section 5) separately.…”

Spectral Analysis, Differential Equations and Mathematical Physics: A Festschrift in Honor of Fritz Gesztesy’s 60th Birthday

“…Now insert the definitions and compute the limit to obtain 12) which in turn implies the theorem. This seminal result initiated the development of Weyl-Titchmarsh theory at a singular endpoint and has triggered a number of results recently [24,25,26,27,28,31,32,33,93,92,94,95,99].…”

KdV Equation in the Quarter–Plane: Evolution of the Weyl Functions and Unbounded Solutions

“…We note that the high energy asymptotics of the Weyl functions was actively studied (see [6,7,17,21] and references therein) following the seminal papers [12,13]. Though the low energy asymptotics of the Weyl functions is used in the present paper, the high energy asymptotics (namely, an important result on asymptotics of the Weyl function in terms of the values of u and its derivatives at x = 0 from [6]) jointly with the evolution of the Weyl function could also prove useful for the analysis of the initial-boundary conditions.…”