The similarity problem for indefinite Sturm–Liouville operators and the HELP inequality

Abstract

Abstract. We study two problems. The first one is the similarity problem for the indefinite Sturm-Liouville operatorare even and positive a.e. on (−b, b).The second object is the so-called HELP inequalitywhere the coefficientsw,r ∈ L 1 loc [0, b) are positive a.e. on (0, b). Both problems are well understood when the corresponding Sturm-Liouville differential expression is regular. The main objective of the present paper is to give criteria for both the validity of the HELP inequality and the similarity to a self-adjoint operator in the singular case. Namely, we establish new criteria formulated in terms of the behavior of the corresponding Weyl-Titchmarsh m-functions at 0 and at ∞. As a biproduct of this result we show that both problems are closely connected. Namely, the operator A is similar to a selfadjoint one precisely if the HELP inequality withw = r andr = w is valid.Next we characterize the behavior of m-functions in terms of coefficients and then these results enable us to reformulate the obtained criteria in terms of coefficients. Finally, we apply these results for the study of the two-way diffusion equation, also known as the time-independent Fokker-Plank equation.

The inverse spectral problem for indefinite strings

“…Is there an equally concise analogue of M. G. Krein's solution of the inverse spectral problem? Although there are various results about spectral theory for (1.1) when ω is allowed to be indefinite (we only mention [2,6,8,28,29,47,68] and the references therein), there is still no satisfactory answer to these questions. A first guess could suggest that instead of the class of Stieltjes functions (which are, roughly speaking, determined by not having singularities on the negative real axis) one obtains the entire class of Herglotz-Nevanlinna functions (which may have singularities on the whole real axis).…”

Coupling of definitizable operators in Krein spaces

“…The proof is based on the K. Veselić criterion of regularity [25,26] adapted to the case of definitizable operators in [27]. In the case when A + and A − are Hilbert space symmetric operators, similar results were obtained in [23] and [28].…”

On a necessary aspect for the Riesz basis property for indefinite Sturm‐Liouville problems

“… Remark Theorem 3.6 is concerned with the particular case of the general HELP inequality . However, Theorem 3.6 remains true under the following assumptions on coefficients in : is regular at both endpoints; the spectral problem subject to the Neumann boundary conditions has a nonnegative spectrum; the functions from also satisfy the Neumann boundary condition at . The case of a singular endpoint is addressed in .…”

“…Moreover, using Pyatkov's interpolation criterion , Parfenov found a necessary and sufficient condition for the Riesz basis property under the assumption that r is odd. Let us mention that a different proof of this fact is given in . Notice that the problem on the Riesz basis property for is still open if the oddness assumption is dropped.…”