Homogeneous Rank One Perturbations

Homogeneous Rank One Perturbations Jan Derezin´ski 0 1 0 Communicated by Claude Alain Pillet. Received: November 11, 2016. Accepted: April 4, 2017 1 Jan Derezin ́ski Department of Mathematical Methods in Physics Faculty of Physics University of Warsaw Pasteura 5 02-093 Warsaw Poland A holomorphic family of closed operators with a rank one perm turbation given by the function x 2 is studied. The operators can be used in a toy model of renormalization group. Rank one perturbations can be used to illustrate various interesting mathematical concepts. For instance, they can be singular: the perturbation is not an operator, and an infinite renormalization may be needed. Rank one perturbations are often applied to model physical phenomena. Our paper is devoted to a special class of exactly solvable rank one perturbations, which are both singular and physically relevant. We consider the Hilbert space L2[0, ∞[. The starting point is the operator of multiplication by x ∈ [0, ∞[, denoted by X. We try to perturb it by a rank one operator m involving the function x 2 . Thus, we try to define an operator formally given by 1. Introduction Note that we allow m and λ to be complex. In particular, (1.1) is usually non-Hermitian. The function x m2 is never square integrable, and therefore, the perturbation is always singular. (1.1) is very special. Formally, X is homogeneous of degree 1 and its perturbation is homogeneous of degree m. We will see that in order to define a closed operator on L2[0, ∞[ one needs to restrict m by the condition −1 < Rem < 1. Besides, a special treatment is needed in the case m = 0. One obtains two holomorphic families of closed operators, Hm,λ and H0ρ. λ and ρ The financial support of the National Science Center, Poland, under the Grant UMO2014/15/B/ST1/00126, is gratefully acknowledged. The author thanks Serge Richard for useful discussions. (As compared with the notation of [4], we add a tilde to distinguish from the operators considered in this paper). These operators have continuous spectrum in [0, ∞[ of multiplicity 1. They can be diagonalized with help of the so-called Hankel transformation Fm, whose kernel has a simple expression in terms of the Bessel function Jm. As shown in [7], for −1 < Rem < 1 there exists a two-parameter holomorphic family of closed operators that can be associated with the differential expression on the right-hand side of (1.2). They correspond to mixed boundary conditions at zero and are denoted H˜m,κ. The case m = 0 needs special treatment, and one introduces a family of H˜0ν . As we show in our paper, the operators H˜m,κ, resp. H˜0ν , are equivalent (similar) to the operators Hm,λ and ρ H0 , where κ and ν are linked by a simple relation with λ and ρ, see Theorems 4.1 and 4.2. The operators H˜m,κ and H˜0ν are very well motivated—they constitute natural classes of Schro¨dinger operators, which are relevant for many problems in mathematical physics. However, their theory looks complicated—it requires the knowledge of some special functions, more precisely, Bessel-type functions and the Gamma function. On the other hand, the theory of Hm,λ and H0ρ does not involve special functions at all—it uses only trigonometric functions and the logarithm. The paper is organized as follows. In Sect. 2 we recall the theory of singular rank one perturbations. It is sometimes called the Aronszajn–Donoghue theory and goes back to [2, 3, 5]. It is described in particular in [1, 6, 12]. We discuss also the scattering theory in the context of rank one perturbations. Here the basic reference is [13]. Note, however, that we do not assume that the perturbation is selfadjoint, and most of the literature on this subject is restricted to the selfadjoint case. A notable exception is the articles [10, 11], where nonself-adjoint perturbations of self-adjoint operators are studied. Section 3 is the main part of our paper. Here we construct and study the ρ operators Hm,λ and H0 . ρ In Sect. 4 we describe the relationship of the operators Hm,λ and H0 with Schro¨dinger operators with inverse square potentials H˜m,κ and H˜0ν . There exists large literature for such Schro¨dinger operators, see e.g., [8], it is, however, usually restricted to the self-adjoint case. The general case is studied in [4] and especially [7]. In “Appendix” we collect some integrals that are used in our paper. 2. General Theory of Rank One Perturbations 2.1. Preliminaries We consider the Hilbert space L2[0, ∞[ with the scalar product 0 In addition, it is also equipped with the bilinear form (f |g) := f |g := f (x)g(x)dx. f (x)g(x)dx. 0 Thus, we use round brackets for the sesquilinear scalar product and angular brackets for the closely related bilinear form. Note that in some sense the latter plays a more important role in our paper (and in similar exactly solvable problems) than the former. If B is an operator then B∗ denotes the usual Hermitian adjoint of B, whereas B# denotes the transpose of B, that is, its (...truncated)