On radial Schr odinger operators with a Coulomb potential: General - PDF document

On radial Schr¨ odinger operators with a Coulomb potential: General boundary conditions Jan Derezi´ nski Department of Mathematical Methods in Physics, Faculty of Physics University of Warsaw, Pasteura 5, 02-093, Warszawa, Poland email: jan.derezinski@fuw.edu.pl J´ er´ emy Faupin Institut Elie Cartan de Lorraine, Universit´ e de Lorraine UFR MIM, 3 rue Augustin Fresnel. 57073 Metz Cedex 03, France email: jeremy.faupin@univ-lorraine.fr Quang Nhat Nguyen, Serge Richard ∗ Graduate school of mathematics, Nagoya University, Chikusa-ku, Nagoya 464-8602, Japan email: nguyen.quang.nhat@d.mbox.nagoya-u.ac.jp, richard@math.nagoya-u.ac.jp June 23, 2020 Abstract This paper presents the spectral analysis of 1-dimensional Schr¨ odinger operator on the half-line whose potential is a linear combination of the Coulomb term 1 /r and the centrifugal term 1 /r 2 . The coupling constants are allowed to be complex, and all possible boundary conditions at 0 are considered. The resulting closed operators are organized in three holomorphic families. These operators are closely related to the Whittaker equation. Solutions of this equation are thoroughly studied in a large appendix to this paper. Various special cases of this equation are analyzed, namely the degenerate , the Laguerre and the doubly degenerate cases. A new solution to the Whittaker equation in the doubly degenerate case is also introduced. Dedicated to Prof. Franciszek Hugon Szafraniec ∗ Supported by the grant Topological invariants through scattering theory and noncommutative geom- etry from Nagoya University, and by JSPS Grant-in-Aid for scientific research (C) no 18K03328, and on leave of absence from Univ. Lyon, Universit´ e Claude Bernard Lyon 1, CNRS UMR 5208, Institut Camille Jordan, 43 blvd. du 11 novembre 1918, F-69622 Villeurbanne cedex, France. 1

Contents 1 Introduction 2 2 The Whittaker operator 5 2.1 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 Zero-energy eigenfunctions of the Whittaker operator . . . . . . . . . . . . 6 2.3 Maximal and minimal operators . . . . . . . . . . . . . . . . . . . . . . . 7 2.4 Families of Whittaker operators . . . . . . . . . . . . . . . . . . . . . . . . 9 3 Spectral theory 13 3.1 Point spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.2 Green’s functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.3 Holomorphic families of closed operators . . . . . . . . . . . . . . . . . . . 23 Blowing up the singularities at m = 0 and at m = ± 1 3.4 . . . . . . . . . . . 28 2 3.5 Eigenprojections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 A The Whittaker equation 33 A.1 General theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 A.2 The Laguerre cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 A.3 The degenerate case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 A.4 The doubly degenerate case . . . . . . . . . . . . . . . . . . . . . . . . . . 39 A.5 Recurrence relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 A.6 Integral identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 A.7 The trigonometric type Whittaker equation . . . . . . . . . . . . . . . . . 44 A.8 Integral identities in the trigonometric case . . . . . . . . . . . . . . . . . 44 B The Bessel equation 45 B.1 The modified Bessel equation . . . . . . . . . . . . . . . . . . . . . . . . . 45 B.2 Recurrence relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 B.3 Integral identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 B.4 The degenerate case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 B.5 The half-integer case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 B.6 The standard Bessel equation . . . . . . . . . . . . . . . . . . . . . . . . . 49 B.7 The zero eigenvalue Whittaker equation . . . . . . . . . . . . . . . . . . . 50 B.8 Integrals for zero eigenvalue solutions of the Whittaker equation . . . . . 51 1 Introduction This paper is devoted to 1-dimensional Schr¨ odinger operators with Coulomb and cen- trifugal potentials. These operators are given by the differential expressions � 1 α − 1 x 2 − β � L β,α := − ∂ 2 x + x. (1.1) 4 The parameters α and β are allowed to be complex valued. We shall study realizations of L β,α as closed operators on L 2 ( R + ), and consider general boundary conditions. 2

The operator given in (1.1) is one of the most famous and useful exactly solvable models of Quantum Mechanics. It describes the radial part of the Hydrogen Hamiltonian. In the mathematical literature, this operator goes back to Whittaker, who studied its eigenvalue equation in [32]. For this reason, we call (1.1) the Whittaker operator . This paper is a continuation of a series of papers [2, 6, 7] devoted to an analysis of exactly solvable 1-dimensional Schr¨ odinger operators. We follow the same philosophy as in [6]. We start from a formal differential expression depending on complex parameters. Then we look for closed realizations of this operator on L 2 ( R + ). We do not restrict ourselves to self-adjoint realizations – we look for realizations that are well-posed , that is, possess non-empty resolvent sets. This implies that they satisfy an appropriate boundary condition at 0, depending on an additional complex parameter. We organize those operators in holomorphic families. Before describing the holomorphic families introduced in this paper, let us recall the main constructions from the previous papers of this series. In [2, 6] we considered the operator � 1 α − 1 � L α := − ∂ 2 x + x 2 . (1.2) 4 As is known, it is useful to set α = m 2 . In [2] the following holomorphic family of closed realizations of (1.2) was introduced: H m , with − 1 < Re( m ) , 1 2 + m . defined by L m 2 with boundary conditions ∼ x It was proved that for Re( m ) ≥ 1 the operator H m is the only closed realization of L m 2 . In the region − 1 < Re( m ) < 1 there exist realizations of L m 2 with mixed boundary conditions. As described in [6], it is natural to organize them into two holomorphic families: H m,κ , with − 1 < Re( m ) < 1 , m � = 0 , κ ∈ C ∪ {∞} , 2 + m + κx 1 1 2 − m , defined by L m 2 with boundary conditions ∼ x and H ν 0 , with ν ∈ C ∪ {∞} , 1 2 � � defined by L 0 with boundary conditions ∼ x ν + ln( x ) . Note that related investigations about these operators have also been performed in [30, 31]. In [7] and in the present paper we study closed realizations of the differential operator (1.1) on L 2 ( R + ). Again, it is useful to set α = m 2 . In [7] we introduced the family H β,m , with β ∈ C , − 1 < Re( m ) , β 1 2 + m � � defined by L β,m 2 with boundary conditions ∼ x 1 − 1 + 2 mx . It was noted in this reference that this family is holomorphic except for a singularity at 0 , − 1 � � ( β, m ) = , which corresponds to the Neumann Laplacian. 2 3