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ISRN Mathematical Physics
VolumeΒ 2012Β (2012), Article IDΒ 230245, 18 pages
http://dx.doi.org/10.5402/2012/230245
Research Article

On Pointlike Interaction between Three Particles: Two Fermions and Another Particle

Institute for Information Transmission Problems, Russian Academy of Sciences, Bolshoy Karetnyi 19, Moscow, Russia

Received 3 April 2012; Accepted 29 April 2012

Academic Editors: J.Β Banasiak and M.Β Martins

Copyright Β© 2012 Robert Adol'fovich Minlos. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The problem of construction of self-adjoint Hamiltonian for quantum system consisting of three pointlike interacting particles (two fermions with mass 1 plus a particle of another nature with mass π‘š>0) was studied in many works. In most of these works, a family of one-parametric symmetrical operators {π»πœ€,πœ€βˆˆβ„1} is considered as such Hamiltonians. In addition, the question about the self-adjointness of π»πœ€ is equivalent to the one concerning the self-adjointness of some auxiliary operators {𝒯𝑙,𝑙=0,1,…} acting in the space 𝐿2(ℝ1+,π‘Ÿ2π‘‘π‘Ÿ). In this work, we establish a simple general criterion of self-adjointness for operators 𝒯𝑙 and apply it to the cases 𝑙=0 and 𝑙=1. It turns out that the operator 𝒯𝑙=0 is self-adjoint for any π‘š, while the operator 𝒯𝑙=1 is self-adjoint for π‘š>π‘š0, where the value of π‘š0 is given explicitly in the paper.

1. Introduction and Statement of the Problem

This paper is continuation of works [1–4] studying the problem of construction of Hamiltonian for a quantum system which consists of two fermions with mass 1 interacting pointwise with a particle of another nature having mass π‘š.

Originally, the construction of such Hamiltonian begins with introduction of the symmetric operator: 𝐻01=βˆ’2ξ‚€1π‘šΞ”π‘¦+Ξ”π‘₯1+Ξ”π‘₯2(1.1) acting in a Hilbert space β„‹=𝐿2(ℝ3)βŠ—πΏasym2(ℝ3×ℝ3). Here, π‘₯1, π‘₯2βˆˆβ„3 are the positions of fermions, 𝑦 is the position of a separate particle, and Δ𝑦, Ξ”π‘₯1, and Ξ”π‘₯2 are Laplacians with respect to 𝑦, π‘₯1, and π‘₯2, respectively. The domain of definition of 𝐻0, 𝐷(𝐻0)βŠ‚β„‹ consists of smooth rapidly decreasing functions πœ“(𝑦,π‘₯1,π‘₯2)βˆˆβ„‹ on infinity, antisymmetrical with respect to π‘₯1, π‘₯2 and satisfying the following conditions: πœ“ξ€·π‘¦,π‘₯1,π‘₯2ξ€Έ||π‘₯𝑖=𝑦=0,𝑖=1,2.(1.2) Usually, some family {π»πœ€,πœ€βˆˆβ„1} of symmetric extensions of the operator 𝐻0 is proposed as a possible β€œtrue” Hamiltonian of the system (the so-called Ter-Martirosian-Skornyakov extensions, see [5]). These extensions were constructed in [1–4]. For some values of mass π‘š, the extensions of Ter-Martirosian-Skornyakov are self-adjoint (for all values of the parameter πœ€); however, for the other values of π‘š they are only symmetric with nonzero deficiency indexes (equal for all πœ€). It turns out (see [3]) that the self-adjointness of all operators {π»πœ€} is equivalent to the one for some auxiliary symmetric operator 𝒯 acting in the space 𝐿2(ℝ3) (see below). This operator commutes with the operators {π‘ˆπ‘”,π‘”βˆˆπ‘‚3} of the representation of the rotation group 𝑂3 that acts in 𝐿2(ℝ3) by the usual formula: ξ€·π‘ˆπ‘”π‘“ξ€Έξ€·π‘”(π‘˜)=π‘“βˆ’1π‘˜ξ€Έ,π‘”βˆˆπ‘‚3,π‘“βˆˆπΏ2ℝ3ξ€Έ.(1.3) Let us denote by β„‹π‘™βŠ‚πΏ2(ℝ3) the maximal subspace, where the representation (1.3) is multiplied by the irreducible representation of 𝑂3 with weight 𝑙, 𝑙=0,1,2,… (see [6]). Evidently, the space ℋ𝑙 is invariant with respect to the operator 𝒯, and the restriction 𝒯𝑙=𝒯|ℋ𝑙 of this operator to the space ℋ𝑙 is symmetric operator. The operator 𝒯 is self-adjoint if all the operators {𝒯𝑙,𝑙=0,1,…} are self-adjoint. In this paper, we find general simple conditions of self-adjointness of 𝒯𝑙 and the form of the defect subspaces (with small exclusions) when these conditions are broken. Then, we apply these conditions to the cases 𝑙=0 and 𝑙=1 and get that the operator 𝒯𝑙=0 is self-adjoint for all values of π‘š>0, while the operator 𝒯𝑙=1 is self-adjoint for π‘š>π‘š0 and has nonzero deficiency indexes for π‘šβ‰€π‘š0, the constant π‘š0>0 is indicated below (see (5.4)).

By the way, we note that the value of π‘š0 obtained in this paper differs from that one given by mistake in [2].

2. A Short Explanation of the Constructions from Papers [1–3]

(1) After Fourier transformation: πœ“ξ€·π‘¦,π‘₯1,π‘₯2ξ€Έξ€·βŸΆξ‚πœ“π‘ž,π‘˜1,π‘˜2ξ€Έ=12πœ‹9/2ξ€œξ€·β„3ξ€Έ3πœ“ξ€·π‘¦,π‘₯1π‘₯2ξ€Έξ€½ξ€·π‘˜expβˆ’π‘–(π‘ž,𝑦)βˆ’π‘–1,π‘₯1ξ€Έξ€·π‘˜βˆ’π‘–2,π‘₯2𝑑𝑦𝑑π‘₯1𝑑π‘₯2≑(β„±πœ“)π‘ž,π‘˜1,π‘˜2ξ€Έ,(2.1) and change of variables: 𝑃=π‘ž+π‘˜1+π‘˜2,𝑝𝑗=π‘ƒπ‘š+2βˆ’π‘˜π‘—,𝑗=1,2,(2.2) the operator 𝐻0=ℱ𝐻0β„±βˆ’1,(2.3) can be represented as a tensor sum: 𝐻0=𝐻0(1)+π‘šξ‚π»π‘š+10(2),(2.4) where 𝐻0(1) is a self-adjoint operator in 𝐿2(ℝ3): 𝐻0(1)𝑓𝑃(𝑃)=2π‘š+2𝑓(𝑃),π‘ƒβˆˆβ„3,π‘“βˆˆπΏ2ℝ3ξ€Έ,(2.5) and 𝐻0(2) acts in 𝐿asym2(ℝ3×ℝ3) by formula 𝐻0(2)𝑔𝑝1,𝑝2𝑝=𝐺1,𝑝2𝑔𝑝1,𝑝2ξ€Έ,π‘”βˆˆπΏasym2ℝ3×ℝ3ξ€Έ,(2.6) with 𝐺𝑝1,𝑝2ξ€Έ=𝑝21+𝑝22+2ξ€·π‘π‘š+11,𝑝2ξ€Έ>0.(2.7) The operator 𝐻0(2) is symmetric, and its domain is 𝐷𝐻0(2)=ξ‚»π‘”βˆˆπΏasym2ℝ3×ℝ3ξ€ΈβˆΆξ€œβ„3𝑔𝑝1,𝑝2𝑑𝑝𝑗=0,𝑗=1,2,(2.8)

(2) the deficiency subspace β„›βˆ’1βŠ‚πΏasym2(ℝ3×ℝ3) of the operator 𝐻0(2) consists of the functions of the form: π‘ˆβ„˜ξ€·π‘1,𝑝2ξ€Έ=β„˜ξ€·π‘1ξ€Έξ€·π‘βˆ’β„˜2𝐺𝑝1,𝑝2ξ€Έ+1,(2.9) where the function β„˜(𝑝) belongs to Hilbert space ξƒ―ξ€œβ„’=β„˜βˆΆβ„3||||β„˜(𝑝)2βˆšπ‘2ξƒ°,+1𝑑𝑝<∞(2.10) with inner product βŸ¨β„˜1,β„˜2ξ€·π‘ˆβŸ©=β„˜1,π‘ˆβ„˜2𝐿2(ℝ3×ℝ3)β‰‘ξ€·π‘Šβ„˜1,β„˜2𝐿2(ℝ3).(2.11) Here π‘Š is some positive operator acting in 𝐿2(ℝ3) (see [3]). The domain of the operator (𝐻0(2))βˆ—, that is, a conjugate to 𝐻0(2), is 𝐷𝐻0(2)ξ‚βˆ—ξ‚=ξƒ―π‘”βˆˆπΏasym2ℝ3×ℝ3ξ€Έξ€·π‘βˆΆπ‘”1,𝑝2𝑝=𝑓1,𝑝2ξ€Έ+π‘ˆβ„˜ξ€·π‘1,𝑝2ξ€Έ+π‘ˆπœ“ξ€·π‘1,𝑝2𝐺𝑝1,𝑝2ξ€Έξƒ°+1,(2.12) where ξ‚π»π‘“βˆˆπ·(0(2)), β„˜,πœ“βˆˆβ„’. In addition, the operator (𝐻0(2))βˆ— acts by the formula: 𝐻0(2)ξ‚βˆ—π‘”ξ‚ξ€·π‘1,𝑝2𝑝=𝐺1,𝑝2𝑔𝑝1,𝑝2ξ€Έβˆ’ξ€·β„˜ξ€·π‘1ξ€Έξ€·π‘βˆ’β„˜2,ξ€Έξ€Έ(2.13) where β„˜ is defined by (2.12).

The following asymptotics holds for vectors ξ‚π»π‘”βˆˆπ·((0(2))βˆ—)π‘β†’βˆž: ξ€œ||𝑝1||<𝑁𝑔𝑝1,𝑝2𝑑𝑝1=4πœ‹π‘β„˜ξ€·π‘2𝑝+𝑏2ξ€Έ+π‘œ(1).(2.14) Here 𝑏(𝑝)=βˆ’(π‘‡β„˜)(𝑝)+(π‘Šπœ“)(𝑝),(2.15) where the operator π‘Š is defined in (2.11), and (π‘‡β„˜)(𝑝) is given by the following expression (πœ‡=2/(π‘š+1)) (π‘‡β„˜)(𝑝)=2πœ‹2ξƒŽξ‚΅πœ‡1βˆ’24𝑝2ξ€œ+1β„˜(𝑝)+ℝ3β„˜(𝑑)𝐺(𝑑,𝑝)+1𝑑𝑑,(2.16) defined on the set: 𝐷(𝑇)=β„˜βˆˆπΏ2ℝ3ξ€ΈβˆΆ||𝑝||β„˜(𝑝)∈𝐿2ℝ3.ξ€Έξ€Ύ(2.17) The above-mentioned Ter-Martirosian-Skornyakov's extension ξ‚π»πœ€(2) of the operator 𝐻0(2) is obtained by requiring 𝑏(𝑝)=πœ€β„˜(𝑝),(2.18) where πœ€βˆˆβ„1 is an arbitrary parameter.

Lemma 2.1. The operator 𝑇 defined in the space 𝐿2(ℝ3) by (2.16) is symmetric, and the self-adjointness of the operators π»πœ€ (for all πœ€) is equivalent to the self-adjointness of the operator 𝑇 (see [2, 3, 5]).

The operator 𝑇 can be represented as a sum of two operators: 𝑇=𝒯+𝑇′,(2.19) where the symmetric operator 𝒯 (with the domain 𝐷(𝒯)=𝐷(𝑇)) acts as follows: (π’―β„˜)(𝑝)=2πœ‹2ξƒŽπœ‡1βˆ’24||𝑝||ξ€œβ„˜(𝑝)+ℝ3β„˜(𝑑)𝑑𝑑𝐺(𝑑,𝑝)(2.20) and 𝑇′ is a bounded self-adjoint operator. Since the deficiency indexes of 𝑇 coincide with the ones of 𝒯 (see [7]), we shall study the conditions of self-adjointness for the operator 𝒯;

(3) as we said, the space β„‹π‘™βŠ‚πΏ2(ℝ3) is invariant with respect to 𝒯; it has the form: ℋ𝑙=𝐿2ℝ1+,π‘Ÿ2ξ€Έπ‘‘π‘ŸβŠ—πΏπ‘™2(𝑆),(2.21) where 𝐿𝑙2(𝑆)βŠ‚πΏ2(𝑆) is the space of spherical functions of weight 𝑙 (see [6]) on the unit sphere π‘†βŠ‚β„3. In addition, the operator 𝒯𝑙=𝒯|ℋ𝑙 has the form 𝒯𝑙=π‘€π‘™βŠ—πΈπ‘™,(2.22) where 𝐸𝑙 is the unit operator in 𝐿𝑙2(𝑠), and 𝑀𝑙 acts in 𝐿2(ℝ1+,π‘Ÿ2π‘‘π‘Ÿ) by the formula: 𝑀𝑙𝑓(π‘Ÿ)=2πœ‹2ξƒŽπœ‡1βˆ’24ξ€œπ‘Ÿπ‘“(π‘Ÿ)+2πœ‹1βˆ’1𝑑π‘₯π‘ƒπ‘™ξ€œ(π‘₯)∞0(π‘Ÿβ€²)2𝑓(π‘Ÿβ€²)π‘‘π‘Ÿβ€²π‘Ÿ2+(π‘Ÿβ€²)2,+πœ‡π‘Ÿπ‘Ÿβ€²π‘₯(2.23) on the domain 𝐷𝑀𝑙≑𝑉=π‘’βˆˆπΏ2ℝ1+,π‘Ÿ2ξ€Έπ‘‘π‘ŸβˆΆπ‘Ÿπ‘’(π‘Ÿ)∈𝐿2ℝ1+,π‘Ÿ2.π‘‘π‘Ÿξ€Έξ€Ύ(2.24) Here 𝑃𝑙(π‘₯), 𝑙=0,1,2,…, π‘₯∈[βˆ’1,1], are orthogonal polynomials (Legendre polynomials) satisfying 𝑃𝑙(1)=1: 𝑃𝑙1(π‘₯)=2𝑙𝑑𝑙!𝑙𝑑π‘₯𝑙π‘₯2ξ€Έβˆ’1𝑙,π‘₯∈(βˆ’1,1).(2.25) The operators {𝑀𝑙,𝑙=0,1,…} are symmetric in 𝐿2(ℝ1+,π‘Ÿ2π‘‘π‘Ÿ), and the self-adjointness of 𝑀𝑙 is equivalent to the self-adjointness of 𝒯𝑙. Later on, we shall study the operators 𝑀𝑙 and derive a condition of self-adjointness.

3. Preparatory Constructions

For every function π‘’βˆˆπ‘‰βŠ‚πΏ2(ℝ1+,π‘Ÿ2π‘‘π‘Ÿ), we consider the family of functions π”₯𝑒(𝑒)=𝛼=π‘Ÿπ›Ό[]𝑒,π›Όβˆˆ0,1,𝑒0ξ€Ύ=𝑒,(3.1) which we call a chain (with initial element 𝑒=𝑒0 and the final one 𝑒1). All functions π‘’π›Όβˆˆπ”₯(𝑒) belong to 𝐿2(ℝ1+,π‘Ÿ2π‘‘π‘Ÿ) and have a uniformly bounded norm: ‖‖𝑒𝛼‖‖2≀‖‖𝑒0β€–β€–2+‖‖𝑒1β€–β€–2[].,π›Όβˆˆ0,1(3.2) Consider the unitary map (Mellin's transformation [8]): πœ”βˆΆπΏ2ℝ1+,π‘Ÿ2ξ€Έπ‘‘π‘ŸβŸΆπΏ2ℝ11,π‘‘π‘ βˆΆπ‘“(π‘Ÿ)βŸΆπ‘“(𝑠)=βˆšξ€œ2πœ‹βˆž0π‘Ÿβˆ’π‘–π‘ +1/2𝑓(π‘Ÿ)π‘‘π‘Ÿ,π‘ βˆˆβ„1(3.3) and its inverse: ξ‚€πœ”βˆ’1𝑓1(π‘Ÿ)=βˆšξ€œ2πœ‹βˆžβˆ’βˆžπ‘Ÿπ‘–π‘ βˆ’3/2𝑓(𝑠)𝑑𝑠.(3.4) For every set of functions π΅βŠ‚πΏ2(ℝ1+,π‘Ÿ2π‘‘π‘Ÿ), we denote by ξ‚π΅βŠ‚πΏ2(ℝ1,𝑑𝑠) the set of their Mellin's transformations: 𝐡=πœ”π΅.(3.5) For every chain π”₯(𝑒), we denote by Γ𝑒 the family of functions: Γ𝑒=𝛾π”₯(𝑒)=𝛼([]ξ€Ύ,𝑠),π›Όβˆˆ0,1(3.6) where 𝛾𝛼(𝑠)=(πœ”π‘’π›Ό)(𝑠), π‘’π›Όβˆˆπ”₯(𝑒). The family Γ𝑒 can be represented as a function Γ𝑒(𝑧) of a complex variable 𝑧=𝑠+𝑖𝛼 in the strip: 𝐼=π‘§βˆˆβ„‚1[]ξ€Ύ,Ξ“βˆΆβ„‘π‘§βˆˆ0,1𝑒1(𝑧)=βˆšξ€œ2πœ‹βˆž0π‘Ÿβˆ’π‘–π‘ βˆ’1/2+𝛼1𝑒(π‘Ÿ)π‘‘π‘Ÿ=βˆšξ€œ2πœ‹βˆž0π‘Ÿβˆ’π‘–π‘§π‘’(π‘Ÿ)π‘‘π‘Ÿ.(3.7) The function Γ𝑒 is said to be associated with the chain π”₯(𝑒), and its values {𝛾𝛼(𝑠)} on the lines πœ‰π›Ό={𝑧=𝑠+𝑖𝛼,π‘ βˆˆβ„1,0≀𝛼≀1}βŠ‚πΌ are called the sections of Γ𝑒.

Proposition 3.1. For every chain π”₯(𝑒), π‘’βˆˆπ‘‰, the associated function {Γ𝑒(𝑧),π‘§βˆˆπΌ} is continuous in a closed strip 𝐼 and analytic inside this strip. Moreover, its sections {𝛾𝛼} satisfy the following inequality: sup0≀𝛼≀1‖‖𝛾𝛼‖‖(β‹…)𝐿2(ℝ1)<∞.(3.8) Inversely, any function {Ξ“(𝑧),π‘§βˆˆπΌ} which possesses these properties is associated with some (unique) chain π”₯(𝑣)βˆΆΞ“=Γ𝑣, π‘£βˆˆπ‘‰. Let call this chain generated by Ξ“. In addition, the functions {𝑣𝛼,π›Όβˆˆ[0,1]} of the chain π”₯(𝑣) are obtained by the inverse Mellin's transformation from the sections of Ξ“={𝛾𝛼}: 𝑣𝛼=πœ”βˆ’1𝛾𝛼.(3.9)

The proof of this proposition can be obtained by using the arguments given in the book by Paley and Wiener (see [9], Chapter I), which are related to the Fourier transformation of functions analytical in a strip in a complex plane. It is not difficult to reformulate these arguments in terms of Mellin's transformation.

Note that the estimate (3.8) for {𝛾𝛼} follows from the estimate (3.2) and the unitary Mellin's transformation. Denote by 𝒒 a linear space of functions Ξ“ satisfying conditions of Proposition 3.1. Let us introduce two maps: Ω∢π”₯(𝑒)βŸΆΞ“π‘’βˆˆπ’’,Ξ©βˆ’1βˆΆΞ“π‘’βŸΆπ”₯(𝑒).(3.10) Let 𝑁(𝑧), π‘§βˆˆπΌ, be a bounded, continuous function in the strip 𝐼, which is analytic inside 𝐼. This function generates the family ξ‚πœ…π‘π›Ό,π›Όβˆˆ[0,1] of bounded operators in 𝐿2(ℝ1) which act as multiplication on the functions 𝑛𝑁𝛼(𝑠)=𝑁(𝑧)|𝑧=𝑠+𝑖𝛼, π‘ βˆˆβ„1, 0≀𝛼≀1: ξ‚€ξ‚πœ…π‘π›Όπœ“ξ‚(𝑠)=𝑛𝑁𝛼(𝑠)πœ“(𝑠),πœ“βˆˆπΏ2ℝ1ξ€Έ.(3.11) Evidently, for any Ξ“βˆˆπ’’, the function 𝑁(𝑧)Ξ“(𝑧) belongs to 𝒒. If the chain π”₯(𝑒) is generated by Ξ“=Γ𝑒 and the chain π”₯(𝑣) is generated by 𝑁(𝑧)Ξ“(𝑧)=Γ𝑣(𝑧), then 𝑣𝛼=πœ…π‘π›Όπ‘’π›Ό[],π›Όβˆˆ0,1,π‘’π›Όβˆˆπ”₯(𝑒),(3.12) where πœ…π‘π›Ό=πœ”βˆ’1ξ‚πœ…π‘π›Όπœ”.(3.13) Denote by Ξ  the following self-adjoint operator in 𝐿2(ℝ1+,π‘Ÿ2π‘‘π‘Ÿ): (Π𝑓)(π‘Ÿ)=π‘Ÿπ‘“(π‘Ÿ),(3.14) with the domain 𝐷(Ξ )=𝑉.

It is clear that for any π‘’βˆˆπ‘‰, the power Π𝛽, 0≀𝛽≀1 of the operator Ξ  is applicable to an element π‘’π›Όβˆˆπ”₯(𝑒) if 𝛽+𝛼≀1 and Π𝛽𝑒𝛼=𝑒𝛼+𝛽.(3.15) For the function Γ𝑒 that is associated with π”₯(𝑒),the action of the operator Π𝛽=πœ”Ξ π›½πœ”βˆ’1 on the sections {𝛾𝛼} of Γ𝑒 has the form: Π𝛽𝛾𝛼=𝛾𝛼+𝛽.(3.16) (again if 𝛼+𝛽≀1).

4. The Operator 𝑀𝑙

The operator 𝑀𝑙 (see (2.23)) can be represented as 𝑀𝑙=Ξ 1/2πœ…π‘™Ξ 1/2,(4.1) where πœ…π‘™=πœ…π‘™1/2 is an operator in 𝐿2(ℝ1+,π‘Ÿ2π‘‘π‘Ÿ) acting by the formula: ξ€·πœ…π‘™1/2𝑓(π‘Ÿ)=2πœ‹2ξƒŽπœ‡1βˆ’24ξ€œπ‘“(π‘Ÿ)+2πœ‹1βˆ’1𝑑π‘₯𝑃0π‘™ξ€œ(π‘₯)∞0ξ€·π‘Ÿξ…žξ€Έ2π‘“ξ€·π‘Ÿξ…žξ€Έπ‘‘π‘Ÿξ…ž(π‘Ÿπ‘Ÿξ…ž)1/2ξ€·π‘Ÿ2+(π‘Ÿξ…ž)2+πœ‡π‘₯π‘Ÿπ‘Ÿξ…žξ€Έ.(4.2)

Lemma 4.1. Operator πœ…π‘™1/2 is bounded and self-adjoint in 𝐿2(ℝ1+,π‘Ÿ2π‘‘π‘Ÿ).

Proof. Pass to the operator: ξ‚πœ…π‘™1/2=πœ”πœ…π‘™1/2πœ”βˆ’1,(4.3) acting in 𝐿2(ℝ1). It follows from calculations in [2, 3] that ξ‚πœ…π‘™1/2 is the operator of multiplication on the function: 𝑛𝑙1/2(𝑠)=2πœ‹2βŽ›βŽœβŽœβŽξƒŽπœ‡1βˆ’24+πœ†π‘™1/2⎞⎟⎟⎠,(𝑠)(4.4) where πœ†π‘™1/2⎧βŽͺ⎨βŽͺβŽ©ξ€œ(𝑠)=10𝑃𝑙(π‘₯)ch(𝑠𝑣(π‘₯))𝑑π‘₯βˆ’ξ€œch(π‘ πœ‹/2)cos(𝑣(π‘₯))foreven𝑙,10𝑃𝑙(π‘₯)sh(𝑠𝑣(π‘₯))𝑑π‘₯sh(π‘ πœ‹/2)cos(𝑣(π‘₯))forodd𝑙,(4.5) and 𝑣(π‘₯)=arcsinπœ‡π‘₯/2, 0≀π‘₯≀1. As we see the function 𝑛𝑙1/2(𝑠), π‘ βˆˆβ„1, is bounded and real. The lemma is proved.

We see from (4.4) and (4.5) that the functions 𝑛𝑙1/2(𝑠) and πœ†π‘™1/2 are continued up to bounded, analytical functions 𝑁𝑙(𝑧) and Λ𝑙(𝑧) correspondingly, defined in the strip 𝐼={π‘§βˆˆβ„‚1βˆΆβˆ’1/2≀ℑ𝑧≀1/2}. Let us define the functions 𝑁𝑙(𝑧)=𝑁𝑙(π‘§βˆ’π‘–/2) which we shall consider in the strip 𝐼={π‘§βˆˆβ„‚βˆΆ0≀ℑ𝑧≀1}. The operator ξ‚πœ…π‘™1/2 coincides with the operator ξπ‘ξ‚πœ…π‘™1/2 from the family 𝑁{ξ‚πœ…π‘™π›Ό} generated by the function 𝑁𝑙 (see (3.11)). Any other operator of this family acts as multiplication on the function: ̂𝑛𝑙𝛼𝑁(𝑠)=𝑙|||(𝑧)𝑧=𝑠+𝑖𝛼.(4.6) Denote by πœ…π‘™π›Ό the operators πœ…π‘™π›Ό=πœ”βˆ’1ξ‚πœ…π‘π‘™π›Όπœ”,(4.7) acting in 𝐿2(ℝ1+,π‘Ÿ2π‘‘π‘Ÿ).

Note that ξ€·πœ…π‘™π›Όξ€Έβˆ—=πœ…π‘™π‘–βˆ’π›Ό.(4.8) It is convenient to represent the operator 𝑀𝑙 in form of three sequential maps π‘€π‘™βˆΆπ‘’0ξ€·π‘’βˆˆπ”₯0ξ€ΈβŸΆΞ 1/2𝑒0=𝑒1/2βŸΆπœ…π‘™1/2𝑒1/2=𝑣1/2⟢Π1/2𝑣1/2=𝑣1∈π”₯(𝑣),(4.9) where 𝑣=𝑣0,𝑣1/2,𝑣1 are elements of the chain π”₯(𝑣) generated by the function Γ𝑣=ξπ‘π‘™Ξ“π‘’βˆˆπ’’. Note that the chain (4.9) can be rewritten in the following way: 𝑒0ξ€·π‘’βˆˆπ”₯0Ω→Γ𝑒0βŸΆΞ“π‘£=𝑁𝑙Γ𝑒0Ξ©βˆ’1β†’π”₯(𝑣)βŸΆπ‘£1∈π”₯(𝑣).(4.10) From (4.1) and self-adjointness of πœ…π‘™1/2 it follows that the operator 𝑀𝑙 with the domain 𝐷(𝑀𝑙)=𝑉 is symmetric. For any π›Όβˆˆ[0,1], a representation of 𝑀𝑙 similar to (4.1) is valid: 𝑀𝑙=Ξ 1βˆ’π›Όπœ…π‘™π›ΌΞ π›Ό(4.11) as well as decomposition like (4.9).

Let us now describe the domain 𝐷(π‘€βˆ—π‘™)βŠ‡π‘‰ of the operator π‘€βˆ—π‘™ conjugated to 𝑀𝑙. Let π‘”βˆˆπ·(π‘€βˆ—π‘™) be a function from 𝐷(π‘€βˆ—π‘™) and β„Ž=π‘€βˆ—π‘™π‘”βˆˆπΏ2(ℝ1+,π‘Ÿ2π‘‘π‘Ÿ). Then for every π‘’βˆˆπ‘‰=𝐷(𝑀𝑙), we can write 𝑀𝑙=ξ€·πœ…π‘’,𝑔𝑙1ξ€Έ=ξ‚€ξ€·πœ…Ξ π‘’,𝑔Π𝑒,𝑙1ξ€Έβˆ—π‘”ξ‚=𝑒,Ξ πœ…π‘™0𝑔=(𝑒,β„Ž).(4.12) Here we use the representation (4.11) for 𝛼=1 and the equality (4.8). Denote 𝑓(π‘Ÿ)=β„Ž(π‘Ÿ)βˆ’(Ξ πœ…π‘™0𝑔)(π‘Ÿ) and apply the following evident assertion.

Lemma 4.2. Let a measurable function 𝑓(π‘Ÿ) satisfies condition ξ€œβˆž0𝑓(π‘Ÿ)𝑒(π‘Ÿ)π‘Ÿ2π‘‘π‘Ÿ=0,(4.13) for any π‘’βˆˆπ‘‰. Then 𝑓=0.

From this and (4.12), it follows that Ξ πœ…π‘™0𝑔=β„Ž.(4.14) Hence 𝑀0β‰‘πœ…π‘™0π‘”βˆˆπ‘‰,(4.15) and β„Ž=𝑀1∈π”₯(𝑀0) is the final element of the chain π”₯(𝑀0). Thus the domain 𝐷(π‘€βˆ—π‘™) of the operator π‘€βˆ—π‘™ is π·ξ€·π‘€βˆ—π‘™ξ€Έ=ξ€½π‘”βˆˆπΏ2ℝ1+,π‘Ÿ2ξ€Έπ‘‘π‘ŸβˆΆπœ…π‘™0ξ€Ύ.π‘”βˆˆπ‘‰(4.16) In the case when the operator πœ…π‘™0 has the inverse one, (πœ…π‘™0)βˆ’1, which is equivalent to the condition: ̂𝑛𝑙0(𝑠)β‰ 0,foranyπ‘ βˆˆβ„1,(4.17) the following equality is true: π·ξ€·π‘€βˆ—π‘™ξ€Έ=ξ€·πœ…π‘™0ξ€Έβˆ’1𝑉.(4.18) Let ξ‚‹π‘€βˆ—π‘™=πœ”π‘€βˆ—π‘™πœ”βˆ’1 be an operator in 𝐿2(ℝ1) with domain 𝑀𝐷(βˆ—π‘™)=πœ”π·(π‘€βˆ—π‘™). Then for ξ‚‹π‘€Μƒπ‘”βˆˆπ·(βˆ—π‘™), the following representation holds true: ̃𝑔(𝑠)=̂𝑛𝑙0ξ€Έ(𝑠)βˆ’1𝑀0𝑁(𝑠)=𝑙(𝑧)βˆ’1Γ𝑀0||||(𝑧)𝑧=𝑠,(4.19) if condition (4.17) is fulfilled. Here 𝑀0(𝑠)=(πœ”π‘€0)(𝑠) where 𝑀0 is defined in (4.15).

Remarks. (1) Note that the function 𝑁𝑙(𝑧) is invariant with respect to reflection of the complex plane around the point 𝑧=𝑖/2: π‘§βŸΆπ‘§βˆ—=βˆ’π‘§+𝑖.(4.20) Under this reflection, the strip 𝐼 is mapped onto itself; hence, for every zero π‘§βˆˆπΌ(𝑧≠𝑖/2) of the function 𝑁𝑙, there exists another zero, π‘§βˆ—βˆˆπΌ, of 𝑁𝑙 with the same multiplicity. The multiplicity of 𝑧=𝑖/2=π‘§βˆ— is even;

(2) Since 𝑁𝑙(𝑧)β†’2πœ‹2√1βˆ’πœ‡2/4>0 as π‘§β†’βˆž inside 𝐼, the function 𝑁𝑙(𝑧) has finite number of zeros inside 𝐼.

We can now formulate the main criterion of self-adjointness of the operator 𝑀𝑙.

Theorem 4.3. The operator 𝑀𝑙 is self-adjoint if and only if the function 𝑁𝑙(𝑧) has no zeros in the closed strip 𝐼.

Proof. (1) Assume 𝑁𝑙(𝑧)β‰ 0 in the strip 𝐼. Then (𝑁𝑙)βˆ’1(𝑧) is bounded and continuous on 𝐼 and analytical inside 𝐼. Let ξ‚ξ‚‹π‘€Μƒπ‘”βˆˆπ·(βˆ—π‘™). Since ̂𝑛𝑙(𝑠)β‰ 0 for π‘ βˆˆβ„1, the representation (4.19) holds true. Since 𝑁𝑙(𝑧)βˆ’1Γ𝑀0(𝑧)=Ξ“π‘£βˆˆπ’’,π‘£βˆˆπ‘‰,(4.21) the element 𝑔=πœ”βˆ’1Μƒπ‘”βˆˆπ·(π‘€βˆ—π‘™) coincides with π‘£βˆˆπ‘‰, that is, 𝐷(π‘€βˆ—π‘™)=𝑉=𝐷(𝑀𝑙); it means the self-adjointness of 𝑀𝑙;
(2) assume now the function 𝑁𝑙(𝑧) has zeros 𝑧1,…,π‘§π‘˜βˆˆπΌ. Consider first the case when all zeros are lying inside 𝐼 and their multiplicities are equal to 𝑝1,…,π‘π‘˜, respectively. Again, let ξ‚ξ‚‹π‘€Μƒπ‘”βˆˆπ·(βˆ—π‘™). Since ̂𝑛𝑙(𝑠)β‰ 0, the representation (4.19) holds true. The function (𝑁𝑙(𝑧))βˆ’1Γ𝑀0(𝑧) is meromorphic in 𝐼 with poles 𝑧1,…,π‘§π‘˜ having the order 𝑝1,…,π‘π‘˜ respectively. For this function the usual canonical representation [10] is true: 𝑁𝑙(𝑧)βˆ’1Γ𝑀0(𝑧)=𝐿𝑀0(𝑧)+π‘˜ξ“π‘π‘›=1π‘›ξ“π‘š=1π‘π‘š(𝑛)𝑀0ξ€Έξ€·π‘§βˆ’π‘§π‘›ξ€Έπ‘š,(4.22) where 𝐿𝑀0(𝑧) is bounded, continuous function on 𝐼, and analytical inside 𝐼, and the coefficients π‘π‘š(𝑛)=π‘π‘š(𝑛)(𝑀0) depend on 𝑀0.
Lemma  4.4.  The function 𝐿𝑀0(𝑧) in (4.22) belongs to the space 𝒒.
The proof of this lemma is given in The appendix.
From (4.19) and (4.22), for 𝑔=πœ”βˆ’1Μƒπ‘”βˆˆπ·(π‘€βˆ—π‘™), we have 𝑔(π‘Ÿ)=𝑣(π‘Ÿ)+π‘š,π‘›π‘π‘š(𝑛)ξ‚΅πœ”βˆ’11ξ‚΅ξ‚΅β‹…βˆ’π‘§π‘›ξ‚Άπ‘šξ‚Άξ‚Ά(π‘Ÿ),(4.23) where the function π‘£βˆˆπ‘‰ is defined from relation 𝐿𝑀0(𝑧)=Γ𝑣𝑑(𝑧)βˆˆπ’’,π‘š,𝑛(π‘Ÿ)∢=πœ”βˆ’11ξƒ©ξƒ©ξ€·β‹…βˆ’π‘§π‘›ξ€Έξƒͺπ‘šξƒͺ(π‘Ÿ)=π΄π‘š(𝑛)π‘Ÿβˆ’3/2βˆ’π‘‘π‘›+𝑖𝑠𝑛(lnπ‘Ÿ)π‘šβˆ’1πœ’(π‘Ÿ),(4.24) where π΄π‘š(𝑛) is an absolute constant, 𝑧𝑛=𝑠𝑛+𝑖𝑑𝑛, 0<𝑑𝑛<1 and ξ‚»πœ’(π‘Ÿ)=1,π‘Ÿ>1,0,π‘Ÿβ‰€1.(4.25) Since linearly independent functions π‘‘π‘š,π‘›βˆˆπ·(π‘€βˆ—π‘™) do not belong to 𝑉, due to (4.23), they form the basis in the defect subspace 𝔙 of the operator 𝑀𝑙 (see [7]). Since the dimension of the subspace 𝔙 is equal to βˆ‘π‘˜1𝑝𝑛 and the operator 𝑀𝑙 is real, its deficiency indexes 𝑛± are equal and have the form: 𝑛+=π‘›βˆ’=12π‘˜ξ“1𝑝𝑛.(4.26) (It follows from Remarks that the sum βˆ‘π‘˜1𝑝𝑛 is even). Consider now the case when one of the zeros of 𝑁𝑙(𝑧), say, 𝑧0=𝑠0βˆˆβ„1, lies on the boundary of 𝐼 and has multiplicity 𝑝 (in addition, there is a zero π‘§βˆ—0=𝑠0+𝑖). In this case, in a neighborhood of 𝑧0, the function 𝑁𝑙(𝑧) has the form: 𝑁𝑙(𝑧)=π‘§βˆ’π‘§0𝑝𝑄(𝑧),(4.27) where 𝑄(𝑧) is analytic in this neighborhood. Consider the function, 1𝐺(𝑧)=ξ€·ξ€·βˆ’π‘–π‘§βˆ’π‘§0ξ€Έξ€Έ1/31(𝑧+2𝑖)2,(4.28) whereby (βˆ’π‘–π‘€)1/3 for ℑ𝑀>0, we mean the branch of a many-valued function (βˆ’π‘–π‘€)1/3 that takes positive values on the positive part of the imaginary axis. Evidently, the function 𝐺(𝑧) is analytic in the strip 𝐼 and satisfies condition (3.8). However, this function is discontinuous at 𝑧0 and does not belong to 𝒒. In addition, the function 𝑁𝑙(z)𝐺(𝑧) now belongs to 𝒒 as follows from (4.27) and (4.28). Thus ||̃𝑔(𝑠)=𝐺(𝑧)𝑧=π‘ βˆˆξ‚π‘‰=πœ”π‘‰,(4.29) but ̂𝑛𝑙𝑁(𝑠)̃𝑔(𝑠)=𝑙|||(𝑧)𝐺(𝑧)𝑧=π‘ βˆˆξ‚π‘‰.(4.30) Consequently, 𝑔=πœ”Μƒπ‘”βˆˆπ‘‰ but πœ…π‘™0π‘”βˆˆπ‘‰, that is, π‘”βˆˆπ·(π‘€βˆ—π‘™). Thus 𝐷(π‘€βˆ—π‘™)≠𝑉, and the operator 𝑀𝑙 has nonzero deficiency indexes. Theorem 4.3 is proved.

5. The Operators 𝑀𝑙 in the Cases 𝑙=0 and 𝑙=1

Here, we apply Theorem 4.3 to the cases 𝑙=0 and 𝑙=1.

Theorem 5.1. (1) For 𝑙=0, the operator 𝑀𝑙=0  is self-adjoint for any π‘š>0;
(2) the operator 𝑀𝑙=1 is self-adjoint for π‘š>π‘š0 and has nonzero deficiency indexes for π‘šβ‰€π‘š0. In addition, for π‘š<π‘š0 these indexes are equal to (1,1). The constant π‘š0 is a unique zero of (5.4).

Proof. We need the following properties of the functions Λ𝑙=0(𝑧) and Λ𝑙=1(𝑧), π‘§βˆˆπΌ.
Lemma  5.2. (1) For any 𝑙=0,1,2,…  the function Λ𝑙(𝑧)  is invariant with respect to reflection (4.20);
(2) The point 𝑧=𝑖/2βˆˆπΌβ€‰β€‰is a nondegenerate critical point for both functions Λ𝑙=0  and Λ𝑙=1;
(3) These functions take real values on the line: Μ‚πœ‰1/2=𝑖𝑧=𝑠+2,π‘ βˆˆβ„1,(5.1)and on the segment: Μ‚πœ={𝑧=𝑖𝑑,0≀𝑑≀1}.(5.2)Outside the set Μ‚πœ‰π΅=1/2βˆͺΜ‚πœ,  both functions take nonreal values;
(4) the real values of Λ𝑙, 𝑙=0,1,  are between 0  and Λ𝑙Λ(0)=𝑙(𝑖). Every value of Λ𝑙|𝐡—except Λ𝑙(𝑖/2) β€”is taken exactly at two points;
(5) the extreme values of Λ𝑙, 𝑙=0,1,   Λ𝑙Λ(0)=𝑙(𝑖)  are given byΛ𝑙=0√(0)=82πœ‹2πœ‡βˆ’1ξ‚€1sin2πœ‡arcsin2Λ>0,𝑙=1(0)=βˆ’323√2πœ‹2πœ‡βˆ’2sin3ξ‚€12πœ‡arcsin2ξ‚β‰‘βˆ’π‘ž(πœ‡)<0,(5.3)
(6) the function π‘ž(πœ‡)  increases monotonically on the interval  0<πœ‡<2.
The proof of this lemma is given in The appendix.
Corollary  5.3.  (1) The zeros of  𝑁𝑙(𝑧), 𝑙=0,1 can only lie in the set 𝐡;
(2) 𝑁𝑙=0(𝑧)|𝐡>0   for any value of πœ‡, and therefore the operator 𝑀𝑙=0  is self-adjoint for all π‘šβˆˆ(0,2);
(3) The function 𝑁𝑙=1(𝑧)|𝐡  is positive if 2πœ‹2√1βˆ’πœ‡2/4>π‘ž(πœ‡)  and vanishes at some point π‘§βˆˆπ΅ (and also at π‘§βˆ—βˆˆπ΅) if 2πœ‹2√1βˆ’πœ‡2/4β‰€π‘ž(πœ‡).
In Figure 1, the curves corresponding to the functions 2πœ‹2√1βˆ’πœ‡2/4 and π‘ž(πœ‡) are depicted. We see that they intersect at a unique point with abscissa πœ‡0∈(0,2) which satisfies the following equation: 2πœ‹2ξƒŽπœ‡1βˆ’204ξ€·πœ‡=π‘ž0ξ€Έ.(5.4)
Thus, for π‘š>π‘š0=2/πœ‡0βˆ’1 the operator 𝑀𝑙=1 is self-adjoint, and for π‘š<π‘š0 it has deficiency indexes (1,1). For π‘š=π‘š0, the operator 𝑀𝑙=1 is not self-adjoint as well. Theorem 5.1 is proved.

230245.fig.001
Figure 1

Appendix

Proof of Lemma 4.4. The function (𝑁𝑙(𝑧))βˆ’1, π‘§βˆˆπΌ admits the canonical representation (see [10]) 𝑁𝑙(𝑧)βˆ’1=𝑄𝑙(𝑧)+π‘˜ξ“π‘π‘›=1π‘›ξ“π‘š=1π‘Žπ‘š(𝑛)ξ€·π‘§βˆ’π‘§π‘›ξ€Έπ‘š,(A.1) where 𝑧1,…,π‘§π‘˜βˆˆπΌ are zeros of 𝑁𝑙(𝑧) (with multiplicities 𝑝1,…,π‘π‘˜), π‘Žπ‘š(𝑛) are constants, π‘Žπ‘(𝑛)𝑛≠0, and 𝑄𝑙(𝑧) is a bounded, continuous analytic function in 𝐼. From this, it follows that for any π‘£βˆˆπ‘‰, 𝑄𝑙(𝑧)Γ𝑣(𝑧)βˆˆπ’’. Consider some term of the sum (A.1) and write π‘Žπ‘š(𝑛)ξ€·π‘§βˆ’π‘§π‘›ξ€Έπ‘šΞ“π‘£ξƒ©π‘ƒ(𝑧)=(𝑛)π‘š,𝑣(𝑧)+π‘šξ“π‘‘=1𝑐(𝑛)π‘šβˆ’π‘‘ξ€·π‘§βˆ’π‘§π‘›ξ€Έπ‘‘ξƒͺπ‘Žπ‘š(𝑛),(A.2) where 𝑃(𝑛)π‘š,𝑣1(𝑧)=ξ€·π‘§βˆ’π‘§π‘›ξ€Έπ‘šξƒ©Ξ“π‘£(𝑧)βˆ’π‘šξ“π‘‘=1𝑐(𝑛)π‘šβˆ’π‘‘ξ€·π‘§βˆ’π‘§π‘›ξ€Έπ‘šβˆ’π‘‘ξƒͺ,𝑐𝑑(𝑛)=𝑐𝑑(𝑛)(1𝑣)=Γ𝑑!𝑣(𝑑)𝑧𝑛,𝑑=0,1,….(A.3) It is clear that 𝑃(𝑛)π‘š,𝑣(𝑧) is bounded, continuous analytic function in 𝐼. We are going to show that this function belongs to 𝒒. Let π‘‚βˆˆπΌ be a small neighborhood of 𝑧𝑛 and πœ’π‘‚(𝑧) the characteristic function of 𝑂. Obviously, the bounded function πœ’π‘‚π‘ƒ(𝑛)π‘š,𝑣 satisfies condition (3.8). Every term of the sum ξ€·1βˆ’πœ’π‘‚ξ€Έπ‘ƒ(𝑛)π‘š,𝑣Γ(𝑧)=𝑣(𝑧)ξ€·π‘§βˆ’π‘§π‘›ξ€Έπ‘šξ€·1βˆ’πœ’π‘‚ξ€Έβˆ’π‘šξ“π‘‘=1𝑐(𝑛)π‘šβˆ’π‘‘(𝑣)ξ€·π‘§βˆ’π‘§π‘›ξ€Έπ‘‘ξ€·1βˆ’πœ’π‘‚ξ€Έ(A.4) satisfies this condition as well.
Thus for fixed 𝑧𝑛 and π‘£βˆˆπ‘‰, π‘π‘›ξ“π‘š=1π‘Žπ‘š(𝑛)Γ𝑣(𝑧)ξ€·π‘§βˆ’π‘§π‘›ξ€Έπ‘š=𝐾𝑣(𝑛)(𝑧)+𝑝𝑛𝑑=1𝑏𝑑(𝑛)(𝑣)ξ€·π‘§βˆ’π‘§π‘›ξ€Έπ‘‘,(A.5) where 𝐾𝑣(𝑛)(𝑧)=π‘π‘›ξ“π‘š=1π‘Žπ‘š(𝑛)𝑃(𝑛)π‘š,𝑣𝑏(𝑧),(A.6)𝑑(𝑛)(𝑣)=π‘π‘›ξ“π‘š=1π‘Žπ‘š(𝑛)𝑐(𝑛)π‘šβˆ’π‘‘(𝑣),𝑑=1,…,𝑝𝑛.(A.7) Thus, we get the representation (4.22) where 𝐿(𝑀0)(𝑧)=𝑄𝑙(𝑧)Γ𝑀0(𝑧)+π‘˜ξ“π‘›=1𝐾𝑀(𝑛)0(𝑧)βˆˆπ’’,(A.8) and the coefficients π‘π‘š(𝑛)(𝑀0) are given by formula (A.7). Lemma 4.4 is proved.

Proof of Lemma 5.2. (1) It is more convenient to consider the functions 𝑁𝑙(𝑧) and Λ𝑙(𝑧) in the strip 𝐼={π‘§βˆΆ|ℑ𝑧|<1/2} instead of the functions 𝑁𝑙(𝑧) and Λ𝑙(𝑧) in the strip 𝐼. Similarly, instead of the reflection π‘§β†’π‘§βˆ— we consider the reflection π‘§β†’βˆ’π‘§ around the point 𝑧0=0. It is clear that the functions Λ𝑙(𝑧), 𝑙=0,1,2,… are invariant with respect to the change π‘§β†’βˆ’π‘§, and it means the invariance of Λ𝑙 with respect to reflection (4.20);
(2) it follows from (4.5) that 𝑧=0 is a nondegenerated critical point of Λ𝑙=0 and Λ𝑙=1, if we note that 0<𝑣(π‘₯)β‰€πœ‹/2. Correspondingly, 𝑧=𝑖/2 is a nondegenerated critical point for Λ𝑙(𝑧), 𝑙=0,1. The real axis πœ‰0={𝑧=𝑠;π‘ βˆˆβ„1} coincides with the saddle-point line at 𝑧=0 (see [10]) for Λ𝑙=0 and βˆ’Ξ›π‘™=1. More precisely, these functions take real values on πœ‰0 and decrease monotonically to zero as |𝑠| increases from zero to infinity. On the contrary, Λ𝑙=0 and βˆ’Ξ›π‘™=1 increase monotonically along imaginary axis as |𝑑| increases from zero to 1/2. The monotonicity of Λ𝑙=0 along real axis follows from (4.5), equality 𝑃0(π‘₯)≑1, and inequality ξ‚΅ch(𝑣(π‘₯)𝑠)ξ‚Άch((πœ‹/2)𝑠)ξ…žπ‘ πœ‹<βˆ’2sh(πœ‹/2βˆ’π‘£(π‘₯))𝑠(ch((πœ‹/2)𝑠))2<0,(A.9) for 𝑠>0 and a similar inequality for 𝑠<0. The proof of monotonicity of Λ𝑙=1 along real axis, and also monotonicity of both functions along imaginary axis is analogous if we note that 𝑃𝑙=1(π‘₯)≑π‘₯ on (0,1). Thus the functions Λ𝑙, 𝑙=0,1, take all values between 0 and Λ𝑙(𝑖/2)=Λ𝑙(βˆ’π‘–/2) and every value except Λ𝑙(0) which is taken exactly twice;
(3) we will show now that the values of functions Λ𝑙(𝑧), 𝑙=0,1, on the set 𝐼⧡𝐡 are nonreal. Let us represent this set as a union of four sets, 𝐼𝑖, 𝑖=1,2,3,4 as shown in Figure 2.
We consider the case 𝑙=0; the case 𝑙=1 is similar. Figure 3 shows the disposition of lines of levels for function 𝐾0(𝑧)=β„œΞ›π‘™=0(𝑧) which pass through the points 𝑖 and βˆ’π‘– between lines 𝛽 and π›½βˆ—, 𝛽={π‘§βˆΆπΎ0(𝑧)=0,ℑ𝑧>0}, π›½βˆ—={π‘§βˆΆπΎ0(𝑧)=0,ℑ𝑧<0}.
All these lines have common tangents at points 𝑖 and βˆ’π‘–, and the line 𝛽 (resp. π›½βˆ—) lies above (resp., below) the strip 𝐼. The picture represented in Figure 3 is obtained by detailed study of the explicit formula for Λ𝑙=0: Λ𝑙=0(𝑧)=4πœ‹2πœ‡sh(𝑧arcsin(πœ‡/2)),𝑧ch(π‘§β‹…πœ‹/2)(A.10) together with the proof that the lines 𝛽 and π›½βˆ— do not intersect the strip 𝐼. This proof is given below.
From Figure 3, we see that the set 𝐼1 lies inside the shaded domain π‘ˆ that is bounded by the real semiaxis πœ‰+0={π‘§βˆΆπ‘§=𝑠,𝑠>0}, the segment (0,𝑖/2) on the imaginary axis and the part of line 𝛽 which lies in the right half-plane. From (A.10), it is easy to see that the function 𝑀=Λ𝑙=0(𝑧) maps the boundary πœ•π‘ˆ of the domain π‘ˆ into the boundary of the right lower quadrant 𝑀={π‘€βˆΆβ„œπ‘€>0,ℑ𝑀<0} of the plain 𝑀. Hence, the domain π‘ˆ is mapped inside this quadrant, that is, all values of the function Λ𝑙=0 in π‘ˆ are nonreal. It means the absence of real values of Λ𝑙=0 in 𝐼1. For the domains 𝐼2, 𝐼3, and 𝐼4, the proof is similar. Let us now prove that 𝛽 and π›½βˆ— do not intersect the line πœ‰1/2. It is sufficient to prove that β„œΞ›π‘™=0>0 on the line πœ‰1/2={π‘§βˆΆπ‘§=𝑠+𝑖/2,π‘ βˆˆβ„1} or, which is the same, that β„œch(𝑧𝑣(π‘₯))||||ch(π‘§πœ‹/2)𝑧=𝑠+𝑖/2>0,(A.11) for any π‘ βˆˆβ„1 and π‘₯∈(0,1). Write []ch(𝑠+𝑖/2)𝑣(π‘₯)[]=ch(𝑠+𝑖/2)πœ‹/2ch(𝑠𝑣(π‘₯))cos(𝑣(π‘₯)/2)+𝑖sh(𝑠𝑣(π‘₯))sin(𝑣(π‘₯)/2)ch(π‘ πœ‹/2)cos(πœ‹/4)+𝑖sh(π‘ πœ‹/2)sin(πœ‹/4)=𝐷(𝑠,π‘₯).(A.12)
Let 𝑠>0. Then the values of numerator and denominator of 𝐷(𝑠,π‘₯) lie in the right upper quadrant of a complex plain, and hence βˆ’πœ‹/2<arg𝐷(𝑠,π‘₯)<πœ‹/2, that is, β„œπ·(𝑠,π‘₯)>0. Similarly (A.11) can be proved in the case 𝑠<0 and for Λ𝑙=0|𝑧=π‘ βˆ’π‘–/2;
(4) let us find the values Λ𝑙(𝑖/2), 𝑙=0,1:(I) the case 𝑙=0: Λ𝑙=0(𝑖/2)=2πœ‹2ξ€œ10cos(𝑣(π‘₯)/2)cos𝑣(π‘₯)cos(πœ‹/4)𝑑π‘₯.(A.13) After the change 𝑣(π‘₯)=πœ‰, the integral (A.13) becomes 4√2πœ‹2πœ‡ξ€œ0arcsinπœ‡/2ξ‚΅πœ‰cos2ξ‚Ά8βˆšπ‘‘πœ‰=2πœ‡πœ‹2ξ‚€1sin2πœ‡arcsin2;(A.14)(II)The case 𝑙=1: Λ𝑙=1𝑖2=βˆ’2πœ‹2ξ€œ10π‘₯sin(𝑣(π‘₯)/2)𝑑π‘₯.cos𝑣(π‘₯)sin(πœ‹/4)(A.15) The same change 𝑣(π‘₯)=πœ‰ reduces to the integral βˆ’8√2πœ‹2πœ‡2ξ€œ0arcsinπœ‡/2ξ‚΅πœ‰sinπœ‰sin2ξ‚Άβˆšπ‘‘πœ‰=βˆ’3223πœ‹2πœ‡2sin3ξ‚€12πœ‡arcsin2;(A.16)
(5) let us show that the function: π‘ž(πœ‡)=2πœ‹2ξ€œ10π‘₯sin(𝑣(π‘₯)/2)cos𝑣(π‘₯)sin(πœ‹/4)𝑑π‘₯(A.17) decreases monotonically as πœ‡ changes from 0 to 2. We have ξ‚΅sin(𝑣(π‘₯)/2)ξ‚Άcos𝑣(π‘₯)ξ…žπœ‡β‰₯0(A.18) because the numerator of (A.18) increases, while the denominator decreases with the growth of πœ‡. This implies that π‘žβ€²(πœ‡)β‰₯0,(A.19) that is, π‘ž(πœ‡) increases monotonically. Lemma 5.2 is proved.

230245.fig.002
Figure 2
230245.fig.003
Figure 3

Acknowledgment

This work is supported by RFBR Grant 11-01-00485a.

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