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.

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.

Acknowledgment

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