Nonself-Adjoint Second-Order Difference Operators in Limit-Circle Cases

Abstract

We consider the maximal dissipative second-order difference (or discrete Sturm-Liouville) operators acting in the Hilbert space ℓ²_𝑤(ℤ) (ℤ:={0,±1,±2,…}), that is, the extensions of a minimal symmetric operator with defect index (2,2) (in the Weyl-Hamburger limit-circle cases at ±∞). We investigate two classes of maximal dissipative operators with separated boundary conditions, called “dissipative at −∞” and “dissipative at ∞.” In each case, we construct a self-adjoint dilation of the maximal dissipative operator and its incoming and outgoing spectral representations, which make it possible to determine the scattering matrix of the dilation. We also establish a functional model of the maximal dissipative operator and determine its characteristic function through the Titchmarsh-Weyl function of the self-adjoint operator. We prove the completeness of the system of eigenvectors and associated vectors of the maximal dissipative operators.

1. Introduction

Method of contour integration of the resolvent is one of the general methods of the spectral analysis of nonself-adjoint (dissipative) operators. It is related to a fine estimate of the resolvent on expanding contours which separates the spectrum. The feasibility of this method is restricted to weak perturbations of self-adjoint operators and operators having sparse discrete spectrum. Since there are no asymptotics of the solutions for a wide class of singular problems, this method cannot be applied properly.

It is well known [1–4] that the theory of dilations with application of functional models gives an adequate approach to the spectral theory of dissipative (contractive) operators. In this theory, a key role is played by the characteristic function, which carries the full information on the spectral properties of the dissipative operator. Thus, the dissipative operator becomes the model in the incoming spectral representation of the dilation. The completeness problem of the system of eigenvectors and associated vectors is solved by the factorization of the characteristic function. The computation of the characteristic functions of dissipative operators is preceded by the construction and investigation of the self-adjoint dilation and the corresponding scattering problem, in which the characteristic function is realized as the scattering matrix [5]. The adequacy of this approach for dissipative Jacobi operators and second-order difference (or discrete Sturm-Liouville) operators has been indicated in [6–9].

In this paper, we consider the maximal dissipative second-order difference (or discrete Sturm-Liouville) operators acting in the Hilbert space ℓ²_𝑤(ℤ), that is the extensions of a minimal symmetric operator with defect index (2,2) (in the Weyl-Hamburger limit-circle cases at ±∞). We investigate two classes of maximal dissipative operators with separated boundary conditions, called “dissipative at −∞” and “dissipative at ∞.” In each of these cases we construct a self-adjoint dilation of the maximal dissipative operator and its incoming and outgoing spectral representations, which make it possible to determine the scattering matrix of the dilation according to the scheme of Lax and Phillips [5]. By means of the incoming spectral representation, we establish a functional model of the maximal dissipative operator and construct its characteristic function using the Titchmarsh-Weyl function of the self-adjoint operator. Finally, on the basis of the results obtained for the characteristic functions, we prove the theorems on completeness of the system of eigenvectors and associated vectors (or root vectors) of the maximal dissipative second-order difference operators.

2. Preliminaries

Let 𝑦={𝑦_𝑛} be a sequence of complex numbers 𝑦_𝑛(𝑛∈ℤ) and ℓ₁𝑦 denote the sequence with components (ℓ₁𝑦)_𝑛. We consider the following second-order difference (or discrete Sturm-Liouville) equation on the whole line:ℓ₁𝑦_𝑛∶=−𝑎_𝑛−1𝑦_𝑛−1+𝑏_𝑛𝑦_𝑛−𝑎_𝑛𝑦_𝑛+1=𝜆𝑤_𝑛𝑦_𝑛, (2.1) where 𝜆 is a complex spectral parameter, 𝑤_𝑛>0,𝑎_𝑛≠0, and 𝑎_𝑛,𝑏_𝑛∈ℝ:=(−∞,∞), 𝑛∈ℤ.

If we let 𝑝_𝑛=𝑎_𝑛,𝑞_𝑛=𝑏_𝑛−𝑎_𝑛−𝑎_𝑛−1, and Δ𝑥_𝑛=𝑥_𝑛+1−𝑥_𝑛, (2.1) can be written in Sturm-Liouville form as follows:−Δ𝑝_𝑛−1Δ𝑦_𝑛−1+𝑞_𝑛𝑦_𝑛=𝜆𝑤_𝑛𝑦_𝑛,𝑛∈ℤ. (2.2)

For arbitrary sequences 𝑦={𝑦_𝑛} and 𝑧={𝑧_𝑛},𝑛∈ℤ, we denote by [𝑦,𝑧] the sequence with components [𝑦,𝑧]_𝑛 defined as:𝑦,𝑧_𝑛∶=𝑎_𝑛𝑦_𝑛𝑧_𝑛+1−𝑦_𝑛+1𝑧_𝑛,𝑛∈ℤ. (2.3) Let 𝑚,𝑛∈ℤ with 𝑛<𝑚. Then we have the Green’s formula:^𝑚_𝑗=𝑛ℓ₁𝑦_𝑗𝑧_𝑗−𝑦_𝑗ℓ₁𝑧_𝑗=𝑦,𝑧_𝑚−𝑦,𝑧_𝑛−1. (2.4)

For any sequence 𝑦={𝑦_𝑛}, let ℓ𝑦 denote the sequence with components (ℓ𝑦)_𝑛 given by (ℓ𝑦)_𝑛=(1/𝑤_𝑛)(ℓ₁𝑦)_𝑛, 𝑛∈ℤ. We denote by ℓ²_𝑤(ℤ)(𝑤∶={𝑤_𝑛},𝑛∈ℤ) the Hilbert space of all complex sequences 𝑦={𝑦_𝑛}, 𝑛∈ℤ such that ∑^∞_𝑛=−∞𝑤_𝑛|𝑦_𝑛|²<∞, with the inner product (𝑦,𝑧)=∑^∞_𝑛=−∞𝑤_𝑛𝑦_𝑛𝑧_𝑛. Next, we denote by 𝐷 the set of all vectors 𝑦∈ℓ²_𝑤(ℤ) such that ℓ𝑦∈ℓ²_𝑤(ℤ). We define a maximal operator 𝐿 on 𝐷 by setting 𝐿𝑦=ℓ𝑦.

It follows from Green’s formula (2.4) that the limits [𝑦,𝑧]_∞=lim_𝑛→∞[𝑦,𝑧]_𝑛 and [𝑦,𝑧]_−∞=lim_{𝑛→−∞}[𝑦,𝑧]_𝑛 exist and are finite for arbitrary vectors 𝑦,𝑧∈𝐷. Therefore, taking the limit as 𝑛→−∞ and 𝑚→∞ in (2.4), for all 𝑦,𝑧∈𝐷, we have𝐿𝑦,𝑧−𝑦,𝐿𝑧=𝑦,𝑧_∞−𝑦,𝑧_−∞. (2.5)

Denote by 𝐿₀ the closure of the symmetric operator 𝐿^′₀ defined by 𝐿^′₀𝑦=𝐿𝑦 on the linear set 𝐷^′₀ of finite sequences (i.e., vectors having only finitely many nonzero components) 𝑦={𝑦_𝑛}(𝑛∈ℤ). The minimal operator 𝐿₀ is symmetric and 𝐿^∗₀=𝐿. The computation of the defect index of 𝐿₀ can be reduced to the computation of the defect index for the half-line case. In fact, ℓ²_𝑤(ℤ) is the orthogonal sum of the space ℓ²_𝑤(ℕ₋), (ℕ₋={−1,−2,−3,…}) and ℓ²_𝑤(ℕ₀) (ℕ₀={0,1,2,…}) which are imbedded in the natural way in ℓ²_𝑤(ℤ). Denote by 𝐿⁻₀(𝐿₋) and 𝐿⁺₀(𝐿₊) the minimal (maximal) operators generated by ℓ₋ and ℓ₊ in the spaces ℓ²_𝑤(ℕ₋) and ℓ²_𝑤(ℕ₀), respectively, and 𝐷^∓₀(𝐷_∓) is a domain of 𝐿^∓₀(𝐿_∓), where (ℓ_∓𝑦)_𝑛∶=(ℓ𝑦)_𝑛, 𝑛∈ℤ0𝑥00082⧵{−1,0}, (ℓ₋𝑦)₋₁∶=(1/𝑤₋₁)(−𝑎₋₂𝑦₋₂+𝑏₋₁𝑦₋₁), (ℓ₊𝑦)₀∶=(1/𝑤₀)(𝑏₀𝑦₀−𝑎₀𝑦₁). Then it is easy to see that the equality def𝐿₀=def𝐿⁻₀+def𝐿⁺₀ is satisfied for the defect number def𝐿₀:=dim{(𝐿₀−𝜆𝐼)𝐷(𝐿₀)}^⟂, Im𝜆≠0, of 𝐿₀. This shows that the defect index of 𝐿₀ has the form (𝑘,𝑘), where 𝑘=0,1 or 2. For defect index (0,0) the operator 𝐿₀ is self-adjoint, that is, 𝐿^∗₀=𝐿₀=𝐿.

Assume that the symmetric operator 𝐿₀ has defect index (2,2). There are several sufficient conditions that guarantee Weyl-Hamburger limit-circle cases at ±∞ (i.e., the operator 𝐿₀ has defect index (2,2), see [10–17]). The domain of 𝐿₀ consists of precisely those vectors 𝑦∈𝐷 satisfying the condition𝑦,𝑧_∞−𝑦,𝑧_−∞=0,∀𝑧∈𝐷. (2.6)

Denote by 𝑃⁽¹⁾(𝜆)={𝑃_𝑛⁽¹⁾(𝜆)} and 𝑃⁽²⁾(𝜆)={𝑃_𝑛⁽²⁾(𝜆)},𝑛∈ℤ the solutions of (2.1) satisfying the initial conditions:𝑃⁽¹⁾₋₁(𝜆)=0,𝑃₀⁽¹⁾(𝜆)𝑃=1,₍₂₎₋₁(𝜆)1=−𝑎₋₁,𝑃₀⁽²⁾(𝜆)=0. (2.7)

The Wronskian of the two solutions 𝑦={𝑦_𝑛} and 𝑧={𝑧_𝑛},𝑛∈ℕ of (2.1) is defined as 𝑊_𝑛(𝑦,𝑧)∶=𝑎_𝑛(𝑦_𝑛𝑧_𝑛+1−𝑦_𝑛+1𝑧_𝑛), so that 𝑊_𝑛(𝑦,𝑧)=[𝑦,𝑧]_𝑛,𝑛∈ℤ. The Wronskian of the two solutions of (2.1) is independent of 𝑛, and the two solutions of this equation are linearly independent if and only if their Wronskian is nonzero. It follows from the conditions (2.7) and the constancy of the Wronskian that 𝑊_𝑛(𝑃⁽¹⁾,𝑃⁽²⁾)=1,𝑛∈ℤ. Consequently, 𝑃⁽¹⁾(𝜆) and 𝑃⁽²⁾(𝜆) form a fundamental system of solutions of (2.1), and 𝑃⁽¹⁾(𝜆), 𝑃⁽²⁾(𝜆)∈ℓ²_𝑤(ℤ) for all 𝜆∈ℂ. The theory of difference equations can be seen in [18, 19].

Let 𝑢=𝑃⁽¹⁾(0) and 𝑣=𝑃⁽²⁾(0). Since the vectors 𝑢={𝑢_𝑛} and 𝑣={𝑣_𝑛}(𝑛∈ℤ) are real valued and [𝑢,𝑣]_𝑛=1(𝑛∈ℤ), the following assertion can be verified easily using (2.3).

Lemma 2.1. For arbitrary vectors 𝑦={𝑦_𝑛}∈𝐷 and 𝑧={𝑧_𝑛}∈𝐷, one has the equality: 𝑦,𝑧_𝑛=𝑦,𝑢_𝑛[𝑧,𝑣]_𝑛−𝑦,𝑣_𝑛[𝑧,𝑢]_𝑛,(𝑛∈ℤ∪{−∞,∞}). (2.8)

The domain 𝐷₀ of the operator 𝐿₀ consists of precisely those vectors 𝑦∈𝐷 satisfying the boundary conditions:𝑦,𝑢_−∞=𝑦,𝑣_−∞=𝑦,𝑢_∞=𝑦,𝑣_∞=0. (2.9)

Let us consider the following linear maps from 𝐷 into ℂ²Γ₁𝑦=𝑦,𝑣_−∞𝑦,𝑢_∞,Γ₂𝑦=𝑦,𝑢_−∞𝑦,𝑣_∞,𝑦∈𝐷. (2.10) Then we have the following result (see [8]).

Theorem 2.2. For any contraction 𝐾 in ℂ² the restriction of the operator 𝐿 to the set of vectors 𝑦∈𝐷 satisfying the boundary conditions (𝐾−𝐼)Γ₁𝑦+𝑖(𝐾+𝐼)Γ₂𝑦=0, (2.11) or (𝐾−𝐼)Γ₁𝑦−𝑖(𝐾+𝐼)Γ₂𝑦=0 (2.12) is, respectively, a maximal dissipative or a accretive extension of the operator 𝐿₀. Conversely, every maximal dissipative (accretive) extension of 𝐿₀ is the restriction of 𝐿 to the set of vectors 𝑦∈𝐷 satisfying (2.11) (2.12), and the contraction 𝐾 is uniquely determined by the extension. These conditions give a self-adjoint extension if and only if 𝐾 is unitary. In the latter case (2.11) and (2.12) are equivalent to the condition (cos𝑆)Γ₁𝑦−(sin𝑆)Γ₂𝑦=0, where 𝑆 is a self-adjoint (Hermitian matrix) operator in ℂ². The general form of dissipative and accretive extensions of the operator 𝐿₀ is given by the conditions 𝐾Γ₁𝑦+𝑖Γ₂𝑦=Γ₁𝑦−𝑖Γ₂𝑦,Γ₁𝑦+𝑖Γ₂(𝐾),𝐾Γ𝑦∈𝐷₁𝑦−𝑖Γ₂𝑦=Γ₁𝑦+𝑖Γ₂𝑦,Γ₁𝑦−𝑖Γ₂𝑦∈𝐷(𝐾), (2.13) respectively, where 𝐾 is a linear operator in ℂ² with ‖𝐾𝑓‖≤‖𝑓‖,𝑓∈𝐷(𝐾)⊆ℂ². The general form of symmetric extensions is given by the formulae (2.13), where 𝐾 is an isometric operator.
In particular, if 𝐾 is a diagonal matrix, the boundary conditions 𝑦,𝑣_−∞−ℎ₁𝑦,𝑢_−∞=0,𝑦,𝑢_∞−ℎ₂𝑦,𝑣_∞=0, (2.14)(2.15) with Imℎ₁≥0 or ℎ₁=∞, and Imℎ₂≥0 or ℎ₂=∞(Imℎ₁≤0 or ℎ₁=∞, and Imℎ₂≤0 or ℎ₂=∞) describe all the maximal dissipative (maximal accretive) extensions of 𝐿₀ with separated boundary conditions. The self-adjoint extensions of 𝐿₀ are obtained precisely when Imℎ₁=0 or ℎ₁=∞, and Imℎ₂=0 or ℎ₂=∞. Here for ℎ₁=∞(ℎ₂=∞)  condition (2.14) (2.15) should be replaced by [𝑦,𝑢]_−∞=0([𝑦,𝑣]_∞=0).

In what follows, we will study the dissipative operators 𝐿^∓_ℎ1ℎ2 generated by ℓ and the boundary conditions (2.14) and (2.15) of two types: “dissipative at −∞,” that is, when either Imℎ₁>0 and Imℎ₂=0  or ℎ₂=∞; “dissipative at ∞,” when Imℎ₁=0  or ℎ₁=∞ and Imℎ₂>0.

3. Self-Adjoint Dilations of the Maximal Dissipative Operators

In order to construct a self-adjoint dilation of the maximal dissipative operator 𝐿⁻_ℎ1ℎ2 in the case of “dissipative at −∞” (i.e., Imℎ₁>0 and Imℎ₂=0 or ℎ₂=∞), we associate with 𝐻∶=ℓ²_𝑤(ℤ) the “incoming” and “outgoing” channels 𝐷₋∶=𝐿²(−∞,0) and 𝐷₊∶=𝐿²(0,∞), we form the orthogonal sum ℋ=𝐷₋⊕𝐻⊕𝐷₊ and we call it the main Hilbert space of the dilation. In the space ℋ, we consider the operator ℒ⁻_ℎ1ℎ2 generated by the expressionℒ𝜑₋,𝑦,𝜑₊=𝑖𝑑𝜑₋𝑑𝜉,ℓ𝑦,𝑖𝑑𝜑₊𝑑𝜍 (3.1) on the set 𝐷(ℒ⁻_ℎ1ℎ2) of vectors ⟨𝜑₋,𝑦,𝜑₊⟩ satisfying the conditions 𝜑₋∈𝑊¹₂(−∞,0), 𝜑₊∈𝑊¹₂(0,∞), 𝑦∈𝐷 and𝑦,𝑣_−∞−ℎ₁𝑦,𝑢_−∞=𝛼𝜑₋(0),𝑦,𝑣_−∞−ℎ₁𝑦,𝑢_−∞=𝛼𝜑₊(0),𝑦,𝑢_∞−ℎ₂𝑦,𝑣_∞=0, (3.2) where 𝑊¹₂ denotes the Sobolev space and 𝛼²∶=2Imℎ₁, 𝛼>0.

Theorem 3.1. The operator ℒ⁻_ℎ1ℎ2 is self-adjoint in ℋ and it is a self-adjoint dilation of the maximal dissipative operator 𝐿⁻_ℎ1ℎ2.

Proof. We assume that 𝑓,𝑔∈𝐷(ℒ⁻_ℎ1ℎ2) with 𝑓=⟨𝜑₋,𝑦,𝜑₊⟩ and 𝑔=⟨𝜓₋,𝑧,𝜓₊⟩. Then using integration by parts and (3.1), we obtain ℒ⁻_ℎ1ℎ2𝑓,𝑔_ℋ=^∞₀𝑖𝜑^′₋𝜓₋𝑑𝜉+𝐿𝑦,𝑧_𝐻+^∞₀𝑖𝜑^′₊𝜓₊𝑑𝜉=𝑖𝜑₋(0)𝜓₋(0)−𝑖𝜑₊(0)𝜓₊(0)+𝑦,𝑧_∞−𝑦,𝑧_−∞+𝑓,ℒ⁻_ℎ1ℎ2𝑔_ℋ. (3.3) If we use the boundary conditions (3.2) for the components of the vectors 𝑓,𝑔 and Lemma 2.1, we see that 𝑖𝜑₋(0)𝜓₋(0)−𝑖𝜑₊(0)𝜓₊(0)+[𝑦,𝑧]_∞−[𝑦,𝑧]_−∞=0. Thus, ℒ⁻_ℎ1ℎ2 is symmetric. Therefore, to prove that ℒ⁻_ℎ1ℎ2 is self-adjoint, it is sufficient to show that (ℒ⁻_ℎ1ℎ2)^∗⊆ℒ⁻_ℎ1ℎ2. Let us take 𝑔=⟨𝜓₋,𝑧,𝜓₊⟩∈𝐷((ℒ⁻_ℎ1ℎ2)^∗) and let (ℒ⁻_ℎ1ℎ2)^∗𝑔=𝑔^∗=⟨𝜓^∗₋,𝑧^∗,𝜓^∗₊⟩∈ℋ so that (ℒ⁻_ℎ1ℎ2𝑓,𝑔)_ℋ=𝑓,𝑔^∗_ℋ,∀𝑓∈𝐷ℒ⁻_ℎ1ℎ2. (3.4) If we choose the components of 𝑓∈𝐷(ℒ⁻_ℎ1ℎ2) properly in (3.4), it becomes easy to show that 𝜓₋∈𝑊¹₂(−∞,0), 𝜓₊∈𝑊¹₂(0,∞), 𝑧∈𝐷, and 𝑔^∗=ℒ𝑔, where the operator ℒ is given by (3.1). As a result, (3.4) takes the form (ℒ𝑓,𝑔)_ℋ=(𝑓,ℒ𝑔)_ℋ, for all 𝑓∈𝐷(ℒ⁻_ℎ1ℎ2). Hence, in the bilinear form (ℒ𝑓,𝑔)_ℋ, the sum of the integral terms must be equal to zero: 𝑖𝜑₋(0)𝜓₋(0)−𝑖𝜑₊(0)𝜓₊(0)+𝑦,𝑧_∞−𝑦,𝑧_−∞=0, (3.5) for all 𝑓=⟨𝜑₋,𝑦,𝜑₊⟩∈𝐷(ℒ⁻_ℎ1ℎ2). In addition, if we solve the boundary conditions (3.2) for [𝑦,𝑢]_−∞ and [𝑦,𝑣]_−∞, we get 𝑦,𝑢_−∞𝑖=−𝛼𝜑₊(0)−𝜑₋(0),𝑦,𝑣_−∞=𝛼𝜑₋(0)−𝑖ℎ₁𝛼𝜑₊(0)−𝜑₋(0). (3.6) It follows from Lemma 2.1 and (3.6) that (3.5) is equivalent to the following equality: 𝑖𝜑₋(0)𝜓₋(0)−𝑖𝜑₊(0)𝜓₊(0)=𝑦,𝑧_−∞−𝑦,𝑧_∞𝑖=−𝛼𝜑₊(0)−𝜑₋(0)[𝑧,𝑣]_−∞−𝛼𝜑₋(0)−𝑖ℎ₁𝛼²𝜑₊(0)−𝜑₋(0)[𝑧,𝑢]_−∞−𝑦,𝑢_∞[𝑧,𝑣]_∞+𝑦,𝑣_∞[𝑧,𝑢]_∞𝑖=−𝛼𝜑₊(0)−𝜑₋(0)[𝑧,𝑣]_−∞−𝛼𝜑₋(0)−𝑖ℎ₁𝛼²𝜑₊(0)−𝜑₋(0)[𝑧,𝑢]_−∞+[𝑧,𝑢]_∞−ℎ₂[𝑧,𝑣]_∞𝑦,𝑣_∞. (3.7) Note that the values 𝜑_±(0) can be any complex numbers. Therefore, when we compare the coefficients of 𝜑_±(0) on the left and right of the last equality we see that the vector 𝑔=⟨𝜓₋,𝑧,𝜓₊⟩ satisfies the boundary conditions [𝑧,𝑣]_−∞−ℎ₁[𝑧,𝑢]_−∞=𝛼𝜓₋(0), [𝑧,𝑣]_−∞−ℎ₁[𝑧,𝑢]_−∞=𝛼𝜓₊(0), [𝑧,𝑢]_∞−ℎ₂[𝑧,𝑣]_∞=0. Consequently, we obtain (ℒ⁻_ℎ1ℎ2)^∗⊆ℒ⁻_ℎ1ℎ2, and hence ℒ⁻_ℎ1ℎ2=(ℒ⁻_ℎ1ℎ2)^∗.
The self-adjoint operator ℒ⁻_ℎ1ℎ2 generates in ℋ a unitary group 𝑈⁻_𝑡=exp[𝑖ℒ⁻_ℎ1ℎ2𝑡],  𝑡∈ℝ. Let 𝑃∶ℋ→𝐻 and 𝑃₁∶𝐻→ℋ denote the mappings acting according to the formulas 𝑃∶⟨𝜑₋,𝑦,𝜑₊⟩→𝑦 and 𝑃₁∶𝑦→⟨0,𝑦,0⟩. Let 𝑍⁻_𝑡=𝑃𝑈⁻_𝑡𝑃₁𝑡≥0. The family {𝑍⁻_𝑡}, 𝑡≥0, of operators is a strongly continuous semigroup of completely nonunitary contractions on 𝐻. (We recall that the linear bounded operator 𝐴 acting in the Hilbert space 𝐻 is called completely nonunitary if invariant subspace 𝑀⊆𝐻 (𝑀≠{0} of operator 𝐴 whose restriction to 𝑀 is unitary, does not exist). Let us denote by 𝐴_ℎ1ℎ2 the generator of this semigroup: 𝐴_ℎ1ℎ2𝑦=lim_𝑡→+0(𝑖𝑡)⁻¹(𝑍⁻_𝑡𝑦−𝑦). The domain of 𝐴_ℎ1ℎ2 consists of all the vectors for which the limit exists. 𝐴_ℎ1ℎ2 is a maximal dissipative operator. The operator ℒ⁻_ℎ1ℎ2 is called the self-adjoint dilation of 𝐴_ℎ1ℎ2 [1–4]. We show that 𝐴_ℎ1ℎ2=𝐿⁻_ℎ1ℎ2, and thus ℒ⁻_ℎ1ℎ2 is a self-adjoint dilation of 𝐿⁻_ℎ1ℎ2. To do this, we first verify the equality [1–4]: 𝑃ℒ⁻_ℎ1ℎ2−𝜆𝐼⁻¹𝑃₁𝑦=𝐿⁻_ℎ1ℎ2−𝜆𝐼⁻¹𝑦,𝑦∈𝐻,Im𝜆<0. (3.8) Denote (ℒ⁻_ℎ1ℎ2−𝜆𝐼)⁻¹𝑃₁𝑦=𝑔=⟨𝜓₋,𝑧,𝜓₊⟩. Then (ℒ⁻_ℎ1ℎ2−𝜆𝐼)𝑔=𝑃₁𝑦, and hence 𝐿𝑧−𝜆𝑧=𝑦, 𝜓₋(𝜉)=𝜓₋(0)𝑒^{−𝑖𝜆𝜉} and 𝜓₊(𝜍)=𝜓₊(0)𝑒^{−𝑖𝜆𝜍}. Since 𝑔∈𝐷(ℒ⁻_ℎ1ℎ2), and hence, 𝜓₋∈𝐿²(−∞,0); it follows that 𝜓₋(0)=0, and, consequently, 𝑧 satisfies the boundary conditions [𝑧,𝑣]_−∞−ℎ₁[𝑧,𝑢]_−∞=0, [𝑧,𝑢]_∞−ℎ₂[𝑧,𝑣]_∞=0. Therefore, 𝑧∈𝐷(𝐿⁻_ℎ1ℎ2) and since a point 𝜆 with Im𝜆<0 cannot be an eigenvalue of a dissipative operator, it follows that 𝑧=(𝐿⁻_ℎ1ℎ2−𝜆𝐼)⁻¹𝑦. Note that 𝜓₊(0) is obtained from the formula 𝜓₊(0)=𝛼⁻¹([𝑧,𝑣]_−∞−ℎ₁[𝑧,𝑢]_−∞). Then ℒ⁻_ℎ1ℎ2−𝜆𝐼⁻¹𝑃₁𝑦=0,𝐿⁻_ℎ1ℎ2−𝜆𝐼⁻¹𝑦,𝛼⁻¹[𝑧,𝑣]_−∞−ℎ₁[𝑧,𝑢]_−∞𝑒^{−𝑖𝜆𝜍}, (3.9) for 𝑦∈𝐻 and Im𝜆<0. By applying 𝑃, one can obtain (3.8).
Now, it is not difficult to show that 𝐴_ℎ=𝐿⁻_ℎ. In fact, it follows from (3.8) that 𝐿⁻_ℎ1ℎ2−𝜆𝐼⁻¹=𝑃ℒ⁻_ℎ1ℎ2−𝜆𝐼⁻¹𝑃₁=−𝑖𝑃^∞₀𝑈⁻_𝑡𝑒^{−𝑖𝜆𝑡}𝑑𝑡𝑃₁=−𝑖^∞₀𝑍⁻_𝑡𝑒^{−𝑖𝜆𝑡}𝑑𝑡=𝐴_ℎ1ℎ2−𝜆𝐼⁻¹,Im𝜆<0, (3.10) and thus 𝐿⁻_ℎ1ℎ2=𝐴_ℎ1ℎ2. Theorem 3.1. is proved.

In order to construct a self-adjoint dilation of the maximal dissipative operator 𝐿⁺_ℎ1ℎ2 in the case “dissipative at ∞” (i.e., Imℎ₁=0 or ℎ₁=∞ and Imℎ₂>0) in ℋ, we consider the operator ℒ⁺_ℎ1ℎ2 generated by the expression (3.1) on the set 𝐷(ℒ⁺_ℎ1ℎ2) of vectors ⟨𝜑₋,𝑦,𝜑₊⟩ satisfying the conditions 𝜑₋∈𝑊¹₂(−∞,0), 𝜑₊∈𝑊¹₂(0,∞), 𝑦∈𝐷 and𝑦,𝑣_−∞−ℎ₁𝑦,𝑢_−∞=0,𝑦,𝑢_∞−ℎ₂𝑦,𝑣_∞=𝛼𝜑₋(0),𝑦,𝑢_∞−ℎ₂𝑦,𝑣_∞=𝛼𝜑₊(0), (3.11) where 𝛼²∶=2Imℎ₂,𝛼>0.

The proof of the next theorem is similar to that of Theorem 3.1.

Theorem 3.2. The operator ℒ⁺_ℎ1ℎ2 is self-adjoint in ℋ and it is a self-adjoint dilation on the maximal dissipative operator 𝐿⁺_ℎ1ℎ2.

4. Scattering Theory of the Dilations and Functional Models of the Maximal Dissipative Operators

The unitary group 𝑈^±_𝑡=exp[𝑖ℒ^±_ℎ1ℎ2𝑡] (𝑡∈ℝ) has a crucial property which enables us to apply the Lax-Phillips scheme [5]. In other words, it has incoming and outgoing subspaces 𝐷₋=⟨𝐿²(−∞,0),0,0⟩ and 𝐷₊=⟨0,0,𝐿²(0,∞)⟩ satisfying the following properties:(1)𝑈^±_𝑡𝐷₋⊂𝐷₋, 𝑡≤0 and 𝑈^±_𝑡𝐷₊⊂𝐷₊, 𝑡≥0;(2)⋂_𝑡≤0𝑈^±_𝑡𝐷₋=⋂_𝑡≥0𝑈^±_𝑡𝐷₊={0};(3)⋃_𝑡≥0𝑈^±_𝑡𝐷₋=⋃_𝑡≤0𝑈^±_𝑡𝐷₊=ℋ; (4)𝐷₋⟂𝐷₊.

Property (4) is obvious. To verify property (1) for 𝐷₊ (the proof for 𝐷₋ is similar), we set 𝑅^±_𝜆=(ℒ^±_ℎ1ℎ2−𝜆𝐼)⁻¹, for all 𝜆 with Im𝜆<0. Then, for any 𝑓=⟨0,0,𝜑₊⟩∈𝐷₊, we have𝑅^±_𝜆𝑓=0,0,−𝑖𝑒^{−𝑖𝜆𝜍𝜍}₀𝑒^{−𝑖𝜆𝑠}𝜑₊(𝑠)𝑑𝑠. (4.1) Hence, we find 𝑅_𝜆𝑓∈𝐷₊. Therefore, if 𝑔⟂𝐷₊, then it follows that0=𝑅^±_𝜆𝑓,𝑔_ℋ=−𝑖^∞₀𝑒^{−𝑖𝜆𝑡}𝑈^±_𝑡𝑓,𝑔_ℋ𝑑𝑡,Im𝜆<0. (4.2)

From this, we conclude that (𝑈^±_𝑡𝑓,𝑔)_ℋ=0 for all 𝑡≥0. Hence 𝑈^±_𝑡𝐷₊⊂𝐷₊, for 𝑡≥0, which completes the proof of property (1).

To prove property (2), we denote by 𝑃⁺∶ℋ→𝐿²(0,∞) and 𝑃⁺₁∶𝐿²(0,∞)→𝐷₊ the mappings acting according to the formulas 𝑃⁺∶⟨𝜑₋,𝑢,𝜑₊⟩→𝜑₊ and 𝑃⁺₁∶𝜑→⟨0,0,𝜑⟩, respectively. Note that the semigroup of isometries 𝑉^±_𝑡=𝑃⁺𝑈^±_𝑡𝑃⁺₁, 𝑡≥0 is a one-sided shift in 𝐿²(0,∞). Indeed, the generator of the semigroup of the one-sided shift 𝑉_𝑡 in 𝐿²(0,∞) is the differential operator 𝑖(𝑑/𝑑𝜉) satisfying the boundary condition 𝜑(0)=0. On the other hand, the generator 𝐴^± of the semigroup of isometries 𝑉^±_𝑡, 𝑡≥0, is the operator 𝐴^±𝜑=𝑃⁺ℒ^±_ℎ1ℎ2𝑃⁺₁𝑓=𝑃⁺ℒ^±_ℎ1ℎ2⟨0,0,𝜑⟩=𝑃⁺⟨0,0,𝑖(𝑑𝜑/𝑑𝜉)⟩=𝑖(𝑑𝜑/𝑑𝜉), where 𝜑∈𝑊¹₂(0,∞) and 𝜑(0)=0. As a semigroup is uniquely determined by its generator, it follows that 𝑉^±_𝑡=𝑉_𝑡, and thus, ⋂_𝑡≥0𝑈^±_𝑡𝐷₊=⟨0,0,⋂_𝑡≥0𝑉_𝑡𝐿²(0,∞)⟩={0}, which verifies the property (2).

The scattering matrix is defined in terms of the spectral representations theory in this scheme of the Lax-Phillips scattering theory. We will continue with their construction and prove property (3) of the incoming and outgoing subspaces along the way.

We recall that the linear operator 𝐴 (with domain 𝐷(𝐴)) acting in the Hilbert space 𝐻 is called completely nonself-adjoint (or simple) if the invariant subspace 𝑀⊆𝐷(𝐴) (𝑀≠{0}) of the operator 𝐴 whose restriction to 𝑀 is self-adjoint, does not exist.

Lemma 4.1. The operator 𝐿^±_ℎ1ℎ2  is completely nonself-adjoint (simple).

Proof. Let 𝐻^′⊂𝐻 be a nontrivial subspace where 𝐿⁻_ℎ1ℎ2 (the proof for 𝐿⁺_ℎ1ℎ2 is similar) induces a self-adjoint operator 𝐿^′ with domain 𝐷(𝐿^′)=𝐻^′∩𝐷(𝐿⁻_ℎ1ℎ2). If 𝑓∈𝐷(𝐿^′), then we get 𝑓∈𝐷(𝐿^′∗) and [𝑦,𝑢]_−∞−ℎ₁[𝑦,𝑣]_−∞=0, [𝑦,𝑢]_−∞−ℎ₁[𝑦,𝑣]_−∞=0. It follows that [𝑦,𝑢]_−∞=0, [𝑦,𝑣]_−∞=0 and 𝑦(𝜆)=0 for the eigenvectors 𝑦(𝜆) of the operator 𝐿⁻_ℎ1ℎ2 that lie in 𝐻^′ and are eigenvectors of 𝐿^′. Since all solutions of (2.1) belong to ℓ²_𝑤(ℤ), we conclude that the resolvent 𝑅_𝜆(𝐿⁻_ℎ1ℎ2) of the operator 𝐿⁻_ℎ1ℎ2 is a Hilbert-Schmidt operator, and hence the spectrum of 𝐿⁻_ℎ1ℎ2 is purely discrete. Using the theorem on expansion in eigenvectors of the self-adjoint operator 𝐿^′, we see that 𝐻^′={0}, that is, the operator 𝐿⁻_ℎ1ℎ2 is simple. The lemma is proved.

To prove property (3) we first setℋ^±₋=_𝑡≥0𝑈^±_𝑡𝐷₋,ℋ^±₊=_𝑡≤0𝑈^±_𝑡𝐷₊, (4.3) and prove the following lemma.

Lemma 4.2. The equality ℋ^±₋+ℋ^±₊=ℋ holds.

Proof. Using property (1) of the subspace 𝐷_±, we can easily show that the subspace ℋ^′_±=ℋ⊖(ℋ^±₋+ℋ^±₊) is invariant with respect to the group {𝑈^±_𝑡} and has the form ℋ^′_±=⟨0,𝐻^′_±,0⟩, where 𝐻^′_± is a subspace in 𝐻. Accordingly, if the subspace ℋ^′_± (and thus, 𝐻^′_± as well) were nontrivial, then the unitary group {𝑈_𝑡^±′}, restricted to this subspace, would be a unitary part of the group {𝑈^±_𝑡}, and thus the restriction 𝐿_ℎ^±′_1ℎ2 of 𝐿^±_ℎ1ℎ2 to 𝐻^′_± would be a self-adjoint operator in 𝐻^′_±. It follows from the simplicity of the operator 𝐿^±_ℎ1ℎ2that 𝐻^′_±={0}, that is, ℋ^′_±={0}. The proof is completed.

Let 𝜑(𝜆) and 𝜓(𝜆) be the solutions of (2.1) satisfying the conditions:𝜑,𝑢_−∞=−1,𝜑,𝑣_−∞=0,𝜓,𝑢_−∞=0,𝜓,𝑣_−∞=1. (4.4)

The Titchmarsh-Weyl function 𝑚_∞ℎ2(𝜆) of the self-adjoint operator 𝐿⁻_∞ℎ2 is determined by the condition [𝜓+𝑚_∞ℎ2𝜑,𝑢]_∞−ℎ₂[𝜓+𝑚_∞ℎ2𝜑,𝑣]_∞=0. Then, we have𝑚_∞ℎ2(𝜆)=−𝜓,𝑢_∞−ℎ₂𝜓,𝑣_∞𝜑,𝑢_∞−ℎ₂𝜑,𝑣_∞. (4.5) The last equality implies that 𝑚_∞ℎ2(𝜆) is a meromorphic function on the complex plane ℂ with a countable number of poles on the real axis, which coincide with the eigenvalues of the self-adjoint operator 𝐿_∞ℎ2. One can also show that the function 𝑚_∞ℎ2(𝜆) has the following properties: Im𝜆Im𝑚_∞ℎ2(𝜆)>0 for Im𝜆≠0 and 𝑚_∞ℎ2(𝜆)=𝑚_∞ℎ2(𝜆) for complex 𝜆 with the exception of the real poles of 𝑚_∞ℎ2(𝜆).

We adopt the following notations: 𝜃(𝜆)=𝜓(𝜆)+𝑚_∞ℎ2(𝜆)𝜑(𝜆),𝑆⁻_ℎ1ℎ2(𝜆)=𝑚_∞ℎ2(𝜆)−ℎ₁𝑚_∞ℎ2(𝜆)−ℎ₁. (4.6)

Let𝑈⁻_𝜆(𝜉,𝜍)=𝑒^{−𝑖𝜆𝜉},𝑚_∞ℎ2(𝜆)−ℎ₁⁻¹𝛼𝜃(𝜆),𝑆⁻_ℎ1ℎ2(𝜆)𝑒^{−𝑖𝜆𝜍}. (4.7) For real values of 𝜆, the vectors 𝑈⁻_𝜆(𝜉,𝜍) do not belong to the space ℋ, but they satisfy the equation ℒ𝑈=𝜆𝑈 and the boundary conditions (3.2). Using 𝑈⁻_𝜆(𝜉,𝜍), we define the transformation 𝐹₋∶𝑓→𝑓₋(𝜆) by (𝐹₋𝑓)(𝜆)∶=𝑓₋(𝜆)∶=(1/√2𝜋)(𝑓,𝑈⁻_𝜆)_ℋ on the vector 𝑓=⟨𝜑₋,𝑦,𝜑₊⟩, where 𝜑₋, 𝜑₊ are smooth, compactly supported functions, and 𝑦={𝑦_𝑛},𝑛∈ℤ, is a finite sequence.

Lemma 4.3. The transformation 𝐹₋ isometrically maps ℋ₋ onto 𝐿²(ℝ). For all vectors 𝑓,𝑔∈ℋ⁻₋, the Parseval equality and the inversion formula hold: 𝑓,𝑔_ℋ=𝑓₋,𝑔_−𝐿2=^∞_−∞𝑓₋(𝜆)𝑔₋1(𝜆)𝑑𝜆,𝑓=√2𝜋^∞−∞𝑓₋(𝜆)𝑈⁻_𝜆𝑑𝜆, (4.8) where 𝑓₋(𝜆)=(𝐹₋𝑓)(𝜆) and 𝑔₋(𝜆)=(𝐹₋𝑔)(𝜆).

Proof. For 𝑓,𝑔∈𝐷₋, 𝑓=⟨𝜑₋,0,0⟩, 𝑔=⟨𝜓₋,0,0⟩, we have 𝑓₋(𝜆)1∶=√2𝜋(𝑓,𝑈⁻_𝜆)_ℋ=1√2𝜋^0−∞𝜑₋(𝜉)𝑒^𝑖𝜆𝜉𝑑𝜉∈𝐻²₋, (4.9) and, by the usual Parseval equality for Fourier integrals, 𝑓,𝑔_ℋ=⁰_−∞𝜑₋(𝜉)𝜓₋(𝜉)𝑑𝜉=^∞_−∞𝑓₋(𝜆)𝑔₋(𝜆)𝑑𝜆=𝐹₋𝑓,𝐹₋𝑔_𝐿2. (4.10) From now on, let 𝐻²_± denote the Hardy classes in 𝐿²(ℝ) consisting of the functions which are analytically extendable to the upper and lower half-planes, respectively.
Let us extend the Parseval equality to the whole ℋ⁻₋. To this end, we consider in ℋ⁻₋ the dense set ℋ^′₋ of vectors obtained from the smooth, compactly supported functions in 𝐷₋∶𝑓∈ℋ^′₋ if 𝑓=𝑈⁻_𝑇𝑓₀, 𝑓₀=⟨𝜑₋,0,0⟩, 𝜑₋∈𝐶^∞₀(−∞,0), where 𝑇=𝑇_𝑓 is a nonnegative number (depending on 𝑓). In this case, if 𝑓,𝑔∈ℋ^′₋, then 𝑈⁻_−𝑇𝑓,𝑈⁻_−𝑇𝑔∈𝐷₋ for 𝑇>𝑇_𝑓 and 𝑇>𝑇_𝑔. Furthermore, the first components of these vectors belong to 𝐶^∞₀(−∞,0). Since the operators 𝑈⁻_𝑡,𝑡∈ℝ, are unitary, the equality 𝐹₋𝑈⁻_−𝑇𝑓=(𝑈⁻_−𝑇𝑓,𝑈⁻_𝜆)_ℋ=𝑒^{−𝑖𝜆𝑇}(𝑓,𝑈⁻_𝜆)_ℋ=𝑒^{−𝑖𝜆𝑇}𝐹₋𝑓 gives us that 𝑓,𝑔_ℋ=𝑈⁻_−𝑇𝑓,𝑈⁻_−𝑇𝑔_ℋ=𝐹₋𝑈⁻_−𝑇𝑓,𝐹₋𝑈⁻_−𝑇𝑔_𝐿2=𝑒^{−𝑖𝜆𝑇}𝐹₋𝑓,𝑒^{−𝑖𝜆𝑇}𝐹₋𝑔_𝐿2=𝐹₋𝑓,𝐹₋𝑔_𝐿2. (4.11) If we take the closure in (4.11), we get the Parseval equality for the whole space ℋ⁻₋. The inversion formula follows from the Parseval equality if all integrals in it are considered as limits in the mean of integrals over finite intervals. In conclusion, we have 𝐹₋ℋ⁻₋=⋃_𝑡≥0𝐹₋𝑈⁻_𝑡𝐷₋=⋃_𝑡≥0𝑒^{−𝑖𝜆𝑡}𝐻²₋=𝐿²(ℝ) which implies that 𝐹₋ maps ℋ⁻₋ onto the whole of 𝐿²(ℝ). The lemma is proved.

Now, we let𝑈⁺_𝜆(𝜉,𝜍)=𝑆⁻_ℎ1ℎ2(𝜆)𝑒^{−𝑖𝜆𝜉},𝑚⁻_∞ℎ2(𝜆)−ℎ₁⁻¹𝛼𝜃(𝜆),𝑒^{−𝑖𝜆𝜍}. (4.12) Note as in the previous case that the vectors 𝑈⁺_𝜆(𝜉,𝜍), for real values of 𝜆, do not belong to the space ℋ. But, 𝑈⁺_𝜆(𝜉,𝜍) satisfies the equation ℒ𝑈=𝜆𝑈, 𝜆∈ℝ, and the boundary conditions (3.2). By means of 𝑈⁺_𝜆(𝜉,𝜍), we consider the transformation 𝐹₊∶𝑓→𝑓₊(𝜆) by setting (𝐹₊𝑓)(𝜆)∶=𝑓₊(𝜆)∶=(1/√2𝜋)(𝑓,𝑈⁺_𝜆)_ℋ on vectors 𝑓=⟨𝜑₋,𝑦,𝜑₊⟩, where 𝜑₋, 𝜑₊ are smooth, compactly supported functions, and 𝑦={𝑦_𝑛}, 𝑛∈ℤ, is a finite sequence. The proof of the next result is similar to that of Lemma 4.3.

Lemma 4.4. The transformation 𝐹₊ isometrically maps ℋ⁻₊ onto 𝐿²(ℝ), and for all vectors 𝑓,𝑔∈ℋ⁻₊, the Parseval equality and the inversion formula hold: (𝑓,𝑔)_ℋ=𝑓₊,𝑔_+𝐿2=^∞_−∞𝑓₊(𝜆)𝑔₊(𝜆)1𝑑𝜆,𝑓=√2𝜋^∞−∞𝑓₊(𝜆)𝑈⁺_𝜆𝑑𝜆, (4.13) where 𝑓₊(𝜆)=(𝐹₊𝑓)(𝜆) and 𝑔₊(𝜆)=(𝐹₊𝑔)(𝜆).

From (4.6), we see that |𝑆⁻_ℎ1ℎ2(𝜆)|=1 for all 𝜆∈ℝ. Therefore, it follows from the explicit formula for the vectors 𝑈⁺_𝜆 and 𝑈⁻_𝜆 that𝑈⁻_𝜆=𝑆⁻_ℎ1ℎ2(𝜆)𝑈⁺_𝜆,(𝜆∈ℝ). (4.14) Lemmas 4.3 and 4.4 imply that ℋ⁻₋=ℋ⁻₊. Together with Lemma 4.2, this results in ℋ=ℋ⁻₋=ℋ⁻₊ and the property (3) of the incoming and outgoing subspaces for 𝑈⁻_𝑡.

Therefore, the transformation 𝐹₋ maps isometrically onto 𝐿²(ℝ) with the subspace 𝐷₋ mapped onto 𝐻²₋ and the operators 𝑈⁻_𝑡 are transformed into the operators of multiplication by 𝑒^𝑖𝜆𝑡, that is, 𝐹₋ is the incoming spectral representation for the group {𝑈⁻_𝑡}. Similarly 𝐹₊ is the outgoing spectral representation for {𝑈⁻_𝑡}. It is seen from (4.14) that we can realize the passage from the 𝐹₊-representation of a vector 𝑓∈ℋ to its 𝐹₋-representation multiplying by the function 𝑆⁻_ℎ1ℎ2(𝜆)∶𝑓₋(𝜆)=𝑆⁻_ℎ1ℎ2(𝜆)𝑓₊(𝜆). According to [5], the scattering function (matrix) of the group {𝑈⁻_𝑡} with respect to the subspaces 𝐷₋ and 𝐷₊, is the coefficient by which the 𝐹₋-representation of a vector 𝑓∈ℋ must be multiplied in order to get the corresponding 𝐹₊-representation: 𝑓₊(𝜆)=𝑆⁻_ℎ1ℎ2(𝜆)𝑓₋(𝜆) and, thus, we have proved the following theorem.

Theorem 4.5. The function 𝑆⁻_ℎ1ℎ2(𝜆) is the scattering matrix of the group {𝑈⁻_𝑡} (of the self-adjoint operator ℒ⁻_ℎ1ℎ2).

Let 𝑆(𝜆) be an arbitrary nonconstant inner function [1–4] on the upper half-plane (the analytic function 𝑆(𝜆) on the upper half-plane ℂ₊ is called inner function on ℂ₊ if |𝑆(𝜆)|≤1 for 𝜆∈ℂ₊ and |𝑆(𝜆)|=1 for almost all 𝜆∈ℝ). Let 𝒦=𝐻²₊⊖𝑆𝐻²₊. We can see that 𝒦≠{0} is a subspace of the Hilbert space 𝐻²₊. Now, let us consider the semigroup of the operators 𝑍_𝑡, 𝑡≥0, acting in 𝒦 according to the formula 𝑍_𝑡𝜑=𝑃[𝑒^𝑖𝜆𝑡𝜑], 𝜑∶=𝜑(𝜆)∈𝒦, where 𝑃 denotes the orthogonal projection from 𝐻²₊ onto 𝒦. The generator of the semigroup {𝑍_𝑡} is denoted by 𝑇∶𝑇𝜑=lim_𝑡→+0(𝑖𝑡)⁻¹(𝑍_𝑡𝜑−𝜑), which is a maximal dissipative operator acting in 𝒦 with the domain 𝐷(𝑇) consisting of all vectors 𝜑∈𝒦, so that the limit exists. The operator 𝑇 is called a model dissipative operator (note that this model dissipative operator, which is associated with the names of Lax and Phillips [5], is a special case of a more general model dissipative operator constructed by Sz-Nagy and Foiaş [1, 2]). The basic assertion is that 𝑆(𝜆) is the characteristic function of the operator 𝑇.

Let 𝐾=⟨0,𝐻,0⟩ so that ℋ=𝐷₋⊕𝐾⊕𝐷₊. It can be concluded from the explicit form of the unitary transformation 𝐹₋  thatℋ⟶𝐿²(ℝ),𝑓⟶𝑓₋(𝜆)=𝐹₋𝑓(𝜆),𝐷₋⟶𝐻²₋,𝐷₊⟶𝑆⁻_ℎ1ℎ2𝐻²₊,𝐾⟶𝐻²₊⊖𝑆⁻_ℎ1ℎ2𝐻²₊,𝑈⁻_𝑡𝑓⟶𝐹₋𝑈⁻_𝑡𝐹₋⁻¹𝑓₋(𝜆)=𝑒^𝑖𝜆𝑡𝑓₋(𝜆). (4.15) The formulas (4.15) show that the operator 𝐿⁻_ℎ1ℎ2 is unitarily equivalent to the model dissipative operator with the characteristic function 𝑆⁻_ℎ1ℎ2(𝜆). Since the characteristic functions of unitarily equivalent dissipative operators coincide [1–4], we have proved the theorem below.

Theorem 4.6. The characteristic function of the maximal dissipative operator 𝐿⁻_ℎ1ℎ2 coincides with the function 𝑆⁻_ℎ1ℎ2(𝜆) defined in (4.6).

If 𝑚_ℎ1∞(𝜆) is the Titchmarsh-Weyl function of the self-adjoint operator 𝐿_ℎ1∞, then it can be expressed in terms of the Wronskian of the solutions as follows:𝑚_ℎ1∞(𝜆)=−𝜒,𝑣_∞𝜙,𝑣_∞. (4.16) Here 𝜙(𝜆) and 𝜒(𝜆) are solutions of (2.1) and normalized by𝜙,𝑢_−∞1=−1+ℎ²₁,𝜙,𝑣_−∞ℎ=−¹1+ℎ²₁,𝜒,𝑢_−∞=ℎ₁1+ℎ²₁,𝜒,𝑣_−∞=11+ℎ²₁. (4.17) Let us adopt the following notations:𝑛(𝜆)∶=𝜙,𝑢_∞𝜒,𝑣_∞,𝑚(𝜆)∶=𝑚_ℎ1∞(𝜆),𝑆⁺(𝜆)∶=𝑆⁺_ℎ1ℎ2(𝜆)𝑚(𝜆)𝑛(𝜆)∶=−ℎ₂𝑚(𝜆)𝑛(𝜆)−ℎ₂. (4.18)

Let𝑉⁻_𝜆(𝜉,𝜍)=𝑒^{−𝑖𝜆𝜉},𝛼𝑚(𝜆)𝑚(𝜆)𝑛(𝜆)−ℎ₂𝜒,𝑣_∞⁻¹𝜙(𝜆),𝑆⁺(𝜆)𝑒^{−𝑖𝜆𝜍}. (4.19) One can see that the vector 𝑉⁻_𝜆(𝜉,𝜍) does not belong to ℋ for 𝜆∈ℝ, but 𝑉⁻_𝜆 satisfies the equation ℒ𝑉=𝜆𝑉, 𝜆∈ℝ, and the boundary conditions (3.11). By means of 𝑉⁻_𝜆, we define the transformation 𝐹₋∶𝑓→𝑓₋(𝜆) by (𝐹₋𝑓)(𝜆)∶=𝑓₋(𝜆)∶=(1/√2𝜋)(𝑓,𝑉⁻_𝜆)_ℋ on the vector 𝑓=⟨𝜑₋,𝑦,𝜑₊⟩, where 𝜑₋, 𝜑₊ are smooth, compactly supported functions, and 𝑦={𝑦_𝑛},𝑛∈ℤ, is a finite sequence. The next result can be proved following the steps similar to the proof of Lemma 4.3.

Lemma 4.7. The transformation 𝐹₋ isometrically maps ℋ⁺₋ onto 𝐿²(ℝ). For all vectors 𝑓,𝑔∈ℋ⁺₋, the Parseval equality and the inversion formula hold: 𝑓,𝑔_ℋ=𝑓₋,𝑔_−𝐿2=^∞_−∞𝑓₋(𝜆)𝑔₋1(𝜆)𝑑𝜆,𝑓=√2𝜋^∞−∞𝑓₋(𝜆)𝑈⁻_𝜆𝑑𝜆, (4.20) where 𝑓₋(𝜆)=(𝐹₋𝑓)(𝜆) and 𝑔₋(𝜆)=(𝐹₋𝑔)(𝜆).

Let𝑉⁺_𝜆(𝜉,𝜍)=𝑆⁺(𝜆)𝑒^{−𝑖𝜆𝜉},𝛼𝑚(𝜆)𝑚(𝜆)𝑛(𝜆)−ℎ₂𝜒,𝑣_∞⁻¹𝜙(𝜆),𝑒^{−𝑖𝜆𝜍}. (4.21) The vector 𝑉⁺_𝜆(𝜉,𝜍) does not belong to ℋ for 𝜆∈ℝ. However, 𝑉⁺_𝜆 satisfies the equation ℒ𝑉=𝜆𝑉, 𝜆∈ℝ, and the boundary conditions (3.11). Using 𝑉⁺_𝜆(𝜉,𝜍), let us consider the transformation 𝐹₊∶𝑓→𝑓₊(𝜆) on vectors 𝑓=⟨𝜑₋,𝑦,𝜑₊⟩, in which 𝜑₋,𝜑₊ are smooth, compactly supported functions, and 𝑦={𝑦_𝑛},𝑛∈ℤ, is a finite sequence, by setting (𝐹₊𝑓)(𝜆)∶=𝑓₊(𝜆)∶=(1/√2𝜋)(𝑓,𝑈⁺_𝜆)_ℋ.

Lemma 4.8. The transformation 𝐹₊ isometrically maps ℋ⁺₊ onto 𝐿²(ℝ). For all vectors 𝑓,𝑔∈ℋ⁺₊, the Parseval equality and the inversion formula hold: 𝑓,𝑔_ℋ=𝑓₊,𝑔_+𝐿2=^∞_−∞𝑓₋(𝜆)𝑔₋(𝜆)1𝑑𝜆,𝑓=√2𝜋^∞−∞𝑓₊(𝜆)𝑈⁺_𝜆𝑑𝜆, (4.22) where 𝑓₊(𝜆)=(𝐹₊𝑓)(𝜆) and 𝑔₊(𝜆)=(𝐹₊𝑔)(𝜆).

It is seen from (4.18) that the function 𝑆⁺_ℎ1ℎ2(𝜆) satisfies |𝑆⁺_ℎ1ℎ2(𝜆)|=1 for 𝜆∈ℝ. Therefore, the explicit formula for the vectors 𝑈⁺_𝜆 and 𝑈⁻_𝜆 gives us that𝑉⁻_𝜆=𝑆⁺_ℎ1ℎ2(𝜆)𝑉⁺_𝜆,𝜆∈ℝ. (4.23) Hence, we conclude the equality ℋ⁺₋=ℋ⁺₊ from Lemmas 4.7 and 4.8. Together with Lemma 4.2, we get ℋ=ℋ⁺₋=ℋ⁺₊. We can see from (4.23) that the passage from the 𝐹₋-representation of a vector 𝑓∈ℋ to its 𝐹₊-representation is realized as follows: 𝑓₊(𝜆)=𝑆⁺_ℎ1ℎ2(𝜆)𝑓₋(𝜆). Thus, we have proved the following assertion.

Theorem 4.9. The function 𝑆⁺_ℎ1ℎ2(𝜆) is the scattering matrix of the group {𝑈⁺_𝑡} (of the self-adjoint operator ℒ⁺_ℎ1ℎ2).

Using the explicit form of the unitary transformation 𝐹₋, we obtainℋ⟶𝐿²(ℝ),𝑓⟶𝑓₋(𝜆)=𝐹₋𝑓(𝜆),𝐷₋⟶𝐻²₋,𝐷₊⟶𝑆⁺_ℎ1ℎ2𝐻²₊,𝐾⟶𝐻²₊⊖𝑆⁺_ℎ1ℎ2𝐻²₊,𝑈⁺_𝑡𝑓⟶𝐹₋𝑈⁺_𝑡𝐹₋⁻¹𝑓₋(𝜆)=𝑒^𝑖𝜆𝑡𝑓₋(𝜆). (4.24) We conclude from (4.24) that the operator 𝐿⁺_ℎ1ℎ2 is a unitary equivalent to the model dissipative operator with characteristic function 𝑆⁺_ℎ1ℎ2(𝜆), which in turn proves the next theorem.

Theorem 4.10. The characteristic function of the maximal dissipative operator 𝐿⁺_ℎ1ℎ2 coincides with the function 𝑆⁺_ℎ1ℎ2(𝜆) defined by (4.18).

5. Completeness Theorems for the System of Eigenvectors and Associated Vectors of the Maximal Dissipative Operators

We know that the characteristic function of a maximal dissipative operator 𝐿^±_ℎ1ℎ2 carries complete information about the spectral properties of this operator [1–4]. For example, completeness of the system of eigenvectors and associated vectors of the maximal dissipative operators 𝐿^±_ℎ1ℎ2 is guaranteed by the absence of a singular factor of the characteristic function 𝑆^±_ℎ1ℎ2(𝜆) in the factorization 𝑆^±_ℎ1ℎ2(𝜆)=𝑆^±(𝜆)𝐵^±(𝜆) (where 𝐵^±(𝜆) is a Blaschke product).

Let 𝐴 be a linear operator in the Hilbert space 𝐇 with the domain 𝐷(𝐴). The complex number 𝜆₀ is called an eigenvalue of the operator 𝐴 if there exists a nonzero element 𝑦₀∈𝐷(𝐴) satisfying 𝐴𝑦₀=𝜆₀𝑦₀. Such an element 𝑦₀ is called the eigenvector of the operator 𝐴 corresponding to the eigenvalue 𝜆₀. The elements 𝑦₁,𝑦₂,…,𝑦_𝑘 are called the associated vectors of the eigenvector 𝑦₀ if they belong to 𝐷(𝐴) and satisfy 𝐴𝑦_𝑗=𝜆₀𝑦_𝑗+𝑦_𝑗−1, 𝑗=1,2,…,𝑘. The element 𝑦∈𝐷(𝐴),𝑦≠0 is called a root vector of the operator 𝐴 corresponding to the eigenvalue 𝜆₀, if all powers of 𝐴 are defined on this element and (𝐴−𝜆₀𝐼)^𝑚𝑦=0 for some integer 𝑚. The set of all root vectors of 𝐴 corresponding to the same eigenvalue 𝜆₀ with the vector 𝑦=0 forms a linear set 𝑁_𝜆0 and is called the root lineal. The dimension of the lineal 𝑁_𝜆0 is called the algebraic multiplicity of the eigenvalue 𝜆₀. The root lineal 𝑁_𝜆0 coincides with the linear span of all eigenvectors and associated vectors of 𝐴 corresponding to the eigenvalue 𝜆₀. Therefore, the completeness of the system of all eigenvectors and associated vectors of 𝐴 is equivalent to the completeness of the system of all root vectors of this operator.

Theorem 5.1. For all values of ℎ₁ with Imℎ₁>0, except possibly for a single value ℎ₁=ℎ⁰₁, and for fixed ℎ₂(Imℎ₂=0 or ℎ₂=0), the characteristic function 𝑆⁻_ℎ1ℎ2(𝜆) of the maximal dissipative operator 𝐿⁻_ℎ1ℎ2 is a Blaschke product and the spectrum of 𝐿⁻_ℎ1ℎ2 is purely discrete and belongs to the open upper half plane. The operator 𝐿⁻_ℎ1ℎ2(ℎ₁≠ℎ⁰₁) has a countable number of isolated eigenvalues with finite algebraic multiplicity and limit points at infinity, and the system of all eigenvectors and associated vectors (or root vectors) of this operator is complete in the space ℓ²_𝑤(ℤ).

Proof. It can be easily seen from (4.6) that 𝑆⁻_ℎ1ℎ2(𝜆) is an inner function in the upper half-plane and, moreover, it is meromorphic in the whole 𝜆-plane. Then, it can be factorized as 𝑆⁻_ℎ1ℎ2(𝜆)=𝑒^𝑖𝜆𝑐𝐵_ℎ1ℎ2(𝜆),𝑐=𝑐ℎ₁≥0, (5.1) where 𝐵_ℎ1ℎ2(𝜆) is a Blaschke product. It can be inferred from (5.1) that 𝑆⁻_ℎ1ℎ2(𝜆)≤𝑒^{−𝑐(ℎ1)Im𝜆},Im𝜆≥0. (5.2) Further, if we express 𝑚_∞ℎ2(𝜆) in terms of 𝑆⁻_ℎ1ℎ2(𝜆) and then use (4.6), we find 𝑚_∞ℎ2(𝜆)=ℎ₁𝑆⁻_ℎ1ℎ2(𝜆)−ℎ₁𝑆⁻_ℎ1ℎ2(𝜆).−1 (5.3)
If 𝑐(ℎ₁)>0 for a given value ℎ₁ (Imℎ₁>0), then lim_𝑡→+∞𝑆⁻_ℎ1ℎ2(𝑖𝑡)=0 follows from (5.2). Hence, we obtain lim_𝑡→+∞𝑚_∞ℎ2(𝑖𝑡)=ℎ₁ in the light of (5.3). Since 𝑚_∞ℎ2(𝜆) is independent of ℎ₁, 𝑐(ℎ₁) can be nonzero at not more than a single point ℎ₁=ℎ⁰₁ (and, further, ℎ⁰₁=lim_𝑡→+∞𝑚_∞ℎ2(𝑖𝑡)). Hence, the theorem is proved.

The proof of the next result is similar to that of Theorem 5.1.

Theorem 5.2. For all values of ℎ₂ with Imℎ₂>0, except possibly for a single value ℎ₂=ℎ⁰₂, and for fixed ℎ₁(Imℎ₁=0 or ℎ₁=∞), the characteristic function 𝑆⁺_ℎ1ℎ2(𝜆) of the maximal dissipative operator 𝐿⁺_ℎ1ℎ2 is a Blaschke product and the spectrum of 𝐿⁺_ℎ1ℎ2 is purely discrete and belongs to the open upper half-plane. The operator 𝐿⁺_ℎ1ℎ2(ℎ₂≠ℎ⁰₂) has a countable number of isolated eigenvalues with finite algebraic multiplicity and limit points at infinity, and the system of all eigenvectors and associated vectors of this operator is complete in the space ℓ²_𝑤(ℤ).

Since a linear operator 𝐒 acting in a Hilbert space 𝐇 is maximal accretive if and only if −𝐒 is maximal dissipative, all results obtained for maximal dissipative operators can be immediately transferred to maximal accretive operators.