Dissipative Sturm-Liouville Operators with Transmission Conditions

Abstract

In this paper we study dissipative Sturm-Liouville operators with transmission conditions. By using Pavlov’s method (Pavlov 1947, Pavlov 1981, Pavlov 1975, and Pavlov 1977), we proved a theorem on completeness of the system of eigenvectors and associated vectors of the dissipative Sturm-Liouville operators with transmission conditions.

1. Introduction

Spectral theory is one of the main branches of modern functional analysis and it has many applications in mathematics and applied sciences. There has recently been great interest in spectral analysis of Sturm-Liouville boundary value problems with eigenparameter-dependent boundary conditions (see [1–14]). Furthermore, many researchers have studied some boundary value problems that may have discontinuities in the solution or its derivative at an interior point 𝑐 [15–19]. Such conditions which include left and right limits of solutions and their derivatives at 𝑐 are often called “transmission conditions” or “interface conditions.” These problems often arise in varied assortment of physical transfer problems [20].

The spectral analysis of non-self-adjoint (dissipative) operators is based on ideas of the functional model and dilation theory rather than the method of contour integration of resolvent which is studied by Naimark [21], but this method is not effective in studying the spectral analysis of boundary value problem. The functional model technique acts a part on the fundamental theorem of Nagy-Foiaş. In 1960s independently from Nagy-Foiaş [22], Lax and Phillips [23] developed abstract scattering programme that is very important in scattering theory. Pavlov’s functional model [24–28] has been extended to dissipative operators which are finite dimensional extensions of a symmetric operator, and the corresponding dissipative and Lax-Phillips scattering matrix was investigated in some detail [5–14, 22–27, 29, 30]. This theory is based on the notion of incoming and outgoing subspaces to obtain information about analytical properties of scattering matrix by utilizing properties of original unitary group. By combining the results of Nagy-Foiaş and Lax-Phillips, characteristic function is expressed with scattering matrix and the dilation of dissipative operator is set up. By means of different spectral representation of dilation, given operator can be written very simply and functional models are obtained. The eigenvalues, eigenvectors and spectral projection of model operator are expressed obviously by characteristic function. The problem of completeness of the system of eigenvectors is solved by writing characteristic function as factorization.

The purpose of this paper is to study non-self-adjoint Sturm-Liouville operators with transmission conditions. To do this, we constructed a functional model of dissipative operator by means of the incoming and outgoing spectral representations and defined its characteristic function, because this makes it possible to determine the scattering matrix of dilation according to the Lax and Phillips scheme [23]. Finally, we proved a theorem on completeness of the system of eigenvectors and associated vectors of dissipative operators which is based on the method of Pavlov. While proving our results, we use the machinery of [5, 7–10].

2. Self-Adjoint Dilation of Dissipative Sturm-Liouville Operator

Consider the differential expression 𝑙𝑦=−𝑦^′′+𝑞(𝑥)𝑦,𝑥∈[𝑎,𝑐)∪(𝑐,𝑏), (1) where 𝐼₁∶=[𝑎,𝑐),𝐼₂∶=(𝑐,𝑏) and 𝐼=𝐼₁∪𝐼₂, 𝑞(𝑥)is real-valued function on 𝐼, and 𝑞∈𝐿¹_loc(𝐼). The points 𝑎 and 𝑐 are regular and 𝑏 is singular for the differential expression𝑙(𝑦). Moreover 𝑞(𝑐±)∶=lim_{𝑥→𝑐±}𝑞(𝑥) one-sided limits exist and are finite.

To pass from the differential expression𝑙(𝑦) to operators, we introduce the Hilbert space 𝐻=𝐿²(𝐼₁)⊕𝐿²(𝐼₂) with the inner product 𝑓,𝑔_𝐻∶=𝛾₁𝛾₂^𝑐_𝑎𝑓₁𝑔₁𝑑𝑥+𝛿₁𝛿₂^𝑏_𝑐𝑓₂𝑔₂𝑑𝑥, (2) where 𝑓(𝑥)=𝑓₁(𝑥),𝑓₂(𝑥),𝑥∈𝐼₁𝑥∈𝐼₂𝑔(𝑥)=𝑔∈𝐻,₁(𝑥),𝑔₂(𝑥),𝑥∈𝐼₁𝑥∈𝐼₂∈𝐻, (3) and 𝛾₁, 𝛾₂, 𝛿₁, and 𝛿₂ are some real numbers with 𝛾₁𝛾₂>0and 𝛿₁𝛿₂>0.

Let 𝐿₀denote the closure of the minimal operator generated by (1) and by 𝐷₀ its domain. Besides, we denote the set of all functions 𝑓(𝑥) from 𝐻such that 𝑓,𝑓^′∈𝐴𝐶_loc(𝐼),𝑓(𝑐±),𝑓^′(𝑐±) one-sided limits exist and are finite and 𝑙(𝑦)∈𝐻;𝐷 is the domain of the maximal operator 𝐿. Furthermore, 𝐿=𝐿^∗₀[21].

For two arbitrary functions 𝑦(𝑥),  𝑧(𝑥)∈𝐷, we have Green’s formula ^𝑏_𝑎𝑙𝑦𝑧𝑑𝑥−^𝑏_𝑎𝑦𝑙(𝑧)𝑑𝑥=𝛾₁𝛾₂𝑦,𝑧_𝑐−−𝛾₁𝛾₂𝑦,𝑧_𝑎+𝛿₁𝛿₂𝑦,𝑧_𝑏−𝛿₁𝛿₂𝑦,𝑧_𝑐+, (4) where [𝑦,𝑧]_𝑥=𝑦(𝑥)𝑧^′(𝑥)−𝑦^′(𝑥)𝑧(𝑥)(𝑥∈𝐼),[𝑦,𝑧]_𝑏=lim_𝑥→𝑏[𝑦,𝑧]_𝑥 exists and is finite.

Suppose that Weyl’s limit circle case holds for the differential expression 𝑙(𝑦) on 𝐼. There are several sufficient conditions in which Weyl’s limit circle case holds for a differential expression [21]. Denote by 𝜃(𝑥)=𝜃₁(𝑥),𝑥∈𝐼₁𝜃₂(𝑥),𝑥∈𝐼₂,𝜑(𝑥)=𝜑₁(𝑥),𝑥∈𝐼₁𝜑₂(𝑥),𝑥∈𝐼₂, (5) the solutions of the equation 𝑙(𝑦)=𝜆𝑦,𝑥∈𝐼, satisfying the initial conditions 𝜃₁(𝑎,𝜆)=cos𝛼,𝜃^′₁()𝜑𝑎,𝜆=sin𝛼,₁(𝑎,𝜆)=−sin𝛼,𝜑^′₁(𝑎,𝜆)=cos𝛼 (6) and transmission conditions 𝜃₂()=𝛾𝑐+,𝜆¹𝛿₁𝜃₁(𝑐−,𝜆),𝜃^′₂()=𝛾𝑐+,𝜆²𝛿₂𝜃^′₁(),𝜑𝑐−,𝜆₂()=𝛾𝑐+,𝜆¹𝛿₁𝜑₁(𝑐−,𝜆),𝜑^′₂()=𝛾𝑐+,𝜆²𝛿₂𝜑^′₁(𝑐−,𝜆), (7) where 𝛼,𝛾₁,𝛾₂,𝛿₁, and𝛿₂∈ℝ with 𝛾₁𝛾₂>0and 𝛿₁𝛿₂>0.The solutions 𝜃(𝑥,𝜆) and 𝜑(𝑥,𝜆) belong to 𝐻. Let 𝑢(𝑥)=𝑢₁(𝑥)=𝜃₁(𝑥,0),𝑥∈𝐼₁𝑢₂(𝑥)=𝜃₂(𝑥,0),𝑥∈𝐼₂,𝑣(𝑥)=𝑣₁(𝑥)=𝜑₁(𝑥,0),𝑥∈𝐼₁,𝑣₂(𝑥)=𝜑₂(𝑥,0),𝑥∈𝐼₂, (8)𝑢(𝑥), and 𝑣(𝑥) be solutions of the equation 𝑙(𝑦)=0,𝑥∈𝐼, satisfying the initial conditions 𝑢₁(𝑎)=cos𝛼,𝑢^′₁(𝑎)𝑣=sin𝛼,₁(𝑎)=−sin𝛼,𝑣^′₁(𝑎)=cos𝛼 (9) and transmission conditions 𝑢₂()=𝛾𝑐+¹𝛿₁𝑢₁(𝑐−),𝑢^′₂()=𝛾𝑐+²𝛿₂𝑢^′₁(),𝑣𝑐−₂()=𝛾𝑐+¹𝛿₁𝑣₁(𝑐−),𝑣^′₂()=𝛾𝑐+²𝛿₂𝑣^′₁(𝑐−). (10)

All the maximal dissipative extensions ℒ_𝐺 of the operator 𝐿₀ are described by the following conditions (see [4, 15, 18]): 𝑅₁𝑦∶𝛾₁𝑦(𝑐−)−𝛿₁𝑦()𝑅𝑐+=0,₂𝑦∶𝛾₂𝑦^′(𝑐−)−𝛿₂𝑦^′()𝑅𝑐+=0,₃𝑦∶𝑦(𝑎)cos𝛼+𝑦^′(𝑎)𝑅sin𝛼=0,₄𝑦∶𝑦,𝑣_𝑏−𝐺𝑦,𝑢_𝑏=0,Im𝐺>0. (11)

Let us add the “incoming” and “outgoing” subspaces 𝐷₋=𝐿²(−∞,0) and 𝐷₊=𝐿²(0,∞) to 𝐻. The orthogonal sum 𝐻=𝐷₋⊕𝐻⊕𝐷₊ is called main Hilbert space of the dilation.

In the space ℋ, we consider the operator ℒ_𝐺 on the set 𝐷(ℒ_𝐺), its elements consisting of vectors 𝑤=⟨𝜑₋,𝑦,𝜑₊⟩, generated by the expression ℒ_𝐺𝜑₋,𝑦,𝜑₊=𝑖𝑑𝜑₋𝑦𝑑𝜉,𝑙,𝑖𝑑𝜑₊𝑑𝜉 (12) satisfying the conditions 𝜑₋∈𝑊¹₂(−∞,0), 𝜑₊∈𝑊¹₂(0,∞), 𝑦∈𝐻, 𝑅₁(𝑦)=0,𝑅₂(𝑦)=0,𝑅₃(𝑦)=0, [𝑦,𝑣]_𝑏−𝐺[𝑦,𝑢]_𝑏=(𝐶/𝛿₁𝛿₂)𝜑₋(0), [𝑦,𝑣]_𝑏−𝐺[𝑦,𝑢]_𝑏=(𝐶/𝛿₁𝛿₂)𝜑₊(0), where 𝑊¹₂ are Sobolev spaces and 𝐶²∶=2Im𝐺, 𝐶>0.

Theorem 1. The operator ℒ_𝐺 is self-adjoint in ℋand it is a self-adjoint dilation of the operator 𝐿_𝐺(=𝐿_𝐾).

Proof. We first prove that ℒ_𝐺is symmetric in ℋ. Namely, (ℒ_𝐺𝑓,𝑔)_ℋ−(𝑓,ℒ_𝐺𝑔)_ℋ=0. Let 𝑓,𝑔∈𝐷(ℒ_𝐺), 𝑓=⟨𝜑₋,𝑦,𝜑₊⟩ and 𝑔=⟨𝜓₋,𝑧,𝜓₊⟩. Then we have ℒ_𝐺𝑓,𝑔_ℋ−𝑓,ℒ_𝐺𝑔_ℋ=ℒ_𝐺𝜑₋,𝑦,𝜑₊,𝜓₋,𝑧,𝜓₊−𝜑₋,𝑦,𝜑₊,ℒ_𝐺𝜓₋,𝑧,𝜓₊=⁰_−∞𝑖𝜑^′₋𝜓₋𝑑𝜉+𝑙𝑦,𝑧_𝐻+^∞₀𝑖𝜑^′₊𝜓₊𝑑𝜉−⁰_−∞𝑖𝜓^′₋𝜑₋𝑑𝜉−𝑦,𝑙(𝑧)_𝐻−^∞₀𝑖𝜓^′₊𝜑₊=𝑑𝜉_0−∞𝑖𝜑^′₋𝜓₋𝑑𝜉+𝛿₁𝛿₂𝑦,𝑧_𝑏+^∞₀𝑖𝜑^′₊𝜓₊𝑑𝜉−⁰_−∞𝑖𝜓^′₋𝜑₋𝑑𝜉−^∞₀𝑖𝜓^′₊𝜑₊𝑑𝜉=𝑖𝜓₋(0)𝜑₋(0)−𝑖𝜑₊(0)𝜓₊(0)+𝛿₁𝛿₂𝑦,𝑧_𝑏. (13) We obtain by direct computation 𝑖𝜓₋(0)𝜑₋(0)−𝑖𝜑₊(0)𝜓₊(0)+𝛿₁𝛿₂𝑦,𝑧_𝑏=0. (14) Thus, ℒ_𝐺 is a symmetric operator. To prove that ℒ_𝐺 is self-adjoint, we need to show that ℒ_𝐺⊆ℒ^∗_𝐺. Take 𝑔=⟨𝜓₋,𝑧,𝜓₊⟩∈𝐷(ℒ^∗_𝐺). Let ℒ^∗_𝐺𝑔=𝑔^∗=⟨𝜓^∗₋,𝑧^∗,𝜓^∗₊⟩∈ℋ, so that ℒ_𝐺𝑓,𝑔_ℋ=𝑓,ℒ^∗_𝐺𝑔_ℋ=𝑓,𝑔^∗_ℋ. (15) By choosing elements with suitable components as the 𝑓∈𝐷(ℒ_𝐺) in (15), it is not difficult to show that 𝜓₋∈𝑊¹₂(−∞,0), 𝜓₊∈𝑊¹₂(0,∞),𝑔∈𝐷(ℒ), and 𝑔^∗=ℒ_𝐺𝑔; the operator ℒ_𝐺 is defined by (12). Therefore (15) is obtained from(ℒ_𝐺𝑓,𝑔)_ℋ=(𝑓,ℒ_𝐺𝑔)_ℋ for all 𝑓∈𝐷(ℒ^∗_𝐺).Furthermore, 𝑔∈𝐷(ℒ^∗_𝐺) satisfies the conditions 𝑦,𝑣_𝑏+𝐺𝑦,𝑢_𝑏=𝐶𝛿₁𝛿₂𝜑₋(0),𝑦,𝑣_𝑏+𝐺𝑦,𝑢_𝑏=𝐶𝛿₁𝛿₂𝜑₊(0). (16) Hence,𝐷(ℒ^∗_𝐺)⊆𝐷(ℒ_𝐺); that is, ℒ_𝐺=ℒ^∗_𝐺.
The self-adjoint operator ℒ_𝐺 generates on ℋ a unitary group 𝑈_𝑡=exp(𝑖ℒ_𝐺𝑡) (𝑡∈ℝ₊=(0,∞)). Let us denote by 𝑃∶ℋ→𝐻 and 𝑃₁∶𝐻→ℋ the mapping acting according to the formulae 𝑃∶⟨𝜑₋,𝑦,𝜑₊⟩→𝑦 and 𝑃₁∶𝑦→⟨0,𝑦,0⟩. Let 𝑍_𝑡∶=𝑃𝑈_𝑡𝑃₁, 𝑡≥0, by using 𝑈_𝑡. The family {𝑍_𝑡}(𝑡≥0) of operators is a strongly continuous semigroup of completely nonunitary contraction on 𝐻. Let us denote by 𝐵_𝐺 the generator of this semigroup∶𝐵_𝐺𝑦=lim_𝑡→+0(𝑖𝑡)⁻¹(𝑍_𝑡𝑦−𝑦). The domain of 𝐵_𝐺 consists of all the vectors for which the limit exists. The operator 𝐵_𝐺 is dissipative. The operator ℒ_𝐺 is called the self-adjoint dilation of 𝐵_𝐺 (see [10, 21, 30]). We show that 𝐵_𝐺=𝐿_𝐺; hence ℒ_𝐺 is self-adjoint dilation of 𝐵_𝐺. To show this, it is sufficient to verify the equality 𝑃ℒ_𝐺−𝜆𝐼⁻¹𝑃₁𝑦=𝐿_𝐺−𝜆𝐼⁻¹𝑦,𝑦∈𝐻,Imℎ<0. (17) For this purpose, we set (ℒ_𝐺−𝜆𝐼)⁻¹𝑃₁𝑦=𝑔=⟨𝜓₋,𝑧,𝜓₊⟩ which implies that (ℒ_𝐺−𝜆𝐼)𝑔=𝑃₁𝑦, and hence 𝑙(̃𝑧)−𝜆𝑧=𝑦,𝜓₋(𝜉)=𝜓₋(0)𝑒^{−𝑖𝜆𝜉} and 𝜓₊(𝜉)=𝜓₊(0)𝑒^{−𝑖𝜆𝜉}. Since 𝑔∈𝐷(ℒ_𝐺), then 𝜓₋∈𝑊¹₂(−∞,0); it follows that 𝜓₋(0)=0, and consequently 𝑧 satisfies the boundary condition [𝑦,𝑣]_𝑏−𝐺[𝑦,𝑢]_𝑏=0. Therefore 𝑧∈𝐷(𝐿_𝐺), and since point 𝜆 with Im𝜆<0 cannot be an eigenvalue of dissipative operator, then 𝑧=(𝐿_𝐺−𝜆𝐼)⁻¹𝑦. Thus we have ℒ_𝐺−𝜆𝐼⁻¹𝑃₁𝑦=0,𝐿_𝐺−𝜆𝐼⁻¹𝑦,𝛿₁𝛿₂𝐶𝑦,𝑣_𝑏+𝐺𝑦,𝑢_𝑏𝑒^{−𝑖𝜆𝜉} (18) for 𝑦 and Im𝜆<0. By applying onto the mapping 𝑃, we obtain (17), and 𝐿_𝐺−𝜆𝐼⁻¹=𝑃ℒ_𝐺−𝜆𝐼⁻¹𝑃₁=−𝑖𝑃^∞₀𝑈_𝑡𝑒^{−𝑖𝜆𝑡}𝑑𝑡𝑃₁=−𝑖^∞₀𝑍_𝑡𝑒^{−𝑖𝜆𝑡}𝑑𝑡=𝐵_𝐺−𝜆𝐼⁻¹,Im𝜆<0, (19) so this clearly shows that 𝐿_𝐺=𝐵_𝐺.

3. Functional Model of Dissipative Sturm-Liouville Operator

The unitary group {𝑈_𝑡} has an important property which makes it possible to apply it to the Lax-Phillips [23]. It has orthogonal incoming and outgoing subspaces 𝐷₋=⟨𝐿²(−∞,0),0,0⟩ and 𝐷₊=⟨0,0,𝐿²(0,∞)⟩ having 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 clear. To be able to prove property (1) for 𝐷₊ (the proof for 𝐷₋ is similar), we set ℛ_𝜆=(ℒ_𝐺−𝜆𝐼)⁻¹. For all 𝜆, with  Im𝜆<0 and for any 𝑓=⟨0,0,𝜑₊⟩∈𝐷₊, we have ℛ_𝜆𝑓=0,0,−𝑖𝑒^{−𝑖𝜆𝜉𝜉}₀𝑒^𝑖𝜆𝑠𝜑₊(𝑠)𝑑𝑠, (20) as ℛ_𝜆𝑓∈𝐷₊. Therefore, if 𝑔⟂𝐷₊, then 0=ℛ_𝜆𝑓,𝑔_ℋ=−𝑖^∞₀𝑒^{−𝑖𝜆𝑡}𝑈_𝑡𝑓,𝑔_ℋ𝑑𝑡,Im𝜆<0, (21) which implies that (𝑈_𝑡𝑓,𝑔)_ℋ=0 for all 𝑡≥0. Hence, for 𝑡≥0, 𝑈_𝑡𝐷₊⊂𝐷₊, and property (1) has been proved.

In order to prove property (2), we define the mappings 𝑃⁺∶ℋ→𝐿²(0,∞) and 𝑃⁺₁∶𝐿²(0,∞)→𝐷₊ as follows: 𝑃⁺∶⟨𝜑₋,𝑦,𝜑₊⟩→𝜑₊ and 𝑃⁺₁∶𝜑→⟨0,0,𝜑⟩, respectively. We take into consider that the semigroup of isometries 𝑈⁺_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) with the boundary condition 𝜑(0)=0. On the other hand, the generator 𝑆 of the semigroup of isometries 𝑈⁺_𝑡(𝑡≥0) is the operator 𝑆𝜑=𝑃⁺ℒ_𝐺𝑃⁺₁𝜑=𝑃⁺ℒ_𝐺⟨0,0,𝜑⟩=𝑃⁺⟨0,0,𝑖(𝑑/𝑑𝜉)𝜑⟩=𝑖(𝑑/𝑑𝜉)𝜑, where 𝜑∈𝑊¹₂(0,∞) and 𝜑(0)=0. Since a semigroup is uniquely determined by its generator, it follows that 𝑈⁺_𝑡=𝑉_𝑡, and, hence, _𝑡≥0𝑈_𝑡𝐷₊=0,0,_𝑡≤0𝑉_𝑡𝐿²(0,∞)={0}, (22) so the proof is completed.

Definition 2. The linear operator 𝐴 with domain 𝐷(𝐴) acting in the Hilbert space 𝐻 is called completely non-self-adjoint (or simple) if there is no invariant subspace 𝑀⊆𝐷(𝐴)(𝑀≠{0}) of the operator 𝐴 on which the restriction 𝐴 to 𝑀 is self-adjoint.

To prove property (3) of the incoming and outgoing subspaces, let us prove following lemma.

Lemma 3. The operator 𝐿_𝐺 is completely non-self-adjoint (simple).

Proof. Let 𝐻^′⊂𝐻 be a nontrivial subspace in which 𝐿_𝐺 induces a self-adjoint operator 𝐿^′_𝐺 with domain 𝐷(𝐿^′_𝐺)=𝐻^′∩𝐷(𝐿_𝐺). If 𝑓∈𝐷(𝐿^′_𝐺), then 𝑓∈𝐷(𝐿^∗_𝐺) and 𝑑0=𝑒𝑑𝑡^𝑖𝐿^′_𝐺𝑡𝑓²_𝐻=𝑑𝑒𝑑𝑡^𝑖𝐿^′_𝐺𝑡𝑓,𝑒^𝑖𝐿^′_𝐺𝑡𝑓_𝐻=−𝐶²𝑒^𝑖𝐿^′_𝐺^𝑡𝑓,𝑢_𝑏²_𝐻. (23) Consequently, we have [𝑦,𝑢]_𝑏=0.Using this result with boundary condition [𝑦,𝑣]_𝑏+𝐺[𝑦,𝑢]_𝑏=0,we have [𝑦,𝑢]_𝑏=0; that is, 𝑦(𝜆)=0. Since all solutions of 𝑙(𝑦)=𝜆𝑦belong to 𝐿²(0,∞), from this it can be concluded that the resolvent 𝑅_𝜆(𝐿_𝐺) is a compact operator, and the spectrum of 𝐿_𝐺 is purely discrete. Consequently, by the theorem on expansion in the eigenvectors of the self-adjoint operator 𝐿^′_𝐺 we obtain 𝐻^′={0}. Hence the operator 𝐿_𝐺 is simple. The proof is completed.

Let us define 𝐻₋=∪_𝑡≥0𝑈_𝑡𝐷₋, 𝐻₊=∪_𝑡≤0𝑈_𝑡𝐷₊.

Lemma 4. The equality 𝐻₋+𝐻₊=ℋ holds.

Proof. Considering property (1) of the subspace 𝐷₊, it is easy to show that the subspace ℋ^′=ℋ⊝(𝐻₋+𝐻₊) is invariant relative to the group {𝑈_𝑡} and has the form ℋ^′=⟨0,𝐻^′,0⟩, where 𝐻^′ is a subspace in 𝐻. Therefore, if the subspace ℋ^′(andhencealso𝐻^′) was nontrivial, then the unitary group {𝑈^′_𝑡} restricted to this subspace would be a unitary part of the group {𝑈_𝑡}, and hence, the restriction 𝐿^′_𝐺 of 𝐿_𝐺 to 𝐻^′ would be a self-adjoint operator in 𝐻^′. Since the operator 𝐿_𝐺 is simple, it follows that 𝐻^′={0}. The lemma is proved.

Assume that 𝜑()𝜑𝑥,𝜆∶=¹(𝑥,𝜆),𝑥∈𝐼₁𝜑₂(𝑥,𝜆),𝑥∈𝐼₂,𝜓()𝜓𝑥,𝜆∶=¹(𝑥,𝜆),𝑥∈𝐼₁𝜓₂(𝑥,𝜆),𝑥∈𝐼₂ (24) are solutions of 𝑙(𝑦)=𝜆𝑦 satisfying the conditions 𝜑₁(𝑎,𝜆)=cos𝛼,𝜑^′₁()𝜓𝑎,𝜆=sin𝛼,₁(𝑎,𝜆)=−sin𝛼,𝜓^′₁()𝜑𝑎,𝜆=cos𝛼,₂()=𝛾𝑐+,𝜆¹𝛿₁𝜑₁(𝑐−,𝜆),𝜑^′₂()=𝛾𝑐+,𝜆²𝛿₂𝜑^′₁(𝑐−,𝜆). (25) Let us adopt the following notations: 𝐾(𝜆)=𝜑,𝑣_𝑏𝜓,𝑢_𝑏,𝑀(𝜆)=−𝜓,𝑢_𝑏𝜑,𝑢_𝑏,𝑆_𝐺(𝜆)=(𝑀(𝜆)𝐾(𝜆)+𝐺)𝑀(𝜆)𝐾(𝜆)+𝐺⁻¹, (26)(27) where 𝑀(𝜆) is a meromorphic function on the complex plane ℂ with a countable number of poles on the real axis. Further, it is possible to show that the function 𝑀(𝜆) possesses the following properties: Im𝑀(𝜆)≤0 for all Im𝜆≠0, and 𝑀(𝜆)=𝑀(𝜆) for all 𝜆∈ℂ, except the real poles 𝑀(𝜆).

We set 𝑈⁻_𝜆()=𝑒𝑥,𝜉,𝜁_{−𝑖𝜆𝜉},𝐶𝛿₁𝛿₂𝑀(𝜆)(𝑀(𝜆)𝐾(𝜆)+𝐺)𝜓,𝑢_𝑏⁻¹𝜑(𝑥,𝜆),𝑆_𝐺(𝜆)𝑒^{−𝑖𝜆𝜁}. (28) We note that the vectors 𝑈⁻_𝜆(𝑥,𝜉,𝜁) for real 𝜆 do not belong to the space ℋ. However, 𝑈⁻_𝜆(𝑥,𝜉,𝜁) satisfies the equation ℒ𝑈=𝜆𝑈 and the corresponding boundary conditions for the operator ℒ_ℎ.

By means of vector 𝑈⁻_𝜆(𝑥,𝜉,𝜁), we define the transformation 𝐹₋∶𝑓→𝑓₋(𝜆) by 𝐹₋𝑓(𝜆)∶=𝑓₋(𝜆)1∶=√2𝜋𝑓,𝑈𝜆_ℋ (29) on the vectors 𝑓=⟨𝜑₋,𝑦,𝜑₊⟩ in which 𝜑₋(𝜉),𝜑₊(𝜁),𝑦(𝑥) are smooth, compactly supported functions.

Lemma 5. The transformation 𝐹₋ isometrically maps 𝐻₋ onto 𝐿²(ℝ). For all vectors 𝑓,𝑔∈𝐻₋ the Parseval equality and the inversion formulae hold: 𝑓,𝑔_ℋ=𝑓₋,𝑔_−𝐿2=^∞_−∞𝑓₋(𝜆)𝑔₋(𝜆)1𝑑𝜆,𝑓=√2𝜋^∞−∞𝑓₋(𝜆)𝑈_𝜆𝑑𝜆, (30)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⟩, with Paley-Wiener theorem, we have 𝑓₋(𝜆)=1√2𝜋𝑓,𝑈_𝜆ℋ=1√2𝜋^0−∞𝜑₋(𝜉)𝑒^{−𝑖𝜆𝜉}𝑑𝜉∈𝐻²₋, (31) and by using usual Parseval equality for Fourier integrals, 𝑓,𝑔_ℋ=^∞_−∞𝜑₋(𝜉)𝜓₋(𝜉)=𝑑𝜉_∞−∞𝑓₋(𝜆)𝑔₋(𝜆)𝑑𝜆=𝐹₋𝑓,𝐹₋𝑔_𝐿2. (32) Here, 𝐻²_± denote the Hardy classes in 𝐿²(ℝ) consisting of the functions analytically extendible to the upper and lower half-planes, respectively.
We now extend the Parseval equality to the whole of 𝐻₋. We consider in 𝐻₋ the dense set of 𝐻^′₋ of the vectors obtained as follows 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. If 𝑓,𝑔∈𝐻^′₋, then for 𝑇>𝑇_𝑓 and 𝑇>𝑇_𝑔 we have 𝑈_−𝑇𝑓,𝑈_−𝑇𝑔∈𝐷₋; moreover, the first components of these vectors belong to 𝐶^∞₀(−∞,0).Therefore, since the operators 𝑈_𝑡(𝑡∈ℝ) are unitary, by the equality 𝐹₋𝑈_𝑡𝑓=𝑈_𝑡𝑓,𝑈⁻_𝜆ℋ=𝑒^𝑖𝜆𝑡𝑓,𝑈⁻_𝜆ℋ=𝑒^𝑖𝜆𝑡𝐹₋𝑓, (33) we have 𝑓,𝑔_ℋ=𝑈_−𝑇𝑓,𝑈_−𝑇𝑔_ℋ=𝐹₋𝑈_−𝑇𝑓,𝐹₋𝑈_−𝑇𝑔_𝐿2,𝑒^𝑖𝜆𝑇𝐹₋𝑓,𝑒^𝑖𝜆𝑇𝐹₋𝑔_𝐿2=𝑓,𝑔_𝐿2. (34)(35) By taking the closure (35), we obtain the Parseval equality for the space 𝐻₋. The inversion formula is obtained from the Parseval equality if all integrals in it are considered as limits in the integrals over finite intervals. Finally 𝐹₋𝐻₋=∪_𝑡≥0𝐹₋𝑈_𝑡𝐷₋=∪_𝑡≥0𝑒^𝑖𝜆𝑡𝐻²₋=𝐿²(ℝ); that is, 𝐹₋ maps 𝐻₋ onto the whole of 𝐿²(ℝ). The lemma is proved.

We set 𝑈⁺_𝜆()=𝑆𝑥,𝜉,𝜁_𝐺(𝜆)𝑒^{−𝑖𝜆𝜉},𝐶𝛿₁𝛿₂𝑀(𝜆)𝑀(𝜆)𝐾(𝜆)+𝐺𝜓,𝑢_𝑏⁻¹𝜑(),𝑒𝑥,𝜆_{−𝑖𝜆𝜁}. (36) We note that the vectors 𝑈⁺_𝜆(𝑥,𝜉,𝜁) for real 𝜆 do not belong to the space ℋ. However, 𝑈⁺_𝜆(𝑥,𝜉,𝜁) satisfies the equation ℒ𝑈=𝜆𝑈 and the corresponding boundary conditions for the operator ℒ_ℎ. With the help of vector 𝑈⁺_𝜆(𝑥,𝜉,𝜁), we define the transformation 𝐹₊∶𝑓→𝑓₊(𝜆) by (𝐹₊𝑓)(𝜆)∶=𝑓₊(𝜆)∶=(1/√2𝜋)(𝑓,𝑈⁺_𝜆)_ℋ on the vectors 𝑓=⟨𝜑₋,𝑦,𝜑₊⟩ in which 𝜑₋(𝜉),𝜑₊(𝜁), and 𝑦(𝑥) are smooth, compactly supported functions.

Lemma 6. 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, (37) where𝑓₊(𝜆)=(𝐹₊𝑓)(𝜆)and 𝑔₊(𝜆)=(𝐹₊𝑔)(𝜆).

Proof. The proof is analogous to Lemma 6.

It is obvious that the matrix-valued function 𝑆_𝐺(𝜆) is meromorphic in ℂ and all poles are in the lower half-plane. From (27), |𝑆_𝐺(𝜆)|≤1 for Im𝜆>0, and 𝑆_𝐺(𝜆) is the unitary matrix for all 𝜆∈ℝ. Therefore, it explicitly follows from the formulae for the vectors 𝑈⁻_𝜆 and 𝑈⁺_𝜆 that 𝑈⁺_𝜆=𝑆_𝐺(𝜆)𝑈⁻_𝜆. (38) It follows from Lemmas 6 and 5 that 𝐻₋=𝐻₊. Together with Lemma 5, this shows that 𝐻₋=𝐻₊=ℋ; therefore property (3) has been proved for the incoming and outgoing subspaces.

Thus, the transformation 𝐹₋ isometrically maps 𝐻₋ 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. This means that 𝐹₋ is the incoming spectral representation for the group {𝑈_𝑡}. Similarly, 𝐹₊ is the outgoing spectral representation for the group {𝑈_𝑡}. It follows from (38) that the passage from the 𝐹₋ representation of an element 𝑓∈ℋ to its 𝐹₊representation is accomplished as 𝑓₊(𝜆)=𝑆_𝐺⁻¹(𝜆)𝑓₋(𝜆). Consequently, according to [22], we have proved the following.

Theorem 7. The function 𝑆_𝐺(𝜆) is the scattering matrix of the group {𝑈_𝑡} (of the self-adjoint operator ℒ_𝐺).

Let 𝑆(𝜆) be an arbitrary nonconstant inner function (see [19]) on the upper half-plane (the analytic function 𝑆(𝜆) and the upper half-plane ℂ₊ is called inner function on ℂ₊ if |𝑆_ℎ(𝜆)|≤1 for all 𝜆∈ℂ₊ and |𝑆_ℎ(𝜆)|=1 for almost all 𝜆∈ℝ). Define 𝐾=𝐻²₊⊝𝑆𝐻²₊. Then 𝐾≠{0} is a subspace of the Hilbert space 𝐻²₊. We consider the semigroup of operators 𝑍_𝑡(𝑡≥0) acting in 𝐾 according to the formula 𝑍_𝑡𝜑=𝑃[𝑒^𝑖𝜆𝑡𝜑], 𝜑=𝜑(𝜆)∈𝐾, where 𝑃 is the orthogonal projection from 𝐻²₊ onto 𝐾. The generator of the semigroup {𝑍_𝑡} is denoted by 𝑇𝜑=lim_𝑡→+0(𝑖𝑡)⁻¹𝑍_𝑡𝜑−𝜑, (39) where 𝑇 is a maximal dissipative operator acting in 𝐾 and with the domain 𝐷(𝑇) consisting of all functions 𝜑∈𝐾, such that the limit exists. The operator 𝑇 is called a model dissipative operator. Recall that this model dissipative operator, which is associated with the names of Lax-Phillips [23], is a special case of a more general model dissipative operator constructed by Nagy and Foiaş [22]. 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 follows from the explicit form of the unitary transformation 𝐹₋ under the mapping 𝐹₋ that ℋ→𝐿²(ℝ),𝑓→𝑓₋(𝜆)=𝐹₋𝑓(𝜆),𝐷₋→𝐻²₋,𝐷₊→𝑆_𝐺𝐻²₊,𝐾→𝐻²₊⊝𝑆_𝐺𝐻²₊,𝑈_𝑡→𝐹₋𝑈_𝑡𝐹₋⁻¹𝑓₋(𝜆)=𝑒^𝑖𝜆𝑡𝑓₋(𝜆). (40) The formulas (40) show that operator 𝐿_𝐺 is unitarily equivalent to the model dissipative operator with the characteristic function 𝑆_𝐺(𝜆). We have thus proved the following theorem.

Theorem 8. The characteristic function of the maximal dissipative operator 𝐿_𝐺 coincides with the function 𝑆_𝐺(𝜆) defined by (27).

4. The Spectral Properties of Dissipative Sturm-Liouville Operators

By using characteristic function, the spectral properties of the maximal dissipative operator 𝐿_𝐺(𝐿_𝐾) can be investigated. The characteristic function of the maximal dissipative operator 𝐿_𝐺 is known to lead to information of completeness about the spectral properties of this operator. For instance, the absence of a singular factor 𝑠(𝜆) of the characteristic function 𝑆_𝐺(𝜆) in the factorization det𝑆_𝐺(𝜆)=𝑠(𝜆)𝐵(𝜆)  (𝐵(𝜆) is a Blaschke product) ensures completeness of the system of eigenvectors and associated vectors of the operator 𝐿_𝐺(𝐿_𝐾)in the space 𝐿₂(0,∞) (see [10, 21, 30]). If thecharacteristic function 𝑆_𝐺(𝜆) has nontrivial singular factor, the system of eigenvectors and associated vectors of the operator 𝐿_𝐺(𝐿_𝐾)can fail to be complete. Because 𝑆_𝐺(𝜆)is smooth, the support of the corresponding singular measure 𝜇 must be contained in the set of poles 𝑆_𝐺(𝜆). But in this case the singular measure 𝜇 is a simple step function. If we require 𝑆_𝐺(𝜆)to have no zeros of infinite multiplicity, then 𝜇=0.So the singular factor vanishes. The characteristic function 𝑆_𝐺(𝜆) of the maximal dissipative operator 𝐿_𝐺 has the form 𝑆_𝐺(𝜆)𝑀(𝜆)𝐾(𝜆)∶=+𝐺𝑀(𝜆)𝐾(𝜆)+𝐺, (41) where Im𝐺>0.

Theorem 9. For all the values of 𝐺 with Im𝐺>0, except possibly for a single value 𝐺=𝐺₀, the characteristic function 𝑆_𝐺(𝜆) of the maximal dissipative operator 𝐿_𝐺 is a Blaschke product. The spectrum of 𝐿_𝐺 is purely discrete and belongs to the open upper half-plane. The operator 𝐿_𝐺(𝐺≠𝐺₀) has  a countable number of isolated eigenvalues with finite multiplicity and limit points at infinity. The system of all eigenvectors and associated vectors of the operator 𝐿_𝐺 is complete in the space 𝐻.

Proof. From (35), it is clear that 𝑆_𝐺(𝜆) is an inner function in the upper half-plane, and it is meromorphic in the whole complex 𝜆-plane. Therefore, it can be factored in the form 𝑆_𝐺(𝜆)=𝑒^𝑖𝜆𝑐𝐵_𝐺(𝜆),𝑐=𝑐(𝐺)≥0, (42) where 𝐵_𝐺(𝜆) is a Blaschke product. It follows from (42) that 𝑆_𝐺(𝜆)=𝑒^𝑖𝜆𝑐𝐵_𝐺(𝜆)≤𝑒^{−𝑏(𝐺)Im𝜆},Im𝜆≥0. (43) Further, expressing 𝑛_𝐺(𝜆)∶=𝑀(𝜆)𝐾(𝜆) in terms of 𝑆_𝐺(𝜆), we find from (35) that 𝑛_𝐺(𝜆)=𝐺𝑆_𝐺(𝜆)−𝐺1−𝑆_𝐺(𝜆). (44) For a given value 𝐺 (Im𝐺>0), if  𝑐(𝐺)>0, then (43) implies that lim_𝑡→+∞𝑆_𝐺(𝑖𝑡)=0, and then (44) gives us that lim_𝑡→+∞𝑛_𝐺(𝑖𝑡)=𝐺₀. Since 𝑛_𝐺(𝜆) does not depend on 𝐺, this implies that 𝑐(𝐺) can be nonzero at not more than a single point 𝐺=𝐺₀ (and further 𝐺₀=−lim_𝑡→+∞𝑛_𝐺(𝑖𝑡)). This completes the proof.