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

()

where I₁ : = [a, c), I₂ : = (c, b) and I = I₁ ∪ I₂, q(x) is real-valued function on I, and

. The points a and c are regular and b is singular for the differential expression l(y). Moreover q(c±): = lim _x→c±q(x) one-sided limits exist and are finite.

To pass from the differential expression l(y) to operators, we introduce the Hilbert space H = L²(I₁) ⊕ L²(I₂) with the inner product

()

where

()

and γ₁, γ₂, δ₁, and δ₂ are some real numbers with γ₁γ₂ > 0 and δ₁δ₂ > 0.

Let L₀ denote 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 H such 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, [21].

For two arbitrary functions y(x), z(x) ∈ D, we have Green’s formula

()

where

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

()

the solutions of the equation l(y) = λy, x ∈ I, satisfying the initial conditions

()

and transmission conditions

()

where α, γ₁, γ₂, δ₁, and δ₂ ∈ ℝ with γ₁γ₂ > 0 and δ₁δ₂ > 0. The solutions θ(x, λ) and φ(x, λ) belong to H. Let

()

u(x), and v(x) be solutions of the equation l(y) = 0, x ∈ I, satisfying the initial conditions

()

and transmission conditions

()

All the maximal dissipative extensions ℒ_G of the operator L₀ are described by the following conditions (see [4, 15, 18]):

()

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

, generated by the expression

()

satisfying the conditions

, where

are Sobolev spaces and C² : = 2Im G, C > 0.

Theorem 1. The operator ℒ_G is self-adjoint in ℋ and it is a self-adjoint dilation of the operator .

Proof. We first prove that ℒ_G is symmetric in ℋ. Namely, . Let f, g ∈ D(ℒ_G), and . Then we have

()

We obtain by direct computation

()

Thus, ℒ_G is a symmetric operator. To prove that ℒ_G is self-adjoint, we need to show that

. Take

. Let

, so that

()

By choosing elements with suitable components as the f ∈ D(ℒ_G) in (15), it is not difficult to show that

, and g^* = ℒ_Gg; the operator ℒ_G is defined by (12). Therefore (15) is obtained from

for all

Furthermore,

satisfies the conditions

()

Hence,

; that is,

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 and . 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. 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; hence ℒ_G is self-adjoint dilation of B_G. To show this, it is sufficient to verify the equality

()

For this purpose, we set

which implies that

, and hence

and ψ₊(ξ) = ψ₊(0)e^−iλξ. Since g ∈ D(ℒ_G), then

; it follows that ψ₋(0) = 0, and consequently z satisfies the boundary condition

. Therefore

, and since point λ with Im λ < 0 cannot be an eigenvalue of dissipative operator, then

. Thus we have

()

for

and Im λ < 0. By applying onto the mapping P, we obtain (17), and

()

so this clearly shows that

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)
,
(4)
D₋⊥D₊.

Property (4) is clear. To be able to prove property (1) for D₊ (the proof for D₋ is similar), we set

. For all λ, with Im λ < 0 and for any f = 〈0,0, φ₊〉∈D₊, we have

()

as ℛ_λf ∈ D₊. Therefore, if g⊥D₊, then

()

which implies that

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

as follows:

and

, respectively. We take into consider that the semigroup of isometries

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

(t ≥ 0) is the operator

, where

and φ(0) = 0. Since a semigroup is uniquely determined by its generator, it follows that

V_t, and, hence,

()

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 is completely non-self-adjoint (simple).

Proof. Let H^′ ⊂ H be a nontrivial subspace in which induces a self-adjoint operator with domain . If , then and

()

Consequently, we have

Using this result with boundary condition

we have

; that is,

. Since all solutions of l(y) = λy belong to L²(0, ∞), from this it can be concluded that the resolvent

is a compact operator, and the spectrum of

is purely discrete. Consequently, by the theorem on expansion in the eigenvectors of the self-adjoint operator

we obtain H^′ = {0}. Hence the operator

is simple. The proof is completed.

Let us define , .

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^′(and hence also H^′) was nontrivial, then the unitary group restricted to this subspace would be a unitary part of the group {U_t}, and hence, the restriction of to H^′ would be a self-adjoint operator in H^′. Since the operator is simple, it follows that H^′ = {0}. The lemma is proved.

Assume that

()

are solutions of l(y) = λy satisfying the conditions

()

Let us adopt the following notations:

()

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: Im M(λ) ≤ 0 for all Im λ ≠ 0, and

for all λ ∈ ℂ, except the real poles M(λ).

We set

()

We note that the vectors

for real λ do not belong to the space ℋ. However,

satisfies the equation ℒU = λU and the corresponding boundary conditions for the operator ℒ_h.

By means of vector

, we define the transformation

()

on the vectors

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:

()

where

and

Proof. For f, g ∈ D₋, f = 〈φ₋, 0,0〉, g = 〈ψ₋, 0,0〉, with Paley-Wiener theorem, we have

()

and by using usual Parseval equality for Fourier integrals,

()

Here,

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 of the vectors obtained as follows from the smooth, compactly supported functions in if f = U_Tf₀, f₀ = 〈φ₋, 0,0〉, , where T = T_f is a nonnegative number depending on f. If , 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 Therefore, since the operators U_t(t ∈ ℝ) are unitary, by the equality

()

we have

()

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

; that is, F₋ maps H₋ onto the whole of L²(ℝ). The lemma is proved.

We set

()

We note that the vectors

for real λ do not belong to the space ℋ. However,

satisfies the equation ℒU = λU and the corresponding boundary conditions for the operator ℒ_h. With the help of vector

, we define the transformation

on the vectors

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:

()

where

and

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 Im λ > 0, and S_G(λ) is the unitary matrix for all λ ∈ ℝ. Therefore, it explicitly follows from the formulae for the vectors

and

that

()

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 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 . Consequently, according to [22], we have proved the following.

Theorem 7. The function 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

. Then K ≠ {0} is a subspace of the Hilbert space

. 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

onto K. The generator of the semigroup {Z_t} is denoted by

()

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

()

The formulas (40) show that operator

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 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_K) can be investigated. The characteristic function of the maximal dissipative operator

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 det S_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_K) in the space L₂(0, ∞) (see [10, 21, 30]). If the characteristic function S_G(λ) has nontrivial singular factor, the system of eigenvectors and associated vectors of the operator

(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

has the form

()

where Im G > 0.

Theorem 9. For all the values of G with Im G > 0, except possibly for a single value G = G₀, the characteristic function S_G(λ) of the maximal dissipative operator is a Blaschke product. The spectrum of is purely discrete and belongs to the open upper half-plane. The operator (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 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

()

where B_G(λ) is a Blaschke product. It follows from (42) that

()

Further, expressing n_G(λ) : = M(λ)K(λ) in terms of S_G(λ), we find from (35) that

()

For a given value G (Im G > 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.

References

1 Fulton C. T., Two-point boundary value problems with eigenvalue parameter contained in the boundary conditions, Proceedings of the Royal Society of Edinburgh A. (1977) 77, no. 3-4, 293–308, MR0593172, https://doi.org/10.1017/S030821050002521X, ZBL0376.34008.
10.1017/S030821050002521X
Web of Science® Google Scholar
2 Fulton C. T., Singular eigenvalue problems with eigenvalue parameter contained in the boundary conditions, Proceedings of the Royal Society of Edinburgh A. (1980) 87, no. 1-2, 1–34, https://doi.org/10.1017/S0308210500012312, MR600446.
10.1017/S0308210500012312
Web of Science® Google Scholar
3 Gohberg I. C. and Kreĭn M. G., Introduction to the Theory of Linear Nonselfadjoint Operators, 1969, American Mathematical Society, Providence, RI, USA, MR0246142.
10.1090/mmono/018
Google Scholar
4 Mukhtarov O. S. and Kadakal M., Some spectral properties of one Sturm-Liouville type problem with discontinuous weight, Siberian Mathematical Journal. (2005) 46, no. 4, 681–694.
10.1007/s11202-005-0069-z
Web of Science® Google Scholar
5 Allahverdiev B. P., On dilation theory and spectral analysis of dissipative Schrödinger operators in Weyl′s limit-circle case, Izvestiya Akademii Nauk. (1990) 54, 242–257, English translation: Mathematics of the USSR-Izvestiya, vol. 36, pp. 247–262, 1991.
Google Scholar
6 Shkalikov A. A., Boundary value problems with a spectral parameter in the boundary conditions, Zeitschrift Für Angewandte Mathematik Und Mechanik. (1996) 76, 133–135.
Web of Science® Google Scholar
7 Saltan S. and Allahverdiev B. P., Spectral analysis of nonselfadjoint Schrödinger operators with a matrix potential, Journal of Mathematical Analysis and Applications. (2005) 303, no. 1, 208–219, https://doi.org/10.1016/j.jmaa.2004.08.031, MR2113877, ZBL1124.47029.
10.1016/j.jmaa.2004.08.031
Web of Science® Google Scholar
8 Allahverdiev B. P., A nonself-adjoint singular Sturm-Liouville problem with a spectral parameter in the boundary condition, Mathematische Nachrichten. (2005) 278, no. 7-8, 743–755, https://doi.org/10.1002/mana.200310269, MR2141954, ZBL1089.34023.
10.1002/mana.200310269
Web of Science® Google Scholar
9 Allahverdiev B. P., Spectral analysis of dissipative Dirac operators with general boundary conditions, Journal of Mathematical Analysis and Applications. (2003) 283, no. 1, 287–303, https://doi.org/10.1016/S0022-247X(03)00293-2, MR1994191, ZBL1093.47045.
10.1016/S0022-247X(03)00293-2
Web of Science® Google Scholar
10 Allahverdiev B. P., Dilation and functional model of dissipative operator generated by an infinite Jacobi matrix, Mathematical and Computer Modelling. (2003) 38, no. 10, 989–1001, https://doi.org/10.1016/S0895-7177(03)90101-4, MR2017970, ZBL1047.47023.
10.1016/S0895-7177(03)90101-4
Google Scholar
11 Ongun M. Y. and Allahverdiev B. P., A completeness theorem for a dissipative Schrödinger problem with the spectral parameter in the boundary condition, Mathematische Nachrichten. (2008) 281, no. 4, 541–554, https://doi.org/10.1002/mana.200410623, MR2404297, ZBL1145.34054.
10.1002/mana.200410623
Web of Science® Google Scholar
12 Eryılmaz A., Spectral analysis of q-sturm-liouville problem with the spectral parameter in the boundary condition, Journal of Function Spaces and Applications. (2012) 2012, 17, 736437, https://doi.org/10.1155/2012/736437.
10.1155/2012/736437
Google Scholar
13 Tuna H., Dissipative q-difference operators of the Sturm-Liouville type, Applications of Mathematics. In press.
Google Scholar
14 Tuna H., Functional model of dissipative fourth order differential operators, Analele Stiintice ale Universitatii Ovidius Constanta. In press.
Google Scholar
15 Bairamov E. and Ugurlu E., The determinants of dissipative Sturm-Liouville operators with transmission conditions, Mathematical and Computer Modelling. (2011) 53, no. 5-6, 805–813, https://doi.org/10.1016/j.mcm.2010.10.017, MR2769453, ZBL1217.34129.
10.1016/j.mcm.2010.10.017
Web of Science® Google Scholar
16 Mukhtarov O., Discontinuous boundary value problem with spectral parameter in boundary conditions, Turkish Journal of Mathematics. (1994) 18, no. 2, 183–192, MR1281382, ZBL0862.34016.
Google Scholar
17 Mukhtarov O. S. and Tunç E., Eigenvalue problems for Sturm-Liouville equations with transmission conditions, Israel Journal of Mathematics. (2004) 144, 367–380, https://doi.org/10.1007/BF02916718, MR2121546, ZBL1073.34096.
10.1007/BF02916718
Web of Science® Google Scholar
18 Akdoğan Z., Demirci M., and Mukhtarov O. S., Green function of discontinuous boundary value problem with transmission conditions, Mathematical Methods in the Applied Sciences. (2007) 30, 1719–1738.
10.1002/mma.867
Web of Science® Google Scholar
19 Muhtarov O. S. and Yakubov S., Problems for ordinary differential equations with transmission conditions, Applicable Analysis. (2002) 81, no. 5, 1033–1064, https://doi.org/10.1080/0003681021000029853, MR1948030, ZBL1062.34094.
10.1080/0003681021000029853
Google Scholar
20 Wang A., Sun J., Hao X., and Yao S., Completeness of eigenfunctions of Sturm-Liouville problems with transmission conditions, Methods and Applications of Analysis. (2009) 16, no. 3, 299–312, MR2650798, ZBL1210.34123.
10.1016/j.jmaa.2008.08.008
Web of Science® Google Scholar
21 Naimark M. A., Lineinye Differentsialnye Operatory, 1969, 2nd edition, Nauka, Moscow, Russia, English Translation of 1st ed., vol. 1-2, Ungar, New York, NY, USA, 1967, 1968.
Google Scholar
22 Nagy B. S. and Foiaş C., Analyse Harmonique Des Operateurs De L.Espace De Hilbert, 1967, Masson, Paris, France, English Translation North-Holland, Amsterdam, The Netherlands, 1970.
Google Scholar
23 Lax P. D. and Phillips R. S., Scattering Theory, 1967, Academic Press, New York, Ny, USA, MR0217440.
Google Scholar
24 Pavlov B. S., Dilation theory and spectral analysis of nonselfadjoint differential operators, Mathematical Programming and Related Questions (Proc. Seventh Winter School, Drogobych, 1974), 1974, Central Economic Mathematical Institute, Moscow, Russia, 3–69, MR0634807, ZBL0516.47007.
Google Scholar
25 Pavlov B. S., Dilation theory and spectral analysis of nonselfadjoint differential operators, American Mathematical Society Translations. (1981) 115, 103–142.
Google Scholar
26 Pavlov B. S., Self-adjoint dilatation of a dissipative Schrödinger operator and an eigenfunction expansion, Functional Analysis and Its Applications. (1975) 9, no. 2, 172–173, 2-s2.0-34250395523, https://doi.org/10.1007/BF01075465.
10.1007/BF01075465
Google Scholar
27 Pavlov B. S., Selfadjoint dilation of a dissipative schrödinger operator and eigenfunction expansion, Functional Analysis and Its Applications. (1975) 98, 172–173.
10.1007/BF01075465
Google Scholar
28 Pavlov B. S., Selfadjoint dilation of a dissipative Schrödinger operator and its resolution in terms of eigenfunctions, Mathematics of the USSR-Sbornik. (1977) 31, no. 4, 457–478.
10.1070/SM1977v031n04ABEH003716
Web of Science® Google Scholar
29 Kuzhel A., Characteristic Functions and Models of Nonselfadjoint Operators, 1996, 349, Kluwer Academic, Dordrecht, The Netherlands.
10.1007/978-94-009-0183-4
Google Scholar
30 Kuzhel S., On an inverse problem in the Lax-Phillips scattering scheme for a class of operator-differential equations, St. Petersburg Mathematical Journal. (2001) 13, no. 1, 60–83, MR1819362.
Google Scholar

All articles

Dissipative Sturm-Liouville Operators with Transmission Conditions

Abstract

1. Introduction

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

3. Functional Model of Dissipative Sturm-Liouville Operator

4. The Spectral Properties of Dissipative Sturm-Liouville Operators

References

References

Information

About Wiley Online Library

Help & Support

Opportunities

Connect with Wiley

Dissipative Sturm-Liouville Operators with Transmission Conditions

Abstract

1. Introduction

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

3. Functional Model of Dissipative Sturm-Liouville Operator

4. The Spectral Properties of Dissipative Sturm-Liouville Operators

References

References

Related

Information