Abstract
In this paper the classical Hill problem with complex potentials are extended to the star graph. The definition of the Hill operator on such graph is discussed. The operator is defined with complex, periodic potentials and using special boundary conditions connecting values of the functions at the vertices. An explicit description of the resolvent is given and the spectrum is described exactly, the inverse problem with respect to the reflection coefficients is solved.
Similar content being viewed by others
1 Introduction
The purpose of the present paper is the spectral analysis of a wave propagation in a layered, inhomogeneous medium, such as a branching tube or a system of joined strings.
It is well known [1–4] that wave propagation in a one-dimensional non-conservative medium in a frequency domain is described by the Schrödinger equation,
where R is the real axis, λ is the wave number (known as the momentum), \(\lambda^{2}\) is the energy, \(p(x)\) describes the joint effect of absorption and generation of energy, and \(q(x)\) describes the regeneration of the force density.
As a model of layered, inhomogeneous medium we will use a special type of noncompact graph, star graph, that is, a flexible mathematical construction with single vertex in which a finite number of edges \(N_{k}=[0,\infty )\), \(k=1,2,3,\ldots,n\), are joined. The models which can be obtained by investigating differential operators on the graphs have both features of ordinary and partial differential operators.
In the problem of spectral analysis of a system of branching strings, we have the following correspondence: the strings correspond to the edges of the graphs and the points of the junctions of the strings correspond to the interior vertices.
Then for studying wave propagation on branching strings we must consider the system of equations
with the following boundary conditions at the initial points of the positive half axis satisfied:
in the space
where the notation \(o_{k}\) with subscript k to denote the initial point 0 of the kth positive half axis is used and the direct sum of the spaces is denoted by ⨁. The prime denotes the derivative with respect to space coordinate and λ is a complex number.
We assume that the potentials \(p_{k} (x_{k})\) and \(q_{k} (x _{k})\), \(k=1,2,3\), are of the form
For simplicity in deriving the results, without any loss of generality, in the future we will consider the case \(n=3\).
In particular, spectral analyses of the operator with the periodic potential of the type \(q(x) = \sum_{n = 1}^{\infty} {q_{n} e ^{inx} }\) in \(L_{2}(-\infty , +\infty )\) have been studied by Gasymov [5], Shin [6], Carlson [7, 8], Guillemin and Uribe [9], and Pastur and Tkachenko [10]. As a final remark we mention [2, 11–15]. More information as regards the potentials can be found in [11].
Now we define the space \(L_{2} (G)\)
with scalar product
and we consider the operator \(L_{G}\)
where
with domain
Then the considered problem (1)-(5) can be interpreted as a study of the operator
on noncompact graph introduced as above.
The idea of investigation of quantum particles confined to a graph is rather old. The first justification of quasi-one-dimensional motion of electrons in aromatic compounds was given by Pauling [16] and worked out by Ruedenberg and Scherr [17] in 1953. Within the framework of the proposed approach each chemical bound is replaced by a narrow tube in which the electron moves and from which he cannot get away. Using honey graphs with the - Kirchhoff - boundary condition in combination with the Pauli principle, they reproduced the actual spectra with 10 percent accuracy.
As a result of that, the problem can be considered as a generalization of the classical inverse spectral problem for the Schrödinger operator on the line.
For studying the wave propagation on the graph a second order (ordinary) differential Hamiltonian is used. The Hamiltonian is a Schrödinger operator with zero Dirichlet condition on its boundary. The Dirichlet condition is responsible for confinement of electrons to the vicinity of graph.
The Schrödinger operator is defined on a graph in the following way. On each edge the wave function is a solution of the one-dimensional equation. At each vertex the wave equation must be uniquely defined.
The spectral problems on graphs arise in the investigation of processes in various domains of natural science; from complex molecules to neuron systems. Methods developed by mathematicians make it possible to describe such problems in terms of the differential equations by constructing for these problems an exact analogue of the Sturm-Liouville theory.
Without a claim to completeness of the investigation of inverse problems on graphs we list here the works of Carlson [8], Freiling and Yurko [18], Gerasimenko [19], Kostrykin and Schrader [20], Kuchment [21], and Pivovarchik [22, 23].
The paper consists of three sections.
In Section 1 we introduce the main notions and give a formulation of the direct problem.
In Section 2 the properties of the spectrum are studied. It is proved that the continuous spectrum of the operator fill out the \(\operatorname{Im}\lambda =0\) axis on which there may exist spectral singularities coinciding with the numbers \(n/2, n=\pm 1,\pm 2,\pm 3,\ldots \) . Moreover, there may be a finite number of eigenvalues outside the interval \((-\infty ,+\infty )\).
In Section 3 we give a formulation of the inverse problem and provide a constructive procedure for the solution of the inverse problem.
1.1 Formulation of the direct problem
The spectral problem can be described as follows:
Find the vector \(y_{k}(x_{k},\lambda )=(y_{k1}(x_{1},\lambda ),y_{k2}(x _{2},\lambda ),y_{k3}(x_{3},\lambda ))\) satisfying the Sturm-Liouville equation
on \(N_{k}\), \(k=1,2,3\), coupled at zero by the usual Kirchhoff conditions and complemented with initial conditions for the functions \(y_{k}(x _{k},\lambda )\), \(k = 1,2,3\).
-
(a)
\(y_{k}\) is continuous at the nodes of the graph, i.e., in particular for our graph
$$ y_{k1}(0,\lambda ) = y_{k2}(0,\lambda ) = y_{k3}(0,\lambda ); $$(8) -
(b)
the sum of the derivatives over all the branches emanating from a node, calculated for each node, is zero,
$$ y_{k1}^{\prime }(0,\lambda ) + y_{k2}^{\prime }(0,\lambda ) + y_{k3} ^{\prime }(0, \lambda ) = 0. $$(9)
It is well known (see [2]) that, for each fixed \(k=1,2,3\) on the edge \(N_{k}\), there exists a fundamental system of solutions of equations (7) \(f_{k}^{\pm } ( {x_{k}},\lambda ) \) for \(\lambda \ne \pm n/2\), \(n \in N\), and \(\lambda \ne 0\) with the properties:
where the numbers \(V_{n}^{( \pm k)}\), \(V_{ n\alpha }^{( \pm k)}\) are defined by the following recurrent formulas:
and the series
are convergent.
Let us introduce
It follows from (12) that \(f_{nk}^{\pm} (x_{k} ) \ne 0\) is valid for \(V_{n\alpha }^{ ( { \pm k} ) } \ne 0\).
From (14) it follows that the linearly independent solutions of (7) according to \(\lambda = \pm n/2\), \(n \in N\) can be determined as
According to the expressions for \(f_{k}^{\pm} (x_{k} ,\lambda ) \) we can say that
where \({\varphi_{kn}^{\pm} ( {x_{k} } ) }\), \({\tilde{\varphi} _{kn}^{\pm} ( {x_{k} } ) }\) are periodic functions. Obviously \(\tilde{f}_{nk} ^{\pm }(x_{k} )\) and \(f_{k}^{\mp} (x_{k} ,\lambda )\) are linearly independent solutions of (7) for \(\lambda = \pm n/2\), \(n \in N\).
Linearly independent solutions of equation (7) corresponding to \(\lambda = 0 \) are defined as
As a solution of the problem we will understand a matrix
on the noncompact graph on the basis of the following requirements:
-
1.
$$L_{G}Y = \lambda^{2}Y; $$
-
2.
\(y_{jk} (x_{k} ,\lambda )\) is a solution on the ray \(N_{k}=[0 , \infty ), k = 1,2,3\);
-
3.
$$ y_{jk} (x_{k} ,\lambda ) = T_{jk} (\lambda )f_{k}^{+} (x_{k} , \lambda ), \quad k \ne j; $$(17)
and
$$ y_{kk} (x_{k} ,\lambda ) = f_{k}^{-} (x_{k} , \lambda ) + R_{kk} ( \lambda )f_{k}^{+} (x_{k} ,\lambda ), \quad k = 1,2,3. $$(18)
According to the physical meaning of the solutions \(Y(x,\lambda ) = [y _{jk} (x_{k} ,\lambda )]_{k,j = 1,2,3} \), it is natural to say that \(T_{kj} (\lambda )\) are the transmission coefficients and \(R_{kk} ( \lambda )\) are the reflection coefficients for equation (7).
The coefficients \(T_{kj} (\lambda )\) and \(R_{kk} (\lambda )\) can be found by writing down the boundary conditions (8), (9) for the solution \(y_{jk} (x_{k} ,\lambda )\).
To be specific, suppose \(k=1\), then
We solve these equations for \(R_{11} (\lambda )\), \(T_{12} (\lambda )\) and \(T_{13} (\lambda )\). We note that, for the Wronskian of the solutions, \(W[f_{1} ^{+} (0,\lambda ),f_{1}^{-} (0, \lambda )] = 2i \lambda \), we obtain
where
2 The properties of the spectrum
To study the spectrum of the operator \(L_{G}\) at first we calculate the kernel of the resolvent of the operator \((L_{G}-\lambda^{2} E)\). Note that every solution \(\Psi_{k} ( {x_{k} ,\lambda } ) \) of the problem on the edge \(N_{k}=[0,\infty )\), \(k=1,2,3\), is a linear combination of the functions \(y_{kk} (x_{k} ,\lambda )\), \(y_{jk} (x _{k} ,\lambda )\), \(j \ne k\), \(j,k=1,2,3\), and can be written in the form
where \(C_{0j}^{ ( k ) }(x_{k})\) and \(C_{1j}^{ ( k ) }(x _{k})\) are such that conditions (8)-(9) hold for \(\Psi_{k} ( {x_{k} ,\lambda } ) \).
We will construct the resolvent of the operator \(L_{G}\) for \(\operatorname{Im} \lambda >0\).
To this aim, we solve the problem
in the space \(L_{2}[0_{k},\infty )\). Here \(\varphi_{k}(x_{k})\) is an arbitrary function belonging to the space \(L_{2}[0_{k},\infty )\), \(k = 1,2,3\).
By taking into account the relation
to find \(C_{0j}^{ ( k ) }(x_{k})\) and \(C_{1j}^{ ( k ) }(x _{k})\) we have
where \(x_{k} \in [0_{k},\infty )=N_{k}\) and \(C_{0j}^{ ( k ) }( \infty )\), \(C_{1j}^{ ( k ) }(o_{k})\) are arbitrary numbers.
Then
By virtue of the condition \(\Psi_{k} ( \bullet ,\lambda ) \in L_{2} [0_{k} ,\infty ), y_{kk} (x_{k} ,\lambda ) \notin L_{2} [o_{k} , \infty )\), \(y_{jk} (x_{k} ,\lambda ) \in L_{2} [o_{k} , \infty )\) we find that \(C_{0j}^{ ( k ) }(\infty ) = 0\).
Then
and
Let us denote
then by taking into account (17) we have
For finding the constants \(C_{1}^{ ( k ) }(\lambda )\), \(k=1,2,3\), we will use the boundary conditions (8)-(9) and obtain
The system of equations can be written as
where for \(f_{k}^{+}=f_{k}^{+}(o_{k},\lambda )\), \(y_{kk}= y_{kk} (o _{k} ,\lambda )\)
and
From this we find that
with
and
By using the last relation we can find the coefficients \(C_{1}^{ ( k ) }(\lambda )\), \(k=1,2,3\), as
To be specific, suppose \(k=1\), then
By taking into account (17) we can rewrite equation (20) as
It is readily seen that the function \(\Psi (x,\lambda ) = (\Psi_{1} (x _{1} ,\lambda ),\Psi_{2} (x_{2} ,\lambda ),\Psi_{3} (x_{3} ,\lambda ))\) where
with
and
are sufficiently smooth and satisfy the boundary conditions (8) and (9), i.e. they are contained in the domain of the operator \(L_{G}\). Thus, the constructed ‘spectral’ Green’s function
is the kernel of the resolvent \((L_{G} - \lambda^{2} E)^{ - 1} \), which is an integral operator. The poles of the resolvent (poles of the Green’s function) are eigenvalues of the operator \(L_{G}\) and can be found as zeros of the determinants of the matrices that participate in the construction of the Green’s function.
A point \(\lambda_{0} \in \sigma (L_{G})\) where \(\sigma (L_{G})\) is the set of spectrum of the operator \(L_{G}\) we call a spectral singularity of the operator \(L_{G}\), in the sense of Naimark [24], if it is not an isolated eigenvalue of \(L_{G}\), but \(G(x,t,\lambda ) \to \infty \) as \(\lambda \in \rho (\lambda )\) (\(\rho (\lambda )\) is the set of all regular points of the operator \(L_{G}\)) and \(\lambda \to \lambda_{0}\).
Note that self-adjoint operators have no spectral singularities and for non-self-adjoint operators the spectral singularities correspond to resonance states with vanishing spectral width [25].
Thus, the procedure described above makes it possible to obtain explicitly the resolvent and calculate its poles.
So, we proved the following theorem.
Theorem 1
Assume \(F(\lambda )\) is nonsingular i.e. \(F^{ - 1} (\lambda )\) exists, then for any
the unique solution \(\Psi (x,\lambda ) = (\Psi_{1} (x_{1} ,\lambda ), \Psi_{2} (x_{2} ,\lambda ),\Psi_{3} (x_{3} ,\lambda ))\) of (7), (8)-(9) is given by
where \(G_{jk} (x_{k} ,t_{j} ,\lambda ),j,k = 1,2,3\), are determined by (21)-(22).
Theorem 2
The operator \(L_{G}\) has no real eigenvalue.
Proof
We recall that equation (7) has fundamental solutions \(f_{k}^{\pm } ( {x_{k}},\lambda ) \) in the case \(\lambda \ne 0, \pm n/2\). Then for the case \(\lambda \ne 0, \pm n/2\) the solution of equation (7) on the edge \(N_{k}\), \(k=1,2,3\), can be written in the form
So, the solution of equation (7) belonging to \(L_{2}(G)= \bigoplus_{k = 1}^{3} L_{2} [o_{k} ,\infty )\) and satisfying the conditions (8)-(9) is necessarily has \(C_{1}=0\) and \(C_{2}=0\), \(y_{k}(x_{k},\lambda )=0\). That shows that equation (7) has only a trivial solution belonging to \(L_{2}(G)=\bigoplus _{k = 1}^{3} L_{2} [o_{k} ,\infty )\) for \(\lambda \in ( - \infty , + \infty )\), \(\lambda \ne 0, \pm n/2\).
If as linearly independent solutions (15) and (16) of (7) according to \(\lambda = \pm n/2\) or \(\lambda =0\) are taken instead of \(f_{k}^{ \pm } ( {x_{k}},\lambda ) \) then a similar result also will be valid. So we proved that \(L_{G}\) has no real eigenvalue. □
Theorem 3
The eigenvalues of operator \(L_{G}\) are finite and coincide with the zeros of the function \(\Delta (\lambda )\).
Proof
From equation (10) it is easy to see that for \(j=1,2,3\)
Therefore, as \(\vert \lambda \vert \to \infty \) we obtain \(f_{j}^{+} (0,\lambda ) = C_{j} + o(1)\), \(j=1,2,3\). Then for \(\Delta = \det F(\lambda )\) we get the following asymptotic equalities:
where C is a constant.
This asymptotic equality shows that the eigenvalues of the operator \(L_{G}\) are finite and coincide with the zeros of the function \(\Delta (\lambda )\).
The theorem is proved. □
Theorem 4
The spectrum of the operator \(L_{G}\) consists of the continuum spectrum filling the axis \({ - \infty < \lambda < + \infty }\) on which there may exist spectral singularities coinciding with the numbers \(\pm n/2 \), \(n \in N\).
Proof
In order for all numbers \(\lambda \in ( - \infty , + \infty )\) to belong to the continuous spectra it suffices to show that the operator has no real eigenvalue, the domain of \(R_{L_{G}-\lambda^{2}I}\) (the resolvent set) of the operator \(( L_{G}-\lambda^{2} I ) \) is dense in \(L_{2}(G)\), and the range of \(R_{L_{G}-\lambda^{2}I}\) is not equal to \(L_{2}(G)\).
The absence of real spectra of \(L_{G}\) was proved above in Theorem 2.
To show that the domain of \(R_{L_{G}-\lambda^{2}I}\) (the resolvent set) of the operator \(( L_{G}-\lambda^{2} I ) \) is dense in \(L_{2}(G)\) we must prove that the orthogonal complement of the set \(R_{L_{G}-\lambda^{2} E}\) consists of only the zero element.
It is well known that the orthogonal complement of the set \(R_{L_{G}- \lambda^{2} E}\) coincides with the space of the solutions of the equation \(L^{*}_{G}f=\lambda^{2}f\) where the operator \(L^{*}_{G}\) is adjoint to the operator \(L_{G}\).
Let \(\psi_{k} ( x_{k} ) \in L_{2} [ o_{k} ,+\infty ) \), \(\psi_{k} ( x_{k} ) \ne 0\) and
be satisfied for any \(f_{k} ( x_{k} ) \in D ( L_{G} ) \).
From (23) it follows that \(\psi_{k} ( x_{k} ) \in D ( L ^{*}_{G} ) \) and \(\psi_{k} ( x_{k} ) \) are eigenfunctions of the operator \(L^{*}_{G} \) corresponding to the eigenvalues λ.
In fact \(\overline{\psi_{k} ( x_{k} ) }\) is the solution of the equation
belonging to \(L_{2}(G)\). We found that \(\psi_{k} ( x_{k} ) =0\), since the operator generated by the expression standing at the left hand side of (24) is an operator of type \(L_{G}\).
This contradiction shows that the domain of \(R_{L_{G}-\lambda^{2}I}\) of the operator \(( L_{G}-\lambda^{2} I ) \) is everywhere dense in \(L_{2}( G)\).
Now let us prove that the range of \(R_{L_{G}-\lambda^{2}I}\) is not equal to \(L_{2}(G)\). For this purpose we have to show that there is a function \(f(x)\) from the space \(L_{2}(G)\) for which the equation
has no solution.
Indeed for the compact supported function \(f(x)={ ( f_{1}(x_{1}),f _{2}(x_{2}),f_{3}(x_{3}) ) }\) defined on \(L_{2}(G)\) by
where
is a solution of the following problem:
on \(L_{2}(0,\infty )\) equation (25) has no solution. To prove this fact we assume the contrary. Let equation (25) have a solution belonging to \(L_{2}(G)\). Then from Theorem 2 it follows that for \(x>a\) the function \(y(x,\lambda )\) will be a solution of (25) only under the condition
Then from (25) we obtain
on the other hand it is easy to see that
This contradiction shows that equation (25) has no solution belonging to \(L_{2}(G)\). So it is proved that the range of \(R_{L_{G}-\lambda ^{2}I}\) is not equal to \(L_{2}(G)\).
From (14) we find that the functions
where \(\Phi_{k}^{\pm} (x_{k} ,\lambda )\), have no poles at the points \(\mp n/2\), \(n \in N\).
By using this fact it is easy to show that the Green’s function \(G(x,t,\lambda )\) has poles of first order at the points \(\lambda_{0}= \pm \frac{n}{2},n \in N\). Therefore \(\lambda =\pm \frac{n}{2}\), \(n \in N\) is a spectral singularity of the operator \(L_{G}\).
The theorem is proved. □
3 The inverse spectral problem on star graph
We will consider recovering the differential operator on each fixed edge.
Since the coefficients \(R_{kk}(\lambda )\) may be found by the using matching conditions (8)-(9) at a central vertex, it is natural to formulate the inverse problem as: recovering of the potentials \(p_{k}(x_{k})\) and \(q_{k}(x_{k})\) at each edge by the reflection coefficients \(R_{kk}(\lambda )\).
Inverse problem: Given the spectral data, the reflection coefficients \(R_{kk}(\lambda )\) on each edge \(N_{k}\), construct the potentials \(p_{k}(x_{k})\) and \(q_{k}(x_{k})\) where \(k=1,2,3\).
Theorem 5
In each fixed edge \(k=1,2,3\), \(n \in N\),
are satisfied.
Proof
It is well known that the functions \(f_{k}^{+}(x_{k} ,\lambda )\), \(f_{k}^{-} (x_{k} ,\lambda )\) are linearly independent and their Wronskian is equal to \(2i\lambda \).
Then from (14) it follows that the Wronskian of the functions \(f_{nk}^{\pm} (x_{k} ),f_{k}^{\mp} (x_{k} ,\mp n/2 )\) is equal to zero, and therefore they are linearly dependent. Thus
Comparing the formulas for these functions we see that \(S_{nk}^{ \pm }=V_{nn}^{(\pm k)}\).
Therefore
Taking into account (26) it is easy to verify that
and
Consequently we can find all numbers \(V_{nn}^{( \pm k)}\) from the relations
and
The theorem is proved. □
Now to reconstruct of the potentials \(p_{k}(x_{k})\) and \(q_{k}(x_{k})\) for given \(R_{kk}(\lambda )\), we first attempt to find explicit connections between the sequences \(V_{n,n }^{( \pm k)} \), \(V_{n, \alpha }^{( \pm k)} \) and \(V_{\alpha} ^{(\mp k)}\).
Taking into account (14) we get
or
These relations are the fundamental equations for the reconstruction of \(p_{nk}(x_{k})\) and \(q_{nk}(x_{k})\) from the known \(V_{n \alpha } ^{\pm k}\), \(V_{n}^{\pm k}\).
We propose to make the dependence of \(V_{n \alpha }^{\pm k}\), on \(V_{\alpha }^{\pm k}\) explicit.
The method applied in this paper is the synthesis of methods presented by Pastur and Tkachenko [10] and Jaulent [26], and for the benefit of the reader we reintroduce it here.
Let \(\tilde{V}_{m \alpha + m}^{\pm k}\), \(m, \alpha = 1,2,3,\ldots \) , be a solution of equation (27) corresponding to \(V_{\alpha }^{\pm k}=1\) and \(\hat{V}_{m \alpha + m}^{ \pm k}\) corresponding to \(V_{\alpha }^{\pm k}= \pm i\),
Let \(\gamma_{m \alpha }^{\pm k}\) and \(\beta_{m \alpha }^{ \mp k}\) be functions defined as
Note that the quantities \(\gamma_{m \alpha }^{ \pm k}\) and \(\beta_{m \alpha }^{ \mp k} \) are uniquely determined by the recurrent equation (28) from the known \(V_{nn}^{\pm k}\).
Then we easily obtain the following:
Equation (29) shows that, if we can define the sequences \(V_{n,\alpha }^{( \pm k)} \) and \(V_{\alpha} ^{(\mp k)}\) from the known \(V_{nn}^{ \pm k}\) then the potentials \(p_{k}(x_{k})\) and \(q_{k}(x_{k})\) may be reconstructed uniquely and effectively from (11)-(13).
Theorem 6
All numbers \(V_{n\alpha }^{\pm k}, n > \alpha \) and \(V_{\alpha} ^{( \mp k)}\) may be uniquely determined through the known numbers \(V_{nn}^{\pm k}\).
Proof
In fact, if the given \(V_{n,n }^{( \pm k)} \) uniquely determine all numbers \(V_{\alpha} ^{(\mp k)}\) then the numbers \(V_{n,\alpha } ^{( \mp k)}\) will be determined by (28).
Let us denote
then from (7) we obtain the equation
in which
As a result we obtain equation (31), whose potentials exponentially decrease as \(t_{k} \to \infty \), \(k=1,2,3\).
The procedure of analytic continuation allows one to get corresponding results for equation (7) from the result of equation (31).
Equation (31) with potentials (32) has the solution
and the numbers \(V_{n}^{\pm k}\), \(V_{n\alpha }^{ \pm k }\) are defined by equations (11)-(13).
Then with the help of the (33) we obtain
where \(K_{k}^{\pm} (t_{k},u_{k}) \), \(\Omega_{k} ^{ \pm } ( {t_{k}} ) \) have the form
Rewriting the equality (27) in the form
and denoting
we obtain the Marchenko type equation
From the general theory of differential equations it is known that
By using it we get
On the other hand, we easily derive the relation from (34),
The last relations (39)-(40) give us the following system of equations for finding the dependence of \(V_{n,\alpha }^{( \pm k)} \) and \(V_{\alpha} ^{(\mp k)}\). We have
Then by using (28) we get
Finally from (41) we obtain
Let
then from (42) we obtain
and
Equations (43) and (44) uniquely determined all numbers \(V_{\alpha} ^{( \pm k)} \). Then from (28) all numbers \(V_{n \alpha } ^{( \pm k)} \) are defined.
The theorem is proved. □
Theorem 7
The specification of the spectral data uniquely determines potentials \(p_{k}(x_{k})\), \(q_{k}(x_{k})\) on each edge \(N_{k}\), \(k=1,2,3\).
References
Aktosun, T, Klaus, M, van der Mee, C: Integral equation methods for the inverse problem with discontinuous wave speed. J. Math. Phys. 37(7), 3218-3245 (1996)
Efendiev, RF: Spectral analysis of a class of nonselfadjoint differential operator pencils with a generalized function. Teor. Mat. Fiz. 145(1), 102-107 (2005) (Russian), translation in Theoret. Math. Phys. 145(1), 1457-1461 (2005)
Perera, K, Squassina, M, Yang, Y: A note on the Dancer-Fucik spectra of the fractional p-Laplacian and Laplacian operators. Adv. Nonlinear Anal. 4(1), 13-23 (2015)
Radulescu, V: Finitely many solutions for a class of boundary value problems with superlinear convex nonlinearity. Arch. Math. (Basel) 84(6), 538-550 (2005)
Gasymov, MG: Spectral analysis of a class of second-order nonselfadjoint differential operators. Funkc. Anal. Prilozh. 14(1), 14-19 (1980) (Russian)
Shin, KC: On half-line spectra for a class of non-self-adjoint Hill operators. Math. Nachr. 261/262, 171-175 (2003)
Carlson, R: A note on analyticity and Floquet isospectrality. Proc. Am. Math. Soc. 134(5), 1447-1449 (2006) (electronic)
Carlson, R: Hill’s equation for a homogeneous tree. Electron. J. Differential Equations 1997 (1997) 23 (electronic)
Guillemin, V, Uribe, A: Hardy functions and the inverse spectral method. Commun. Partial Differ. Equ. 8(13), 1455-1474 (1983)
Pastur, LA, Tkachenko, VA: Spectral theory of Schródinger operators with periodic complex-valued potentials. Funct. Anal. Appl. 22(2), 156-158 (1988)
Efendiev, RF: Spectral analysis for one class of second-order indefinite non-self-adjoint differential operator pencil. Appl. Anal. 90(12), 1837-1849 (2011)
Efendiev, RF: The characterization problem for one class of second order operator pencil with complex periodic coefficients. Mosc. Math. J. 7(1), 55-65 (2007)
Efendiev, RF, Orudzhev, HD: Inverse wave spectral problem with discontinuous wave speed. Zh. Mat. Fiz. Anal. Geom. 6(3), 255-265 (2010)
Orudzhev, HD, Efendiev, RF: The spectral analysis of some not self-adjoint operator pencil with a discontinuous coefficient. Reports of the National Academy of Sciences of Ukraine 4, 25-31 (2014)
Orujov, AD: On the spectrum of the quadratic pencil of differential operators with periodic coefficients on the semi-axis. Bound. Value Probl. 2015 (2015) 117
Pauling, L: The Nature of the Chemical Bond and the Structure of Molecules and Crystals: An Introduction to Modern Structural Chemistry, vol. 18. Cornell University Press, Ithaca (1960)
Ruedenberg, K, Scherr, CW: Free-electron network model for conjugated systems. I. Theory. J. Chem. Phys. 21(9), 1565-1581 (1953)
Freiling, G, Yurko, V: Inverse problems for Sturm-Liouville operators on noncompact trees. Results Math. 50(3-4), 195-212 (2007)
Gerasimenko, NI: The inverse scattering problem on a noncompact graph. Teor. Mat. Fiz. 75(2), 187-200 (1988) (Russian), translation in Theoret. Math. Phys. 75(2), 460-470 (1988)
Kostrykin, V, Schrader, R: The generalized star product and the factorization of scattering matrices on graphs. J. Math. Phys. 42(4), 1563-1598 (2001)
Kuchment, P: On some spectral problems of mathematical physics. In: Partial Differential Equations and Inverse Problems. Contemp. Math., vol. 362, pp. 241-276. Amer. Math. Soc., Providence (2004)
Pivovarchik, V: Inverse problem for the Sturm-Liouville equation on a star-shaped graph. Math. Nachr. 280(13-14), 1595-1619 (2007)
Pivovarchik, V: Inverse problem for the Sturm-Liouville equation on a simple graph. SIAM J. Math. Anal. 32(4), 801-819 (2000) (electronic)
Naimark, MA: Linear Differential Operators, 2nd edn. Izdat. ‘Nauka’, Moscow (1969) (Russian) Revised and augmented. With an appendix by VÉ Ljance, 526 pp.
Sinha, A, Roychoudhury, R: Spectral singularity in confined PT symmetric optical potential. J. Math. Phys. 54(11), 112106 (2013) 12 pp.
Jaulent, M: On an inverse scattering problem with an energy-dependent potential. Ann. Inst. Henri Poincaré A, Phys. Théor. 17, 363-378 (1972)
Gerasimenko, NI, Pavlov, BS: A scattering problem on noncompact graphs. Teor. Mat. Fiz. 74(3), 345-359 (1988) (Russian), translation in Theoret. Math. Phys. 74(3), 230-240 (1988)
Kostrykin, V, Schrader, R: Laplacians on metric graphs: eigenvalues, resolvents and semigroups. In: Quantum Graphs and Their Applications. Contemp. Math., vol. 415, pp. 201-225. Amer. Math. Soc., Providence (2006)
Aktosun, T: On the Schrödinger equation with steplike potentials. J. Math. Phys. 40(11), 5289-5305 (1999)
Acknowledgements
We are indebted to an anonymous referee for a detailed reading of the manuscript and useful comments. This work was supported by the research center of Alexandria University.
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
The three authors typed read and approved the final manuscript; also they contributed to each part of this work equally.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Efendiev, R.F., Orudzhev, H.D. & El-Raheem, Z.F. Spectral analysis of wave propagation on branching strings. Bound Value Probl 2016, 215 (2016). https://doi.org/10.1186/s13661-016-0723-3
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s13661-016-0723-3