Open Access

Inverse eigenvalue problems for a discontinuous Sturm-Liouville operator with two discontinuities

Boundary Value Problems20132013:209

DOI: 10.1186/1687-2770-2013-209

Received: 18 June 2013

Accepted: 26 August 2013

Published: 11 September 2013

Abstract

In this paper, we consider a discontinuous Sturm-Liouville operator with parameter-dependent boundary conditions and two interior discontinuities. We obtain eigenvalues and eigenfunctions together with their asymptotic approximate formulas. Then, we give some uniqueness theorems by using Weyl function and spectral data, which are called eigenvalues and normalizing constants for solution of inverse problem.

MSC:34A55, 34B24, 34L05.

Keywords

Sturm-Liouville problem eigenvalues eigenfunctions transmission conditions Weyl function

1 Introduction

It is well known that the theory of Sturm-Liouville problems is one of the most actual and extensively developing fields of theoretical and applied mathematics, since it is an important tool in solving many problems in mathematical physics (see [14]). In recent years, there has been increasing interest in spectral analysis of discontinuous Sturm-Liouville problems with eigenvalue-linearly and nonlinearly dependent boundary conditions [1, 512]. Various physics applications of such problems can be found in [1, 3, 4, 1319] and corresponding bibliography cited therein.

Some boundary value problems with discontinuity conditions arise in heat and mass transfer problems, mechanics, electronics, geophysics and other natural sciences (see [3] also [2029]). For instance, discontinuous inverse problems appear in electronics for building parameters of heterogeneous electronic lines with attractive technical characteristics [20, 30, 31]. Such discontinuity problems also appear in geophysical forms for oscillations of the earth [32, 33]. Furthermore, discontinuous inverse problems appear in mathematics for exploring spectral properties of some classes of differential and integral operators.

Inverse problems of spectral analysis form recovering operators by their spectral data. The inverse problem for the classical Sturm-Liouville operator was studied first by Ambarsumian in 1929 [34] and then by Borg in 1945 [35]. After that, direct and inverse problems for Sturm-Liouville operator have been extended to so many different areas.

We consider a discontinuous Sturm-Liouville problem L with function ρ ( x )
l y : = ρ ( x ) y ( x ) + q ( x ) y ( x ) = λ y ( x ) , x [ a , δ 1 ) ( δ 1 , δ 2 ) ( δ 2 , b ] = Ω ,
(1)
where
ρ ( x ) = { ρ 1 2 , a x < δ 1 , ρ 2 2 , δ 1 < x < δ 2 , ρ 3 2 , δ 2 < x b ,
and ρ 1 , ρ 2 , and ρ 3 are given positive real numbers; q ( x ) L 2 [ Ω , R ] ; λ C is a complex spectral parameter; boundary conditions at the endpoints
l 1 y : = λ ( θ 1 y ( a ) θ 2 y ( a ) ) ( θ 1 y ( a ) θ 2 y ( a ) ) = 0 ,
(2)
l 2 y : = λ ( γ 1 y ( b ) γ 2 y ( b ) ) + ( γ 1 y ( b ) γ 2 y ( b ) ) = 0
(3)
with discontinuity conditions at two points x = δ 1 , x = δ 2
l 3 y : = y ( δ 1 + 0 ) θ 3 y ( δ 1 0 ) γ 3 y ( δ 1 0 ) = 0 ,
(4)
l 4 y : = y ( δ 1 + 0 ) θ 4 y ( δ 1 0 ) γ 4 y ( δ 1 0 ) = 0 ,
(5)
l 5 y : = y ( δ 2 + 0 ) θ 5 y ( δ 2 0 ) γ 5 y ( δ 2 0 ) = 0 ,
(6)
l 6 y : = y ( δ 2 + 0 ) θ 6 y ( δ 2 0 ) γ 6 y ( δ 2 0 ) = 0 ,
(7)
where θ i , γ i and θ j , γ j ( i = 1 , 6 , j = 1 , 2 ) are real numbers and
α 1 = | θ 3 γ 3 θ 4 γ 4 | > 0 , α 2 = | θ 5 γ 5 θ 6 γ 6 | > 0 , β 1 = | θ 1 θ 1 θ 2 θ 2 | > 0 and β 2 = | γ 1 γ 1 γ 2 γ 2 | > 0 .

In the present paper, we construct a linear operator T in a suitable Hilbert space such that problem (1)-(7) and the eigenvalue problem for operator T coincide. We investigate eigenvalues and eigenfunctions together with their asymptotic behaviors of operator T. Besides, we study some uniqueness theorems according to Weyl function and spectral data, which are called eigenvalues and normalizing constants.

2 Operator formulation and spectral properties

We make known the inner product in the Hilbert space H : = H 1 C 2 , where H 1 = ( L 2 ( Ω ) , , 1 ) , denotes the Hilbert space of complex numbers and a self-adjoint operator T defined on H such that (1)-(7) can be dealt with as the eigenvalue problem of operator T. We define an inner product in H by
F , G : = α 1 α 2 ρ 1 2 a δ 1 f ( x ) g ¯ ( x ) d x + α 2 ρ 1 2 δ 1 δ 2 f ( x ) g ¯ ( x ) d x + ρ 3 2 δ 2 b f ( x ) g ¯ ( x ) d x + α 1 α 2 β 1 f 1 g ¯ 1 + 1 β 2 f 2 g ¯ 2
(8)
for
F = ( f ( x ) f 1 f 2 ) H , G = ( g ( x ) g 1 g 2 ) H .
Consider the operator T defined by the domain
D ( T ) = { F H : f ( x ) , f ( x ) A C loc ( Ω ) , l f H 1 , l 3 = l 4 = l 5 = l 6 = 0 , f 1 = θ 1 f ( a ) θ 2 f ( a ) f 2 = γ 1 f ( b ) γ 2 f ( b ) }

such that T F : = ( l f , θ 1 f ( a ) θ 2 f ( a ) , ( γ 1 f ( b ) γ 2 f ( b ) ) ) for F = ( f , θ 1 f ( a ) θ 2 f ( a ) , γ 1 f ( b ) γ 2 f ( b ) ) D ( T ) and also l 3 - l 6 are satisfied for f.

Thus, we can rewrite the considered problem (1)-(7) in the operator form as T F = λ F .

Theorem 1 The operator T is symmetric in H.

Proof Let F , G D ( T ) . By two partial integrations, we get
T F , G = F , T G + α 1 α 2 ( W ( f , g ¯ , δ 1 0 ) W ( f , g ¯ , a ) ) + α 2 ( W ( f , g ¯ , δ 2 0 ) W ( f , g ¯ , δ 1 + 0 ) ) + W ( f , g ¯ , b ) W ( f , g ¯ , δ 2 + 0 ) + α 1 α 2 β 1 ( θ 1 f ( a ) θ 2 f ( a ) ) ( θ 1 g ¯ ( a ) θ 2 g ¯ ( a ) ) 1 β 2 ( γ 1 f ( b ) γ 2 f ( b ) ) ( γ 1 g ¯ ( b ) γ 2 g ¯ ( b ) ) α 1 α 2 β 1 ( θ 1 g ¯ ( a ) θ 2 g ¯ ( a ) ) ( θ 1 f ( a ) θ 2 f ( a ) ) + 1 β 2 ( γ 1 g ¯ ( b ) γ 2 g ¯ ( b ) ) ( γ 1 f ( b ) γ 2 f ( b ) ) ,
where by W ( f , g ; x ) , we denote the Wronskian of the functions f and g as
f ( x ) g ( x ) f ( x ) g ( x ) .
Since f and g satisfy the boundary conditions (2)-(3) and transmission conditions (4)-(7), we obtain
α 1 α 2 β 1 [ ( θ 1 f ( a ) θ 2 f ( a ) ) ( θ 1 g ¯ ( a ) θ 2 g ¯ ( a ) ) ( θ 1 g ¯ ( a ) θ 2 g ¯ ( a ) ) ( θ 1 f ( a ) θ 2 f ( a ) ) ] = α 1 α 2 W ( f , g ¯ , a ) × 1 β 2 [ ( γ 1 g ¯ ( b ) γ 2 g ¯ ( b ) ) ( γ 1 f ( b ) γ 2 f ( b ) ) ( γ 1 f ( b ) γ 2 f ( b ) ) ( γ 1 g ¯ ( b ) γ 2 g ¯ ( b ) ) ] = W ( f , g ¯ , b ) W ( f , g ¯ , δ i + 0 ) = α i W ( f , g ¯ , δ i 0 ) .

Thus, we have T F , G = F , T G , i.e., T is symmetric. □

Lemma 1 Problem (1)-(7) can be considered as the eigenvalue problem of the symmetric operator T.

Corollary 1 All eigenvalues and eigenfunctions of problem (1)-(7) are real, and two eigenfunctions φ ( x , λ 1 ) and φ ( x , λ 2 ) , corresponding to different eigenvalues λ 1 and λ 2 , are orthogonal in the sense of
α 1 α 2 ρ 1 2 a δ 1 φ ( x , λ 1 ) φ ( x , λ 2 ) d x + α 2 ρ 2 2 δ 1 δ 2 φ ( x , λ 1 ) φ ( x , λ 2 ) d x + ρ 3 2 δ 2 b φ ( x , λ 1 ) φ ( x , λ 2 ) d x + α 1 α 2 β 1 ( θ 1 φ ( a , λ 1 ) θ 2 φ ( a , λ 1 ) ) ( θ 1 φ ( a , λ 2 ) θ 2 φ ( a , λ 2 ) ) + 1 β 2 ( γ 1 φ ( b , λ 1 ) γ 2 φ ( b , λ 1 ) ) ( γ 1 φ ( b , λ 2 ) γ 2 φ ( b , λ 2 ) ) = 0 .
We define the solutions
φ ( x , λ ) = { φ 1 ( x , λ ) , x [ a , δ 1 ) , φ 2 ( x , λ ) , x ( δ 1 , δ 2 ) , φ 3 ( x , λ ) , x ( δ 2 , b ] , ψ ( x , λ ) = { ψ 1 ( x , λ ) , x [ a , δ 1 ) , ψ 2 ( x , λ ) , x ( δ 1 , δ 2 ) , ψ 3 ( x , λ ) , x ( δ 2 , b ]
of equation (1) by the initial conditions
φ 1 ( a , λ ) = λ θ 2 θ 2 , φ 1 ( a , λ ) = λ θ 1 θ 1 , φ 2 ( δ 1 , λ ) = θ 3 φ 1 ( δ 1 , λ ) + γ 3 φ 1 ( δ 1 , λ ) , φ 2 ( δ 1 , λ ) = θ 4 φ 1 ( δ 1 , λ ) + γ 4 φ 1 ( δ 1 , λ ) , φ 3 ( δ 2 , λ ) = θ 5 φ 2 ( δ 2 , λ ) + γ 5 φ 2 ( δ 2 , λ ) , φ 3 ( δ 2 , λ ) = θ 6 φ 2 ( δ 2 , λ ) + γ 6 φ 2 ( δ 2 , λ ) ,
(9)
and similarly,
ψ 3 ( b , λ ) = λ γ 2 + γ 2 , ψ 3 ( b , λ ) = λ γ 1 + γ 1 , ψ 2 ( δ 2 , λ ) = γ 6 ψ 3 ( δ 2 , λ ) γ 5 ψ 3 ( δ 2 , λ ) α 2 , ψ 2 ( δ 2 , λ ) = θ 6 ψ 3 ( δ 2 , λ ) θ 5 ψ 3 ( δ 2 , λ ) α 2 , ψ 1 ( δ 1 , λ ) = γ 4 ψ 2 ( δ 1 , λ ) γ 3 ψ 2 ( δ 1 , λ ) α 1 , ψ 1 ( δ 1 , λ ) = θ 4 ψ 2 ( δ 1 , λ ) θ 3 ψ 2 ( δ 1 , λ ) α 1 ,
(10)

respectively.

These solutions are entire functions of λ for each fixed x [ a , b ] and satisfy the relation
ψ ( x , λ n ) = κ n φ ( x , λ n )
for each eigenvalue λ n , where
κ n = θ 2 ψ ( a , λ n ) θ 1 ψ ( a , λ n ) β 1 .

Lemma 2 Let λ = k 2 , k = σ + i ω .

Then the following integral equations and also asymptotic behaviors hold for ν = 0 , 1 :
d ν d x ν φ 1 ( x , λ ) = ( λ θ 2 θ 2 ) d ν d x ν cos k ρ 1 ( x a ) + 1 k ρ 1 ( λ θ 1 θ 1 ) d ν d x ν sin k ρ 1 ( x a ) + ρ 1 k a x d ν d x ν sin k ρ 1 ( x t ) q ( t ) φ 1 ( t , λ ) d t = ( λ θ 2 θ 2 ) d ν d x ν cos k ρ 1 ( x a ) + 1 k ρ 1 ( λ θ 1 θ 1 ) d ν d x ν sin k ρ 1 ( x a ) + O ( | k | ν + 1 e | Im k | ( x a ) ρ 1 ) , d ν d x ν φ 2 ( x , λ ) = ( θ 3 φ 1 ( δ 1 , λ ) + γ 3 φ 1 ( δ 1 , λ ) ) d ν d x ν cos k ρ 2 ( x δ 1 ) + 1 k ρ 2 ( θ 4 φ 1 ( δ 1 , λ ) + γ 4 φ 1 ( δ 1 , λ ) ) d ν d x ν sin k ρ 2 ( x δ 1 ) + ρ 2 k δ 1 x d ν d x ν sin k ρ 2 ( x t ) q ( t ) φ 2 ( t , λ ) d t = ( λ θ 2 θ 2 ) γ 3 k ρ 1 sin k ρ 1 ( δ 1 a ) d ν d x ν cos k ρ 2 ( x δ 1 ) + O ( | k | ν + 2 e | Im k | ( ( δ 1 a ) ρ 1 + ( x δ 1 ) ρ 2 ) ) , d ν d x ν φ 3 ( x , λ ) = ( θ 5 φ 2 ( δ 2 , λ ) + γ 5 φ 2 ( δ 2 , λ ) ) d ν d x ν cos k ρ 3 ( x δ 2 ) + 1 k ρ 3 ( θ 5 φ 2 ( δ 2 , λ ) + γ 6 φ 2 ( δ 2 , λ ) ) d ν d x ν sin k ρ 3 ( x δ 2 ) + ρ 3 k δ 2 x d ν d x ν sin k ρ 3 ( x t ) q ( t ) φ 3 ( t , λ ) d t = ( λ θ 2 θ 2 ) γ 3 γ 5 k 2 ρ 1 ρ 2 sin k ρ 1 ( δ 1 a ) sin k ρ 2 ( δ 2 δ 1 ) d ν d x ν cos k ρ 3 ( x δ 2 ) + O ( | k | ν + 3 e | Im k | ( ( δ 1 a ) ρ 1 + ( δ 2 δ 1 ) ρ 2 + ( x δ 2 ) ρ 3 ) ) .

Lemma 3 Let λ = k 2 , k = σ + i ω .

Then the following integral equations and also asymptotic behaviors hold for ν = 0 , 1 :
d ν d x ν ψ 3 ( x , λ ) = ( λ γ 2 + γ 2 ) d ν d x ν cos k ρ 3 ( x b ) + 1 k ρ 3 ( λ γ 1 + γ 1 ) d ν d x ν sin k ρ 3 ( x b ) ρ 3 k x b d ν d x ν sin k ρ 3 ( x t ) q ( t ) ψ 3 ( t , λ ) d t = ( λ γ 2 + γ 2 ) d ν d x ν cos k ρ 3 ( x b ) + 1 k ρ 3 ( λ γ 1 + γ 1 ) d ν d x ν sin k ρ 3 ( x b ) + O ( | k | ν + 1 e | Im k | ( x b ) ρ 3 ) , d ν d x ν ψ 2 ( x , λ ) = γ 6 ψ 3 ( δ 2 , λ ) γ 5 ψ 3 ( δ 2 , λ ) α 2 d ν d x ν cos k ρ 2 ( x δ 2 ) + 1 k ρ 2 θ 6 ψ 3 ( δ 2 , λ ) θ 5 ψ 3 ( δ 2 , λ ) α 2 d ν d x ν sin k ρ 2 ( x δ 2 ) ρ 2 k x δ 2 d ν d x ν sin k ρ 2 ( x t ) q ( t ) ψ 2 ( t , λ ) d t = ( λ γ 2 + γ 2 α 2 ) ( k ρ 3 γ 5 sin k ρ 3 ( δ 2 b ) d ν d x ν cos k ρ 2 ( x δ 2 ) ) + O ( | k | ν + 2 e | Im k | ( ( δ 2 b ) ρ 3 + ( x δ 2 ) ρ 2 ) ) , d ν d x ν ψ 1 ( x , λ ) = γ 4 ψ 2 ( δ 1 , λ ) γ 3 ψ 2 ( δ 1 , λ ) α 1 d ν d x ν cos k ρ 1 ( x δ 1 ) + 1 k ρ 1 θ 4 ψ 2 ( δ 1 , λ ) θ 3 ψ 2 ( δ 1 , λ ) α 1 d ν d x ν sin k ρ 1 ( x δ 1 ) ρ 1 k x δ 1 d ν d x ν sin k ρ 1 ( x t ) q ( t ) ψ 1 ( t , λ ) d t = ( λ γ 2 + γ 2 α 2 α 1 ) × ( k 2 ρ 2 ρ 3 γ 3 γ 5 sin k ρ 3 ( δ 2 b ) sin k ρ 2 ( δ 1 δ 2 ) d ν d x ν cos k ρ 1 ( x δ 1 ) ) + O ( | k | ν + 3 e | Im k | ( ( δ 2 b ) ρ 3 + ( δ 1 δ 2 ) ρ 2 + ( x δ 1 ) ρ 1 ) ) .
The function Δ ( λ ) is called the characteristic function, and numbers { μ n } n 1 are called the normalizing constants of problem (1)-(7) such that
Δ ( λ ) = λ ( γ 1 φ ( b , λ ) γ 2 φ ( b , λ ) ) + ( γ 1 φ ( b , λ ) γ 2 φ ( b , λ ) ) ,
(11)
μ n : = α 1 α 2 ρ 1 a δ 1 φ 2 ( x , λ n ) d x + α 2 ρ 2 δ 1 δ 2 φ 2 ( x , λ n ) d x + ρ 3 δ 2 b φ 2 ( x , λ n ) d x + α 1 α 2 β 1 ( θ 1 φ ( a , λ n ) θ 2 φ ( a , λ n ) ) 2 + 1 β 2 ( γ 1 φ ( b , λ n ) γ 2 φ ( b , λ n ) ) 2 .
(12)
Lemma 4 The following equality holds for each eigenvalue λ n
α 1 α 2 Δ ˙ ( λ n ) = κ n μ n .
Proof Since
ρ ( x ) ψ ( x , λ ) + q ( x ) ψ ( x , λ ) = λ ψ ( x , λ ) , ρ ( x ) φ ( x , λ n ) + q ( x ) φ ( x , λ n ) = λ φ ( x , λ n ) ,
we get
φ ( x , λ n ) ψ ( x , λ ) ψ ( x , λ ) φ ( x , λ n ) ( | a δ 1 + | δ 1 δ 2 + | δ 2 b ) = ( λ λ n ) ρ 1 2 a δ 1 ψ ( x , λ ) φ ( x , λ n ) d x + ( λ λ n ) ρ 2 2 δ 1 δ 2 ψ ( x , λ ) φ ( x , λ n ) d x + ( λ λ n ) ρ 3 2 δ 2 b ψ ( x , λ ) φ ( x , λ n ) d x .
After that, add and subtract Δ ( λ ) on the left-hand side of the last equality, and by using conditions (2)-(7), we obtain
Δ ( λ ) + ( λ λ n ) ( θ 1 ψ ( a , λ ) θ 2 ψ ( a , λ ) ) ( λ λ n ) ( γ 1 φ ( b , λ n ) γ 2 φ ( b , λ n ) ) + ( 1 α 1 ) ( ψ ( δ 1 , λ ) φ ( δ 1 , λ n ) φ ( δ 1 , λ n ) ψ ( δ 1 , λ ) ) + ( 1 α 2 ) ( ψ ( δ 2 , λ ) φ ( δ 2 , λ n ) φ ( δ 2 , λ n ) ψ ( δ 2 , λ ) ) = ( λ λ n ) ρ 1 2 a δ 1 ψ ( x , λ ) φ ( x , λ n ) d x + ( λ λ n ) ρ 2 2 δ 1 δ 2 ψ ( x , λ ) φ ( x , λ n ) d x + ( λ λ n ) ρ 3 2 δ 2 b ψ ( x , λ ) φ ( x , λ n ) d x ,
or
α 1 α 2 Δ ( λ ) λ λ n = α 1 α 2 ρ 1 2 a δ 1 ψ ( x , λ ) φ ( x , λ n ) d x + α 2 ρ 2 2 δ 1 δ 2 ψ ( x , λ ) φ ( x , λ n ) d x + ρ 3 2 δ 2 b ψ ( x , λ ) φ ( x , λ n ) d x + α 1 α 2 β 1 ( θ 1 ψ ( a , λ ) θ 2 ψ ( a , λ ) ) ( θ 1 φ ( a , λ n ) θ 2 φ ( a , λ n ) ) + 1 β 2 ( γ 1 φ ( b , λ n ) γ 2 φ ( b , λ n ) ) ( γ 1 ψ ( b , λ ) γ 2 ψ ( b , λ ) ) .
For λ λ n , α 1 α 2 Δ ˙ ( λ n ) = κ n μ n is obtained by using the equality
ψ ( x , λ n ) = κ n φ ( x , λ n )

and (12). □

Corollary 2 The eigenvalues of problem L are simple.

Lemma 5 [36]

Let { α i } i = 1 p be the set of real numbers satisfying the inequalities α 0 > α 0 > > α p 1 > 0 , and let { a i } i = 1 p be the set of complex numbers. If a p 0 , then the roots of the equation
e α 0 λ + a 1 e α 1 λ + + a p 1 e α 0 λ + a p = 0
have the form
λ n = 2 π n i α 0 + Ψ ( n ) ( n = 0 , ± 1 , ) ,

where Ψ ( n ) is a bounded sequence.

Now, from Lemma 2 and (11), we can write
Δ ( λ ) Δ 0 ( λ ) = O ( k 6 e | Im k | ( ( δ 1 a ) ρ 1 + ( δ 2 δ 1 ) ρ 2 + ( b δ 2 ) ρ 3 ) ) ,

where Δ 0 ( λ ) = k 7 θ 2 γ 2 γ 3 γ 5 ρ 1 ρ 2 ρ 3 sin k ρ 1 ( δ 1 a ) sin k ρ 2 ( δ 2 δ 1 ) sin k ρ 3 ( b δ 2 ) .

We can see that non-zero roots, namely λ n 0 of the equation Δ 0 ( λ ) = 0 , are real and analytically simple.

Furthermore, it can be proved by using Lemma 5 that
λ n 0 = n π ( δ 1 a ) ρ 1 + ( δ 2 δ 1 ) ρ 2 + ( b δ 2 ) ρ 3 + Ψ n , sup n | Ψ n | < .
(13)
Theorem 2 The eigenvalues { λ n } n 0 have the following asymptotic behavior for sufficiently large n:
λ n = λ n 4 0 + o ( 1 ) .
(14)
Proof Denote
G n : = { λ : k 2 = λ , | k | | k n 0 | + δ } ,
where k n 0 = λ n 0 and δ is a sufficiently small number. The relations
| Δ 0 ( λ ) | C δ | k | 7 e | Im k | ( ( δ 1 a ) ρ 1 + ( δ 2 δ 1 ) ρ 2 + ( b δ 2 ) ρ 3 )
and
Δ ( λ ) Δ 0 ( λ ) = O ( k 6 e | Im k | ( ( δ 1 a ) ρ 1 + ( δ 2 δ 1 ) ρ 2 + ( b δ 2 ) ρ 3 ) )

are valid for λ G n .

Then, by Rouche’s theorem that the number of zeros of Δ 0 ( λ ) coincides with the number of zeros of Δ ( λ ) in G n , namely n + 4 zeros, λ 0 , λ 1 , λ 2 , , λ n + 3 . In the annulus, between G n and G n + 1 , Δ ( λ ) has accurately one positive zero, namely k n 2 : k n = λ n 0 + δ n , for n 1 . So, it follows that λ n + 4 = k n 2 . Applying to Rouche’s theorem in η ε = { k : | k k n 0 | < ε } for sufficiently small ε and sufficiently large n, we get δ n = o ( 1 ) . Finally, we obtain the asymptotic formula
λ n = λ n 4 0 + o ( 1 ) .
Denote
Δ i ( λ ) : = W ( φ i , ψ i , x ) : = φ i ψ i φ i ψ i , x Ω i ( i = 1 , 3 ¯ ) ,

which are independent of x Ω i and are entire functions such that Ω 1 = [ a , δ 1 ) , Ω 2 = ( δ 1 , δ 2 ) , Ω 3 = ( δ 2 , b ] .

It can be easily seen that
Δ ( λ ) : = Δ 3 ( λ ) = α 2 Δ 2 ( λ ) = α 1 α 2 Δ 1 ( λ ) .

 □

Example Let q = 0 , a = 0 , b = π , δ 1 = π 4 , δ 2 = π 2 , θ 3 = γ 4 = 0 , γ 3 = 1 , θ 4 = 1 , θ 5 = γ 6 = 0 , γ 5 = 1 , θ 6 = 1 , γ 2 = 1 , γ 6 = 1 , γ 1 = γ 2 = 0 , θ 2 = 1 θ 1 = 1 , θ 1 = θ 2 = 0 .

Since
Δ ( λ ) = ρ 1 ρ 2 ρ 3 k 7 sin k ρ 1 δ 1 sin k ρ 2 ( δ 2 δ 1 ) sin k ρ 3 ( π δ 2 ) + O ( k 6 e | Im k | ( δ 1 ρ 1 + ( δ 2 δ 1 ) ρ 2 + ( π δ 2 ) ρ 3 ) ) ,
the eigenvalues of the boundary value problem (1)-(7) satisfy the following asymptotic formulae:
λ n 1 = k n 1 = 4 ( n 4 ) ρ 1 + ε n , λ n 2 = k n 2 = 4 ( n 4 ) ρ 2 + ε n , λ n 3 = k n 3 = 2 ( n 4 ) ρ 3 + ε n ,

where ε n = O ( n 1 ) .

3 Inverse problems

In this section, we study the inverse problems for the reconstruction of the boundary value problem (1)-(7) by Weyl function and spectral data.

We consider the boundary value problem L ˜ with the same form of L but with different coefficients q ˜ ( x ) , θ ˜ i , γ ˜ i , δ ˜ j , θ ˜ j , γ ˜ j , i = 1 6 , j = 1 , 2 .

If a certain symbol α denotes an object related to L, then the symbol α ˜ denotes the corresponding object related to L ˜ .

The Weyl function Let Φ ( x , λ ) be a solution of equation (1), which satisfies the condition ( λ θ 1 θ 1 ) Φ ( a , λ ) ( λ θ 2 θ 2 ) Φ ( a , λ ) = 1 and transmissions (4)-(7).

Assume that the function χ ( x , λ ) is the solution of equation (1) that satisfies the conditions χ ( a , λ ) = β 1 1 θ 2 , χ ( a , λ ) = β 1 1 θ 1 and the transmission conditions (4)-(7).

Since W [ χ , φ ] = 1 , the functions χ and φ are linearly independent. Therefore, the function ψ ( x , λ ) can be represented by
ψ ( x , λ ) = θ 2 ψ ( a , λ ) θ 1 ψ ( a , λ ) β 1 φ ( x , λ ) + Δ ( λ ) χ ( x , λ )
or
Φ ( x , λ ) = ψ ( x , λ ) Δ ( λ ) = χ ( x , λ ) + θ 2 ψ ( a , λ ) θ 1 ψ ( a , λ ) β 1 Δ ( λ ) φ ( x , λ )
(15)
that is called the Weyl solution, and
θ 2 ψ ( a , λ ) θ 1 ψ ( a , λ ) β 1 Δ ( λ ) = M ( λ )
(16)

is called the Weyl function.

Theorem 3 If M ( λ ) = M ˜ ( λ ) , then L = L ˜ , i.e., q ( x ) = q ˜ ( x ) , a.e. and θ i = θ ˜ i , γ i = γ ˜ i , i = 1 6 , δ j = δ ˜ j , θ j = θ ˜ j , γ j = γ ˜ j , j = 1 , 2 .

Proof We introduce a matrix P ( x , λ ) = [ P k j ( x , λ ) ] k , j = 1 , 2 by the formula
P ( x , λ ) ( φ ˜ Φ ˜ φ ˜ Φ ˜ ) = ( φ Φ φ Φ )
or
( P 11 ( x , λ ) P 12 ( x , λ ) P 21 ( x , λ ) P 22 ( x , λ ) ) = ( φ Φ ˜ + Φ φ ˜ φ Φ ˜ Φ φ ˜ φ Φ ˜ + φ ˜ Φ φ Φ ˜ φ ˜ Φ ) ,
(17)
where Φ ( x , λ ) = ψ ( x , λ ) Δ ( λ ) .
P 11 ( x , λ ) = χ ( x , λ ) φ ˜ ( x , λ ) φ ( x , λ ) χ ˜ ( x , λ ) + ( M ( λ ) M ˜ ( λ ) ) φ ( x , λ ) φ ˜ ( x , λ ) , P 12 ( x , λ ) = φ ( x , λ ) χ ˜ ( x , λ ) χ ( x , λ ) φ ˜ ( x , λ ) + ( M ˜ ( λ ) M ( λ ) ) φ ( x , λ ) φ ˜ ( x , λ ) .

Thus, if M ( λ ) M ˜ ( λ ) , then the functions P 11 ( x , λ ) and P 12 ( x , λ ) are entire in λ for each fixed x.

Denote G w = { λ : λ = k 2 , | k k δ | > w , δ = 1 , 2 , } and G ˜ w = { λ : λ = k 2 , | k k ˜ δ | > w , δ = 1 , 2 , } , where w is sufficiently small number, k δ and k ˜ δ are square roots of the eigenvalues of the problem L and L ˜ , respectively. It is easily shown that
Φ i ( ν ) ( x , λ ) C w | k | ν ( 2 + i ) e | Im λ | ( ( δ 1 a ) ρ 1 + ( δ 2 δ 1 ) ρ 2 + ( b δ 2 ) ρ 3 ) , x Ω i ( i = 1 , 2 , 3 ) , ν = 0 , 1
(18)
are valid for sufficiently large | λ | , where Ω 1 = [ a , δ 1 ) , Ω 2 = ( δ 1 , δ 2 ) and Ω 3 = ( δ 2 , b ] . Hence, Lemma 2 and (18) yield that
| P 11 ( x , λ ) | C w , | P 12 ( x , λ ) | C w | k | 1 for  λ Ω  and for  λ G w G ˜ w .
(19)
According to (19), and Liouville’s theorem, P 11 ( x , λ ) = C ( x ) and P 12 ( x , λ ) 0 for x [ a , b ] { δ 1 , δ 2 , δ ˜ 1 , δ ˜ 2 } . By virtue of (17), we get
φ ( x , λ ) = C ( x ) φ ˜ ( x , λ ) , Φ ( x , λ ) = C ( x ) Φ ˜ ( x , λ ) .
(20)
It is obvious that
W [ Φ ( x , λ ) , φ ( x , λ ) ] = Φ ( a , λ ) ( θ 1 λ θ 1 ) Φ ( a , λ ) ( θ 2 λ θ 2 ) = ψ ( a , λ ) ( θ 1 λ θ 1 ) ψ ( a , λ ) ( θ 2 λ θ 2 ) Δ ( λ ) 1 ,

and similarly, W [ Φ ˜ ( x , λ ) , φ ˜ ( x , λ ) ] 1 . Thus, we have C 2 ( x ) 1 .

Otherwise, the following asymptotic expressions hold
φ 1 ( x , λ ) = φ ˜ 1 ( x , λ ) = λ 2 e i k ( x a ) ρ 1 ( 1 + o ( 1 ) ) for  x < δ 1  and  x < δ ˜ 1 , φ 2 ( x , λ ) = φ ˜ 2 ( x , λ ) = λ 3 2 2 e i k ( ( δ 1 a ) ρ 1 + ( x δ 1 ) ρ 2 ) ( 1 + o ( 1 ) ) for  δ 1 < x < δ 2  and  δ ˜ 1 < x < δ ˜ 2 , φ 3 ( x , λ ) = φ ˜ 3 ( x , λ ) = λ 2 2 e i k ( ( δ 1 a ) ρ 1 + ( δ 2 δ 1 ) ρ 2 + ( x δ 2 ) ρ 3 ) ( 1 + o ( 1 ) ) for  δ 2 < x  and  δ ˜ 2 < x .
(21)
Without loss of generality, we assume that δ 1 < δ ˜ 1 and δ 2 < δ ˜ 2 . From (20)-(21), we get C ( x ) 1 for x [ a , δ 1 ) ( δ ˜ 1 , δ 2 ) ( δ ˜ 2 , b ] and also
1 λ 1 / 2 ( 1 + o ( 1 ) ) C ( x ) = e ( x δ 1 ) ρ 2 ( 1 + o ( 1 ) ) for  x ( δ 1 , δ ˜ 1 ) , 1 λ 1 / 2 e ( x δ 1 ) ρ 2 ( 1 + o ( 1 ) ) C ( x ) = e ( δ 2 δ 1 ) ρ 2 + ( x δ 2 ) ρ 3 ( 1 + o ( 1 ) ) for  x ( δ 2 , δ ˜ 2 ) .
(22)

As | λ | in (22), we contradict C 2 ( x ) 1 . Therefore, δ 1 = δ ˜ 1 , δ 2 = δ ˜ 2 . Thus, φ ( x , λ ) φ ˜ ( x , λ ) , Φ ( x , λ ) Φ ˜ ( x , λ ) and ψ ( x , λ ) ψ ( x , λ ) ψ ˜ ( x , λ ) ψ ˜ ( x , λ ) . Hence, from equation (1) and transmission conditions (4)-(7), q ( x ) = q ˜ ( x ) , a.e., θ i = θ ˜ i , γ i = γ ˜ i , i = 3 6 , and from (9) and (10), θ i = θ ˜ i , θ j = θ ˜ j , γ j = γ ˜ j , i , j = 1 , 2 . □

Lemma 6 The following representation holds
M ( λ ) = n = 0 α 1 α 2 μ n ( λ n λ ) .
Proof Weyl function M ( λ ) is a meromorphic function with respect to λ, which has simple poles at λ n . Therefore, we calculate
Re s λ = λ n M ( λ ) = θ 2 ψ ( a , λ n ) θ 1 ψ ( a , λ n ) β 1 Δ ˙ ( λ n ) .
Since κ n = θ 2 ψ ( a , λ n ) θ 1 ψ ( a , λ n ) β 1 and Δ ˙ ( λ n ) = κ n μ n α 1 α 2 ,
Re s λ = λ n M ( λ ) = α 1 α 2 μ n .
(23)
Let Γ N = { λ : λ = k 2 , | k | = λ N + ε } , where ε is a sufficiently small number. Consider the contour integral I N ( λ ) = 1 2 π i Γ n M ( μ ) μ λ d μ , λ int Γ N . For λ G w ,
Δ ( λ ) | λ | 7 / 2 C w e | Im k | ( ( δ 1 a ) ρ 1 + ( δ 2 δ 1 ) ρ 2 + ( b δ 2 ) ρ 3 )
satisfies. Using this equality and (16), we get
| M ( λ ) | C w | λ | for  λ G w .
Thus, lim N I N ( λ ) = 0 . As a result, the residue theorem and (23) yield
M ( λ ) = n = 0 α 1 α 2 μ n ( λ n λ ) .

 □

Theorem 4 If λ n = λ ˜ n and μ n = μ ˜ n for all n, then L L ˜ , i.e., q ( x ) = q ˜ ( x ) , a.e., θ i = θ ˜ i , γ i = γ ˜ i , i = 1 6 , δ j = δ ˜ j , θ j = θ ˜ j , γ j = γ ˜ j , j = 1 , 2 . Hence, problem (1)-(7) is uniquely determined by spectral data { λ n , μ n } .

Proof If λ n = λ ˜ n and μ n = μ ˜ n for all n, then M ( λ ) = M ˜ ( λ ) by Lemma 6. Therefore, we get L = L ˜ by Theorem 3.

Let us consider the boundary value problem L 1 that we get the condition θ 2 y ( a , λ ) θ 1 y ( a , λ ) = 0 instead of condition (2) in L. Let { τ n } n 0 be the eigenvalues of the problem L 1 . It is clear that τ n are zeros of
Δ 1 ( τ ) : = θ 2 ψ ( a , τ ) θ 1 ψ ( a , τ ) .

 □

Theorem 5 If λ n = λ ˜ n and τ n = τ ˜ n for all n, then L ( q , δ j , θ k , γ i , γ j ) = L ( q ˜ , δ ˜ j , θ ˜ k , γ ˜ i , γ ˜ j ) , i = 1 6 , k = 3 6 , j = 1 , 2 .

Hence, the problem L is uniquely determined by the sequences { λ n } and { τ n } , except coefficients θ j and θ j .

Proof Since the characteristic functions Δ ( λ ) and Δ 1 ( τ ) are entire of order 1 2 , functions Δ ( λ ) and Δ 1 ( τ ) are uniquely determined up to multiplicative constant with their zeros by Hadamard’s factorization theorem [37]
Δ ( λ ) = C n = 0 ( 1 λ λ n ) , Δ 1 ( τ ) = C 1 n = 0 ( 1 τ τ n ) ,

where C and C 1 are constants dependent on { λ n } and { τ n } , respectively. Therefore, when λ n = λ ˜ n and τ n = τ ˜ n for all n, Δ ( λ ) Δ ˜ ( λ ) and Δ 1 ( τ ) Δ ˜ 1 ( τ ) . Hence, θ 2 ψ ( a , τ ) θ 1 ψ ( a , τ ) = θ ˜ 2 ψ ˜ ( a , τ ) θ ˜ 1 ψ ˜ ( a , τ ) . As a result, we get M ( λ ) = M ˜ ( λ ) by (16). So, the proof is completed by Theorem 3. □

Declarations

Authors’ Affiliations

(1)
Department of Mathematics, Faculty of Sciences, Cumhuriyet University

References

  1. Fulton CT: Two-point boundary value problems with eigenvalue parameter contained in the boundary conditions. Proc. R. Soc. Edinb. A 1977, 77: 293-308. 10.1017/S030821050002521XMathSciNetView Article
  2. Kobayashi M: Eigenvalues of discontinuous Sturm-Liouville problems with symmetric potentials. Comput. Math. Appl. 1989, 18(4):355-364.
  3. Likov, AV, Mikhailov, YA: The Theory of Heat and Mass Transfer. Qosenergaizdat (1963)
  4. Shkalikov AA: Boundary value problems for ordinary differential equations with a parameter in boundary conditions. Tr. Semin. Im. I.G. Petrovskogo 1983, 9: 190-229.MathSciNet
  5. Binding PA, Browne PJ, Seddighi K: Sturm-Liouville problems with eigenparameter dependent boundary conditions. Proc. Edinb. Math. Soc. 1993, 37: 57-72.MathSciNetView Article
  6. Binding PA, Browne PJ, Watson BA: Inverse spectral problems for Sturm-Liouville equations with eigenparameter dependent boundary conditions. J. Lond. Math. Soc. 2000, 62: 161-182. 10.1112/S0024610700008899MathSciNetView Article
  7. Binding PA, Browne PJ, Watson BA: Equivalence of inverse Sturm-Liouville problems with boundary conditions rationally dependent on the eigenparameter. J. Math. Anal. Appl. 2004, 291: 246-261.MathSciNetView Article
  8. Fulton CT: Singular eigenvalue problems with eigenvalue parameter contained in the boundary conditions. Proc. R. Soc. Edinb. A 1980, 87: 1-34. 10.1017/S0308210500012312MathSciNetView Article
  9. Mennicken R, Schmid H, Shkalikov AA: On the eigenvalue accumulation of Sturm-Liouville problems depending nonlinearly on the spectral parameter. Math. Nachr. 1998, 189: 157-170. 10.1002/mana.19981890110MathSciNetView Article
  10. Russakovskii EM: Operator treatment of boundary problems with spectral parameters entering via polynomials in the boundary conditions. Funct. Anal. Appl. 1975, 9: 358-359.MathSciNetView Article
  11. Schmid H, Tretter C: Singular Dirac systems and Sturm-Liouville problems nonlinear in the spectral parameter. J. Differ. Equ. 2002, 181(2):511-542. 10.1006/jdeq.2001.4082MathSciNetView Article
  12. Ozkan AS, Keskin B: Spectral problems for Sturm-Liouville operator with boundary and jump conditions linearly dependent on the eigenparameter. Inverse Probl. Sci. Eng. 2012, 20: 799-808. 10.1080/17415977.2011.652957MathSciNetView Article
  13. Akdoğan Z, Demirci M, Mukhtarov OS: Sturm-Liouville problems with eigendependent boundary and transmissions conditions. Acta Math. Sci. 2005, 25(4):731-740.MathSciNet
  14. Akdoğan Z, Demirci M, Mukhtarov OS: Discontinuous Sturm-Liouville problem with eigenparameter-dependent boundary and transmission conditions. Acta Appl. Math. 2005, 86: 329-334. 10.1007/s10440-004-7466-3MathSciNetView Article
  15. Binding PA, Browne PJ, Watson BA: Sturm-Liouville problems with boundary conditions rationally dependent on the eigenparameter II. J. Comput. Appl. Math. 2002, 148: 147-169. 10.1016/S0377-0427(02)00579-4MathSciNetView Article
  16. Kerimov NB, Memedov KK: On a boundary value problem with a spectral parameter in the boundary conditions. Sib. Mat. Zh. 1999, 40(2):325-335. (English translation: Sib. Math. J. 40(2), 281-290 (1999))
  17. Mukhtarov OS, Kadakal M, Muhtarov FS: On discontinuous Sturm-Liouville problem with transmission conditions. J. Math. Kyoto Univ. 2004, 444: 779-798.MathSciNet
  18. Tunç E, Muhtarov OS: Fundamental solution and eigenvalues of one boundary value problem with transmission conditions. Appl. Math. Comput. 2004, 157: 347-355. 10.1016/j.amc.2003.08.039MathSciNetView Article
  19. Yakubov S: Completeness of Root Functions of Regular Differential Operators. Longman, New York; 1994.
  20. Meschanov VP, Feldstein AL: Automatic Design of Directional Couplers. Sviaz, Moscow; 1980.
  21. Tikhonov AN, Samarskii AA: Equations of Mathematical Physics. Pergamon, Oxford; 1990.
  22. Voitovich NN, Katsenelbaum BZ, Sivov AN: Generalized Method of Eigen-Vibration in the Theory of Diffraction. Nauka, Moscow; 1997.
  23. McNabb A, Anderssen R, Lapwood E: Asymptotic behavior of the eigenvalues of a Sturm-Liouville system with discontinuous coefficients. J. Math. Anal. Appl. 1976, 54: 741-751. 10.1016/0022-247X(76)90193-1MathSciNetView Article
  24. Shieh CT, Yurko VA: Inverse nodal and inverse spectral problems for discontinuous boundary value problems. J. Math. Anal. Appl. 2008, 347: 266-272. 10.1016/j.jmaa.2008.05.097MathSciNetView Article
  25. Willis C: Inverse Sturm-Liouville problems with two discontinuities. Inverse Probl. 1985, 1: 263-289. 10.1088/0266-5611/1/3/010MathSciNetView Article
  26. Yang CF: An interior inverse problem for discontinuous boundary-value problems. Integral Equ. Oper. Theory 2009, 65: 593-604. 10.1007/s00020-009-1693-yView Article
  27. Yang CF, Yang XP: An interior inverse problem for the Sturm-Liouville operator with discontinuous conditions. Appl. Math. Lett. 2009, 22: 1315-1319. 10.1016/j.aml.2008.12.001MathSciNetView Article
  28. Yang Q, Wang W: Asymptotic behavior of a differential operator with discontinuities at two points. Math. Methods Appl. Sci. 2011, 34: 373-383.MathSciNet
  29. Yurko VA: Integral transforms connected with discontinuous boundary value problems. Integral Transforms Spec. Funct. 2000, 10: 141-164. 10.1080/10652460008819282MathSciNetView Article
  30. Litvinenko, ON, Soshnikov, VI: The Theory of Heterogenious Lines and Their Applications in Radio Engineering. Moscow (1964)
  31. Shepelsky DG: The inverse problem of reconstruction of the medium’s conductivity in a class of discontinuous and increasing functions. Adv. Soviet Math. 19. In Spectral Operator Theory and Related Topics. Am. Math. Soc., Providence; 1994:209-232.
  32. Anderssen RS: The effect of discontinuities in density and shear velocity on the asymptotic overtone structure of torsional eigenfrequencies of the earth. Geophys. J. R. Astron. Soc. 1997, 50: 303-309.View Article
  33. Lapwood FR, Usami T: Free Oscillations of the Earth. Cambridge University Press, Cambridge; 1981.
  34. Ambarsumian VA: Über eine frage der eigenwerttheorie. Z. Phys. 1929, 53: 690-695. 10.1007/BF01330827View Article
  35. Borg G: Eine umkehrung der Sturm-Liouvilleschen eigenwertaufgabe. Bestimmung der differentialgleichung durch die eigenwerte. Acta Math. 1946, 78: 1-96. 10.1007/BF02421600MathSciNetView Article
  36. Zhdanovich VF: Formulae for the zeros of Dirichlet polynomials and quasi-polynomials. Dokl. Akad. Nauk SSSR 1960, 135(8):1046-1049.
  37. Titchmarsh EC: The Theory of Functions. Oxford University Press, London; 1939.

Copyright

© Güldü; licensee Springer 2013

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.