0 Boundary Value Problems

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Eigenparameter dependent inverse boundary value problem for a class of Sturm-Liouville operator

Khanlar R Mamedov and F Ayca Cetinkaya*

"Correspondence: faycacetinkaya@mersin.edu.tr Science and Letters Faculty, Department of Mathematics, Mersin University Mersin, 33343,Turkey

Abstract

In this work a Sturm-Liouville operator with piecewise continuous coefficient and spectral parameter in the boundary conditions is considered. The eigenvalue problem is investigated; it is shown that the eigenfunctions form a complete system and an expansion formula with respect to the eigenfunctions is obtained. Uniqueness theorems for the solution of the inverse problem with a Weyl function and spectral data are proved. MSC: 34L10; 34L40; 34A55

Keywords: Sturm-Liouville operator; expansion formula; inverse problem; Weyl function

1 Introduction

We consider the boundary value problem

-y" + q(x)y = X2p (x)y, 0 < x < n, U(y) :=y'(0) + (a! - X2a2)y(0) = 0, y (y) := X2(ßiy\n ) + ß2y(n )) - ßi y (n ) - ß3y(n ) = 0,

(!) (2) (3)

where q(x) e ¿2(0, n) is a real valued function, X is a complex parameter, ai, fy, i = 1,2, j = 1,4 are positive real numbers and

P (x) =

1, 0 < x < a,

a < x < n,

ft Spri

ringer

where 0 < y =1.

Physical applications of the eigenparameter dependent Sturm-Liouville problems, i.e. the eigenparameter appears not only in the differential equation of the Sturm-Liouville problem but also in the boundary conditions, are given in [1-4]. Spectral analyses of these problems are examined as regards different aspects (eigenvalue problems, expansion problems with respect to eigenvalues, etc.) in [5-13]. Similar problems for discontinuous Sturm-Liouville problems are examined in [14-18].

© 2014 Mamedov and Cetinkaya; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.Org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the originalwork is properly credited.

Inverse problems for differential operators with boundary conditions dependent on the spectral parameter on a finite interval have been studied in [19-23]. In particular, such problems with discontinuous coefficient are studied in [24-27].

We investigate a Sturm-Liouville operator with discontinuous coefficient and a spectral parameter in boundary conditions. The theoretic formulation of the operator for the problem is given in a suitable Hilbert space in Section 2. In Section 3, an asymptotic formula for the eigenvalues is given. In Section 4, an expansion formula with respect to the eigenfunctions is obtained and Section 5 contains uniqueness theorems for the solution of the inverse problem with a Weyl function and spectral data.

2 Operator formulation

Let p(x, X) and ty(x, X) be the solutions of (1) satisfying the initial conditions

p(0, X) = 1, p'(0, X)=X2a2- ai, (4)

ty (n, X)=A- X2A, ty '(n, X) = X2ft- As. (5)

For the solution of (1), the following integral representation as i-i±(x) = ±x^/p(x) + a(1 ^ y/p(x)) is obtained similar to [28] for all X:

e(x, X) = 1 ( 1 + —= ) eiX^+(x) + 1 ( 1 - —= \ eiX^(x) + T * K(x, t)eiXt dt, 2\ Jp(x)J 2\ Jp(x)} J-V+(x)

where K(x, •) e fi+(x)). The following properties hold for the kernel K(x, t)

which has the partial derivative Kx belonging to the space L1(-^+(x), fi+(x)) for every x e [0, n]:

dLK <***«> = ÏtP(x^(1 + TM) q(x), (6)

dfxK <x, +°>- iK x- °>=-¿mi1- jpm)qW- (7)

We obtain the integral representation of the solution p(x, X):

f^ (x) f^ (x) ~ sin Xt

p(x, X)=p0(x, X)+ / A(x, t) cos Xtdt + (X2a2 - a^W A (x, t)-dt, (8)

Jo Jo X

A(x, t) = K (x, t) - K (x, -t), A (x, t) = K (x, t) + K (x, -t)

satisfying (6), (7). Let us define

A(X) := p(x, X), ty(x, X)) = p(x, X)ty'(x, X) - p'(x, X)ty(x, X),

which is independent from x e [0, n ]. Substituting x = 0 and x = n into (9) we get

A(X) = -U (f) = V (p).

The function A(X) is entire and has zeros at the eigenvalues of the problem (1)-(3). In the Hilbert space Hp = L2,p(0, n) ® C2 let an inner product be defined by

f,g): = f h(x)g1(x)p(x) dx

+ flg2 + fgB «2 82

(Mx)\ f2 \f3 )

e Hp, g =

(gi{x)\

e Hp, 82:= Aft- ^3^4>0.

We define the operator

L(f ):=

(-fi'(x) + q(x)f1(x)\

f (0) + af(0) ) +fn)/

D(L) = f e Hp :Ji(x),f{(x) e AC[0, n], f e L2 [0, n], f2 = ^(Q)/ = M' (n) + M(n)},

m = p-){-f;>+q(x)fi}-

The boundary value problem (1)-(3) is equivalent to the equation LY = X2 Y .WhenX = Xn are the eigenvalues, the eigenfunctions of operator L are in the form of

$(x, Xn) = ®n :=

p ( x, X n ) a2p(0, Xn) , Xn )+fy2P(n, Xn)/

, n = 1,2.

For any eigenvalue Xn the solutions (4), (5) satisfy the relation f (x, Xn) = knp(x, Xn)

and the normalized numbers of the boundary value problem (1)-(3) are given below:

an := I p2(x, Xn)p(x) dx + a2p2(0, Xn)

+ T (Ap'(n, Xn) + fy2P(n, Xn)) . 82

Lemma 1 The eigenvalues of the boundary value problem (l)-(3) are simple, i.e.

À (k) = 2knknan.

Proof Since

-q>"(x, kn) + q(x)y(x, kn) = k2np(x)q>(x, kn),

-f "(x, k) + q(x)f (x, k) =k2p (x)f (x, k),

we get

d [p(x, kn)f '(x, k)-^'(x, kn)f (x, k)] = (k2n - k2)p(x)p(x, kn)f (x, k). With the help of (2), (3) we get

À(kn)-À(k) = (k - k2) / ^(x, kn)f (x, k)p(x) dx. Jo

Adding

(k2- k2)a2^(0, kn)f (0, k)

(kn - k2)

(P4,v'(n, kn)+^2^(n, kn)) (P4f '(n, k)+ fof (n, k))

to both sides of the last equation and using the relations (10), (11) we have

À(kn) - À(k) = (kn + k)(kn - k)knan.

Taking k ^ kn, we find (12).

3 Asymptotic formulas of the eigenvalues

The solution of (l) satisfying the initial conditions (4) when q(x) = 0 is in the following form:

^0(x, k) = c0(x, k) + (k2a2 - a1)

s0(x, k)

c0(x, k) =

cos kx,

2(1 + -p.) cos kj+(x) + i(1-

0 < x < a, ) cos kj-(x), a < x < n,

s0(x, k) =

sin kx k ,

1 Î1 , 1_\sin_k£+(x) 1

2(1 + -r))5

2 Jp(x)

+ 2(1-

1_\ sin kj (x)

0 < x < a, , a < x < n.

The eigenvalues X°n (n = 0,^1, ^2,...) oftheboundaryvalueproblem (l)-(3) whenq(x) = 0 can be found by using the equation

Ao(k) = (X2^ - P3)Mn, x) - (A - (n, x) = 0

and can be represented in the following way:

X0n = n + t (n), n = 0,Tl, T2,...,

where supn |t(n)| < +<». Roots X°° of the function A0(X) are separated, i.e.,

Lemma 2 The eigenvalues of the boundary value problem (1)-(3) are in the form of

inf xn - X0| = t >0.

0 dn nn .-->

Xn = Xn + 777 + , Xn > 0,

where (dn) is a bounded sequence,

4xnA (X0)

ai - a2

q(t) cos(x°n^ (n)) /p(t)

and {nn} e l2.

Proof From (8), it follows that

P ^ + )

, X) = ^0(n, X)+ / A(n, t) cos Xtdt

The expressions of A(X) and A0(X) let us calculate A(X) - A0(X):

Therefore, for sufficiently large n, on the contours Fn = jx : \X\ = |x0| + 2 we have

|A(X)-A0M| < |A0(X)|.

By the Rouche theorem, we obtain the result that the number of zeros of the function

{A(X)-Ao(X)} + Ao(X) = A(X)

inside the contour Yn coincides with the number of zeros of the function A0(X). Moreover, applying the Rouche theorem to the circle yn(S) = {X: \X - X°n \< 5} we find, for sufficiently large n, that there exists one zero Xn of the function A(X) in yn(S). Owing to the arbitrariness of 5 > 0 we have

Xn = X°° + £n, €n = 0(1), n ^X. (6)

Substituting (16) into (15), as n ^ x taking into account the equality A0(xn) = 0 and the relations sin en^+(n) ^ en^+(n), cos en^+(n) ^ 1, integrating by parts and using the properties of the kernels A(x, t) and A(x, t) we have

^, dn + nn

x^ + €n x^

fl+W fl+W

PI (n ) ç^ (n )

nn = I At(n, t) sinX°ntdt +(ai - a2) / At(n, t) cos X°ntdt. Jo Jn

Let us show that nn e l2. It is obvious that nn can be reduced to the integral

№ + (n)

fi R(t)e'xtdt,

J-1+(n )

where R(t) e L2(-i+(n), ¡i+(n)). Now, take

/1+ (n )

R(t)eixtdt.

1 + (n )

It is clear from [28] (p.66) that {Zn} = Z (Xn) e l2. By virtue of this we have {nn} e l2. The lemma is proved. □

4 Expansion formula with respect to eigenfunctions

Denote

G(x, t; X) := -

y(t, X)^(x, X), t < x, ^ (t, X)ç(x, X), t > x

and consider the function

y(x, X) := j G(x, t; X)f (t)p(t) dt - fj t (x, X) + a|)^ X). (18)

Theorem 3 The eigenfunctions $(x, Xn) of the boundary value problem (1)-(3) form a complete system in L2,p (0, n) ® C2.

Proof With the help of (10) and (12), we can write

t (x, Xn)= y(x, Xn). (19)

Using (17) and (18) we get

ResX=XKy(x,X) = -—-v(x,Xn) V(t,Xn)f (t)p(t) dt

2Xnan 0

1 V(x,Xn)(f1- 7^). (0)

2X n an V k)

Now let f (x) e L2,p (0, n) ® C2 and assume

($(x, Xn),f (x)) = / p(x, Xn)f(x) p (x) dx + ^(0, Xn)f2

(P4V>'(n, Xn) + , Xn))f3

= 0. (21)

Then from (20), we have ResX=Xn y(x, X) = 0. Consequently, for fixed x e [0, n] the function y(x, X) is entire with respect to X. Let us denote

Gs := {X : |X - X^ > 5,n = 0,^1,T2,...},

where 5 is sufficiently small positive number. It is clear that the relation below holds:

| A(X) | > C|X|3e|ImX e Gs, C = cons. (22)

From (18) it follows that for fixed 5 >0 and sufficiently large X* > 0 we have C

|y(x,X)| < —, X e G5, |X|>X*, C = cons. | X|

Using maximum principle for module of analytic functions and Liouville theorem, we get y(x, X) = 0. From this we obtain f (x) = 0 a.e. on [0, n]. Thus we conclude the completeness of the eigenfunctions $(x, Xn) in L2,p (0, n) ® C2. □

Theorem 4 Iff (x) e D(L), then the expansion formula

f(x) = ^2 any(x, Xn) (3)

is valid, where

a" 2a,

- i y(t, Xn)f(t)p(t) dt,

and the series converges uniformly with respect to x e [0,n]. Forf (x) e L2,p(0,n), the series converges in L2pP (0, n), moreover, the Parseval equality holds:

¡■it to I f (x)| p(x) dx = an|an|2.

an ^n |

Proof Since ^(x, X) and t (x, X) are the solutions of the boundary value problem (1)-(3), we have

t (x, X) If [_y"(t, X) + q(t)y(t, X)f (t) ^

У(X, X) = ^^(i0 X2 dt

P(x, X) | /-x [-t"(t, X) + q(t)t(t, X)f(t) dt

A(X) (A X2

f1 -t (x, X) + -A- <p(x, X). (24)

A(X) A(X)

Integrating by parts and taking into account the boundary conditions (2), (3) we obtain y(x, X) = -X2f (x) - X2 [Z1(x, X) + Z2(x, X)]

f1 t (x, X)^^7^ y(x, X), (25)

A(X) A(X)

1 r , 1 r , ,

Z(x, X) = A(X) t (x, X) y q>'(t, X)f(t) dt + A(X)v(x, X) J t (t, X)f(t) dt, Z2(x, X) = Ax [(X2a2 - a1)t (x, X)f(0)] - A^y [(X2ft - A)^ (x, X)f(n)]

t (x, X) i q>(t, X)q(t)f(t) dt + —<p(x, X)/ t (t, X)q(t)f(t) dt. J0 A(x) Jx

If we consider the following contour integral where Tn is a counter-clockwise oriented contour:

In(x) =-(f Xy(x, X) dX,

2ni JVn

and then taking into consideration (20) we get

In(x) = ^ Resx=xn [Xy(x, X)]

to TO X f TO X f

= / anv(x, Xn )+/ --1 (x, Xn) -/ --q> (x, Xn), (26)

tr A (xn) A (xn)

an = — I xn)

■ i q>(t, Xn)f (t)p(t) dt.

On the other hand, with the help of (25) we get

1 £ X' X nfi

In(x) =f (x) - -—; ® [Zi(x, X) + Z2(x, X)] dX + 2_\ —r ^ (x, Xn)

2n 1 JVn „=i A(Xn)

- i-Xhv(x, Xn). (27)

^ A (Xn)

„ ^ A(Xn) Comparing (26) and (27) we obtain

^anV(x, Xn) = f (x) + en(x),

en(x) =--S [Z1(x,X) + Z2(x,X)1 dX.

2ni JrnL

The relations below hold for sufficiently large X* >0 i i C2

max Z2(x,X) <-4, X e Gs, \X\< X*, (28)

xe[0,n ]' 1 \ X \2

max |Z1(x,X)| < X e Gs, \X\<X*. (9)

xe[0,n ]' 1 \ X \2

The validity of

lim max |en (x)| = 0

n^x xe[0,n ]

can easily be seen from (28) and (29). The last equation gives us the expansion formula

f (x) = any(x, Xn).

Since the system of $(x, Xn) is complete and orthogonal in L2,p(0, n) ® C2, the Parseval equality

I f(x)| p(x) dx = an\an\

j0 n=1

holds. □

5 Uniqueness theorems

We consider the statement of the inverse problem of the reconstruction of the boundary value problem (1)-(3) from the Weyl function.

Let the functions c(x, X) and s(x, X) denote the solutions of (1) satisfying the conditions c(0, X) = 1, c'(0, X) = 0, 5(0, X) = 0 and s'(0, X) = 1, respectively, and p(x, X) and ^(x, X) be the solutions of (1) under the initial conditions (4), (5).

Further, let the function $(x, X) be the solution of (1) satisfying U($) = 1 and V($) = 0. We set

M(X) := ^. ( ) A(X)

The functions $(x, X) and M(X) are called the Weyl solution and the Weyl function for the boundary value problem (1)-(3), respectively. The Weyl function is a meromorphic function having simple poles at points Xn, eigenvalues of the boundary value problem of (1)-(3). The Wronskian

W (x):=[cp(x, X), $(x, X))

does not depend on x. Taking x = 0, we get

W (0) = p(0, X)$'(0, X)-p'(0, X)$(0, X) = 1.

Hence,

W (x) = p(x, X), $(x, X)) = 1. (30)

In view of (4) and (5), we get for X = Xn

*(x, X) = ■ (

Using (31) we obtain

M(X) =--X

( ) A(X) ,

where A0(X) = — (0, X) is the characteristic function of the boundary value problem L0:

ly = X2y, 0 < x < n, y(0) = 0, V (y) = 0.

It is clear that

$(x, X) = s(x, X) + M(X)p(x, X). (32)

Theorem 5 The boundary value problem of (1)-(3) is identically denoted by the Weyl function M(X).

Proof Let us denote the matrix P(x, X) = [Pjk(x, X)];-,k=1,2 as p (x, X) $ (x, X)\ / p(x, X) $(x, X) \

P(x, X)

p'(x,X) $'(x,X)/ \p'(x,X) $'(x,X) j

Then we have

p(x, X) = P11(x, X)p (x, X) + P12(x, X)p'(x, X), $(x, X) = P11(x, X)$ (x, X) + P12(x, X)$'(x, X)

P11(x,X) = p(x,X)$'(x,X) -p'(x,X)$(x,X), P12(x, X) = p (x, X)$(x, X)-p(x, X)$ (x, X).

Taking (31) into consideration in (35) we get

P11(x, X) = 1 + —1 (x, X)[p'(x, X) - p'(x, X)] + —p(x, X)[f(x, X) - t '(x, X)], A (X) A (X)

P12(x,X) = —[p(x,X)t(x,X) - p(x,X)t(x,X)]. A(X)

From the estimates as |X|^to

p'(x, X) - p'(x, X)

A(X) t'(x, X) - t'(x, X)

A(X) we have from (36)

| ImX|^+(x)

| ImX|(^+(n)-^+(x))

lim max |P11(x, X)-1= lim max |P12(x, x) = 0

|X|^TO xe[0,n r |X|^TO xe[0,n ]' 1

for X e G5.

Now, if we take into consideration (32) and (35), we have

P11(x, X) = p(x, X)s'(x, X) - pp(x, X)s(x, X) + p'(x, X)p(x, X)[M(X) -M(X)], P12(x,X) = p(x,X)s(x,X) - p(x,X)s(x,X) + p(x,X)p(x,X)[M(X) -M(X)].

Therefore if M(X) = M(X), one has

P11(x, X) = p(x, X)S'(x, X) - s(x, X)p'(x, X), P12(x, X) = p(x, X)s(x, X) - s(x, X)p(x, X).

Thus, for every fixed x functions P11(x, X) and P12(x, X) are entire functions for X. It can easily be seen from (37) that P11(x, X) = 1 and P12(x, X) = 0. Consequently, we get p(x, X) = p (x, X) and $(x, X) = $ (x, X) for every x and X. Hence, we arrive at q(x) = q(x). □

The validity of the equation below can be seen analogously to [29]:

M(X)=M(0) + £ (38)

anXn (X2- X2n)

Theorem 6 The spectral data identically define the boundary value problem (1)-(3).

Proof From (38), it is clear that the function M(X) can be constructed by Xn. Since XXn = Xn for every n e N, we can say that M(X) = M(X). Then from Theorem 5, it is obvious that

L = L. □

Competing interests

The authors declare that they have no competing interests.

Authors' contributions

All authors contributed equally to the manuscript and read and approved the final manuscript.

Acknowledgements

This work is supported by The Scientific and Technological Research Council of Turkey (TÛBÎTAK).

Received: 10 March 2014 Accepted: 29 July 2014 Published online: 25 September 2014

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doi:10.1186/s13661-014-0194-3

Cite this article as: Mamedov and Cetinkaya: Eigenparameter dependent inverse boundary value problem for a class of Sturm-Liouville operator. Boundary Value Problems 2014 2014:194.

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