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0Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2013, Article ID 361989,10 pages http://dx.doi.org/10.1155/2013/361989
Research Article
Inverse Problems for the Quadratic Pencil of the Sturm-Liouville Equations with Impulse
Rauf Kh. Amirov1 and A. Adiloglu Nabiev2
1 Department of Mathematics, Faculty of Sciences, Cumhuriyet University, 58140 Sivas, Turkey
2 Department of Secondary Science and Mathematics Education, Faculty of Education, Cumhuriyet University, 58140 Sivas, Turkey Correspondence should be addressed to Rauf Kh. Amirov; emirov@cumhuriyet.edu.tr
Received 22 November 2012; Revised 2 March 2013; Accepted 4 March 2013 Academic Editor: Juan J. Nieto
Copyright © 2013 R. Kh. Amirov and A. A. Nabiev. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
In this study some inverse problems for a boundary value problem generated with a quadratic pencil of Sturm-Liouville equations with impulse on a finite interval are considered. Some useful integral representations for the linearly independent solutions of a quadratic pencil of Sturm-Liouville equation have been derived and using these, important spectral properties of the boundary value problem are investigated; the asymptotic formulas for eigenvalues, eigenfunctions, and normalizing numbers are obtained. The uniqueness theorems for the inverse problems of reconstruction of the boundary value problem from the Weyl function, from the spectral data, and from two spectra are proved.
1. Introduction
The theory of inverse problems for differential operators occupies an important position in the current developments of the spectral theory of linear operators. Inverse problems of spectral analysis consist in the recovery of operators from their spectral data. One takes for the main spectral data, for instance, one, two, or more spectra, the spectral function, the spectrum, and the normalizing constants, the Weyl function. Different statements of inverse problems are possible depending on the selected spectral data. The already existing literature on the theory of inverse problems of spectral analysis is abundant. The most comprehensive account of the current state of this theory and its applications can be found in the monographs of Marchenko [1], Levitan [2], Beals et al. [3], and Yurko [4].
In the present work we consider some inverse problems for the boundary value problem generated by the differential equation
L xy := y" + [A2 - 2Xp (x) -q(x)]y = 0, x e [0, a) u (a, n]
with the boundary conditions
U(y):=y' (0) = 0, V(y):=y(n) = 0 (2)
and with the jump conditions
y(a + 0) = ay(a-0), y (a + 0) = a-1y' (a-0), (3)
where X is the spectral parameter, p(x) e W2,[0,n], q(x) e L2[0,n] are real functions, a is a real number, and a > 0, a=1,a e (n/2,n). Here we denote by W2f[0,n] the space of functions f(x), x e [0, n], such that the derivatives
n- 1) are absolutely continuous and f n'(x) e
L 2[0,n].
There exist many papers containing a fairly comprehensive analysis of direct and inverse problems of spectral analysis of the Sturm-Liouville equation
ly := -y" + q(x)y = X2y,
a special case (p(x) = 0) of (1). For instance, inverse problems for a regular Sturm-Liouville operator with separated boundary conditions have been investigated in [5] (see also [1-4]).
Some versions of inverse problems for (1) which is a natural generalization of the Sturm-Liouville equation were
fully studied in [6-14]. Namely, the inverse problems for a pencil LA on the half axis and the entire axis were considered in [6-8], where the scattering data, the spectral function, and the Weyl function, respectively, were taken for the spectral data. The problem of the recovery of (1) from the spectra of two boundary value problems with certain separated boundary conditions was solved in [9]. The analysis of inverse spectral problems for (1) with other kinds of separated boundary conditions as well as with periodic and antiperiodic boundary conditions was the subject of [10] (see also [11]) where the corresponding results of the monograph [1] were extended to the case p(x) = 0. The inverse periodic problem for the pencil LA was solved in [12] using another approach. We also point out the paper [14], in which the uniqueness of the recovery of the pencil LA from three spectra was investigated.
Boundary value problems with discontinuities inside the interval often appear in mathematics, physics, and other fields of natural sciences. The inverse problems of reconstructing the material properties of a medium from data collected outside of the medium give solutions to many important problems in engineering and geosciences. For example, in electronics, the problem of constructing parameters of heterogeneous electronic lines is reduced to a discontinuous inverse problem [15, 16]. The reduced mathematical model exhibits the boundary value problem for the equation of type (1) with given spectral information which is described by the desirable amplitude and phase characteristics. Note that the problem of reconstructing the permittivity and conductivity profiles of a one-dimensional discontinuous medium is also closed to the spectral information [17, 18]. Geophysical models for oscillations of the Earth are also reduced to boundary value problems with discontinuity in an interior point [19].
Direct and inverse spectral problems for differential operators without discontinuities have been extensively studied by many authors [20-25]. Some classes of direct and inverse problems for discontinuous boundary value problems in various statements have been considered in [18, 26-32] and other works. Boundary value problems with singularity have been studied in [33-37], and for further discussion see the references therein. Note that the inverse spectral problem for the boundary problem (1)-(3) has never been considered before.
In what follows we denote the boundary value problem (1)-(3) by L(a,a). In Section 2 we derive some integral representations for the linearly independent solutions of (1), and using these, we investigate important spectral properties of the boundary value problem L(a,a). In Section 3 the asymptotic formulas for eigenvalues, eigenfunctions, and normalizing numbers of L(a,a) are obtained. Finally, in Section 4 three inverse problems of reconstructing the boundary value problem L(a,a) from the Weyl function, from the spectral data, and from two spectra are considered and the uniqueness theorems are proved.
2. Integral Representations of Solutions and the Spectral Characteristics
Let fv(x,X) (v = 1,2) be solution of (1) under the initial condition
fv (0,1) = 1, fl (0,X) = Xwv
and discontinuity conditions (3), where wv = (-1)v+1 i.
It is obvious that the functions fv(x, X) satisfy the integral equations
fv (x,X)
= I+ (x) eXWyX + r (x) eXwv{2a-x)
,+ , N I sin X(x - t)
" Jo x
+1+ (x)
- I- (x) J
(x sin X(x - t)
sin X(x + t - 2a) o X
[q(t)+2\p(t)}fv (t,X)dt [q(t)+2Xp(t)} fv (t,X)dt
[q(t) + 2Xp(t)} fv (t,X)dt,
where l±(x) = (l/2)(l(x) ± (1/I(x)) and l(x)=-
1, 0 < x < a a, a < x <n.
Using the integral equations (6) and standard successive approximation methods [7, 9, 11], the following theorem is proved.
Theorem 1. If q(x) e L2 [-b,b], p(x) e W-J [-b,b] (0 < b < n), then the solution fv(x, X) has the form
A] (x,t)eXW]tdt,
fov (X, X) = I+ (X) eXtVyX (X) + I- (X) eXWl(2a-x)R- (x),
R± (X) = e™v№, f (X)=\ p(t)dt,
J(a+a)l 2
and the function A v (x, t) satisfies the inequality
IA v (x,t)\dt<eca(x) -1
a(x)= J [(x-t)\q(t)\+2\p(t)\]dt
c = 2 (a* + \a-\ + l),
± 1( l" a = -(a ± — 2\ a,
Let <p(x, X) be the solution of (1) that satisfies the initial conditions
cp(0,X) = l, <p' (0,X) = 0,
and the jump condition (3).
Then by using Theorem 1, we can formulate the following assertion.
Theorem 2. Let q(x) e L2[0,n], p(x) e W.1 [0,n]. Then there are the functions A(x, t), B(x, t) whose first order partial derivatives are summable on [0,n] foreach x e [0,n] suchthat the representation
f(x,X) = f0 (x, X)+ \ A(x,t) cos Xtdt+\ B(x,t) sin Xtdt Jo Jo
is satisfied, where
<p0 (x, X) = 1+ (x) cos [Xx - (x)]
+ I- (x) cos [X (2a - x) - p- (x)]. Moreover, the relations a+p+ (x) = a+xp (0)
+2 f [A & f) sin ¡3+ (l-)-B (!■, !■) cos ¡3+ tf)] d!;,
2— [A (x, x) cos ß+ (x) + B (x, x) sinß+ (%)] dx
= a+ [q(x) + p2 (x)],
2-^ [A (x, t) cos ß- (x) - B (x, t) sinß- (%)] dx
= a- [q(x) + p2 (%)], At (x,t)\t=0 = B(x,0) = 0
are held.
for all y(x) e W^2[0, a) U (a, n] such that y(x) = 0 and
y (0)JW)-y (n)JW) = 0. (18)
Definition 3. A complex number X0 is called an eigenvalue of the boundary value problem L(a, a) if (1) with X = X0 has a nontrivial solution y0(x) satisfying the boundary conditions (2) and the jump conditions (3). In this case y0(x) is called the eigenfunction of the problem L(a, a) corresponding to the eigenvalue A0. The number of linearly independent solutions of the problem L(a, a) for a given eigenvalue X0 is called the multiplicity of A0.
The following lemmas can be proved analogously to the corresponding assertions in [11].
Lemma 4. The eigenvalues of the boundary value problem L(a, a) are real, nonzero, and simple.
Proof. We define a linear operator L0 in the Hilbert space L2[0,n] as follows. The domain D(L0) consists of all func-(14) tions y(x) e W22[0,n] satisfying the boundary conditions (2) and the jump conditions (3). For y e D(L0), we set L0y = -y" + q(x)y. Integrationbypart yields
t=2a-x+0
If one additionally supposes that q(x) e W2^[0,n], p(x) e w22 [0, n], then the functions A(x,t) and B(x,t) satisfy the system of partial differential equations
A^ (x, t)-q (x) A (x, t) - 2p (x) Bt (x, t) = Att (x, t),
Bxx (x, t)-q (x) B (x, t) + 2p (x) At (x, t) = Btt (x, t).
Conversely, if the second order derivatives of functions A(x, t), B(x, t) are summable on [0, n] for each x e [0, n] and A(x, t), B(x, t) satisfy equalities (16) and conditions (15), then the function <p(x, X) which is defined by (13) is a solution of (1) satisfying initial conditions (12) and discontinuity conditions (3).
One here supposes that the function q(x) satisfies the additional condition
(Loy> y) = \ Loyy (x)dx
j0 {\y (x)\2 + q(x)\y(x)\2)dx.
Since condition (17) holds, it follows that (L0y, y) > 0.
Let X0 be an eigenvalue of the boundary value problem L(a,a) and y0(x) an eigenfunction corresponding to this eigenvalue and normalized by the condition (y0,y0) = 1. By taking the inner product of both sides of the relation y"(x) + [X20 - 2X0p(x) - q(x)]y0(x) = 0 by y0(x), we obtain X0 - 2X0(py0, y0) - (L0y0, y0) = 0 and hence
^0 = (PÏ0> Ï0) ± V(Py0'Ï0 f + (L0y0>y0)■ (19')
The desired assertion follows from the last relation by virtue of (L0y0,y0) > 0 with regard to the fact that p(x) is real.
Let us show that X0 is a simple eigenvalue. Assume that this is not true. Suppose that y1(x) and y2(x) are linearly independent eigenfunctions corresponding to the eigenvalue A0. Then for a given value of A0, each solution y0(x) of (1) will be given as linear combination of solutions y1(x) and y2(x). Moreover it will satisfy boundary conditions (2) and discontinuity conditions (3). However, it is impossible. □
Lemma 5. The problem (1)-(3) does not have associated functions.
Proof. Let y0(x) be an eigenfunction corresponding to eigenvalue X0 and normalized by the condition (y0,y0 ) = 1 of the problem (1)-(3). Suppose that y1(x) is an associated function of eigenfunction y0(x), that is, the following equalities hold:
Kyo - 2XoP (x) yo - Loyo = °>
jo {\y' (4 +q(x)\y(x)\2\dx>0 (17) Aoyi-2Xop(x)yi-L0y, +2{X0-p{x))y0 = 0.
If these equations are multiplied by y1(x) and y0(x), respectively, as inner product, subtracting them side by side and taking into our account that operator L0 is symmetric, the function p(x) and X0 are real, we get X0 = (py0, y0). Due to the condition (6), X0 = (py0, y0) does not agree with (19'). Therefore, the assertion is not true. □
Lemma 6. Eigenfunctions corresponding to different eigenvalues of the problem L(a, a) are orthogonal in the sense of the equality
(Xj +X 2)(yi,y2)-2(pyl,y2) = 0, (21)
where (•, •) denotes the inner product in L 2[0, n].
Lemma 7. Let y(x,X) be a solution of (1) satisfying the condition (18) and the jump conditions (3). Then X is real and nonzero and
\ (X- p(x))\y(x,X)\2dx = 0. (22)
Moreover, the sign ofthe left-hand side of (22) is similar to the sign of X.
Then we have
(cp (x,Xo),f(x,Xo))
= - [y' (n, Xo) y (n, Xo) - y (n, Xo) y (n, Ao)] ci
= --y(n,Xo)
which is a contradiction.
Note that we have also proved that for each eigenvalue there exists only one eigenfunction (up to a multiplicative constant). Therefore there exists sequence such that y(x,Xn) = ¡3n<p(x,Xn).
Let us denote
an ■= \ f2(x,Xn)dx-— \ p(x)f2(x,Xr)dx. (26) Jo X„ Jo
3. Properties of the Spectrum
In this section we investigate some spectral properties of the boundary value problem L(a, a).
Let y(x, X) be a solution of (1) with the conditions y(n,X) = 0, y'(n,X) = 1 and the jump conditions (3). It is clear that function y(x, X) is entire in X for each fixed x.
Denote A(X) = {y(x,X),<p(x,X)), where (y,z) ■= y'z - yz . By virtue of Liouville's formula, the Wronskian {f(x, X), <p(x, X)) does not depend on x. The function A(X) is called the characteristic function of L(a, a). Obviously, the function A(X) is entire in X and it has at most a countable set of zeros [Xn].
Lemma 8. The zeros [Xn] ofthe characteristic function A(X) coincide with the eigenvalues of the boundary value problem L(a, a). Thefunctions <p(x, Xn) and f(x, Xn) are eigenfunctions corresponding to the eigenvalue Xn, and there exists a sequence [pn] such that
Y(x,K ) = ßn<p(x,K), ßn = 0.
Proof. Let A(X0) = 0. Thenbyvirtue of {f(x, A0), <p(x, X0)) = 0, <p(x, X0) = Cf(x, X0) for some constant C. Hence X0 is an eigenvalue and <p(x,X0), f(x,X0) are eigenfunctions related to A0.
Conversely, let X0 be an eigenvalue of L(a, a), show that A(X0) = 0. Assuming the converse suppose that A(X0) = 0. In this case the functions (p(x,X0) and f(x,X0) are linearly independent. Then y(x,X0) = ct(p(x,X0) + c2f(x,X0) is a general solution of the problem L(a, a). If q = 0, we can write
(p (x, Xo) = -y (x, Xo) - — f (x, Xo).
The numbers {an} are called normalized numbers of the boundary value problem L(a, a).
Lemma 9. The equality A(Xn) = -2Xnanpn holds. Here A(X) = (d/dX)A(X).
Proof. If we differentiate the equalities
- f" (x, X) + [2Xp (x) + q (x)] (f (x, X) = X2f (x, X),
- y" (x, X) + [2Xp (x) + q (x)] f (x, X) = X2f (x, X)
with respect to X, we get
- <p" (x, X) + [2Xp (x) + q (x)] q> (x, X) = X2<p (x, X) +2[X- p (x)] <f (x, X),
- y" (x, X) + [2Xp (x) + q (x)] f (x, X) = X2f (x, X) +2[X- p (x)] f (x, X).
By virtue of these equalities we have
{(p (x, X) y' (x, X) - <p' (x, X) y (x, A)}
= 2[X- p (x)] f (x, X) y (x, X),
— {<p (x, X) y' (x, X) - f (x, X) y (x, A)}
= 2[X- p (%)] f (x, X) y (x, X).
If the last equations are integrated from x to n and from 0 to x, respectively, by the discontinuity conditions we obtain
(<p(ç,a)y' (s,\)-<p' tf.WUC = -2 f (A-p(ï))<p(ï,A)y(ï,A)dï,
(<p(-,A)f' (£,,A)-cp' (ï,A)y(ï,A))\XQ
= -2 I* (A-p(ï))<p(ï, A) y (ï, A) dÇ.
If we add the last equalities side by side, we get
W[<p(ï,A),y(ï,A)]+W[<p(ï,A),y(ï,A)]
-2 j1 (A-p(ï))<p(ï,A)y(ï,A)dï
À (A) = -2f(A-p (V) cp (I-, A) y (Ç, A) df. (32)
For A ^ An, this yields
À (An) = -2 I (An - p (V) cp (Ç, An) Y (H, An) dÇ
= -2pn I (*« -P(Ï))V (iK)dt
= -2A Jn cp2 (Ç,Xn)dt
-I p(t)cp2 &An)dt A„ Jo
= -2X n ß„ a„.
The lemma is proved.
Let A0(A) = a+ cos[An-p+(n)]+a cos[A(2a-n)+p (ri)] and {A0n} are zeros of A 0(A).
Lemma 10. The roots of the characteristic equation A 0(A) = 0 are separate, that is
inf \A° - Ao\ = ß>0.
n=m\ n m\ 1
Proof. Let An - fi+(n) = x. Then, A(2a -n) + f$~(n) = kx + b, where k = (2a - n)/n, b = j3+(n)((2a - n)/n) + fi-(n). Since a e (n/2, ri), then k e (0,1). Using these notations we can rewrite the equation A0(A) = 0 in the following form:
A cos x = cos (kx + b).
Here A = -(a+/a-) which implies that |A| > 1. Preliminarily show that there are no multiple roots of (35). Assuming the converse we suppose x0 to be a multiple root of (35). Then
A sin xo = k sin (kxo + b)
holds. Now (35)and(36) imply that A2 = 1-(1-k2 )sin2(kx0 + b) < 1 which is a contradiction. Therefore, (35) has no multiple roots.
Further assuming (34) not to be true let [x'p} and {x} be increasing sequences of roots of (35) such that x'p = x'p and
\xp -xp\ = 0.
If we assume that x'p = 2npn + r'p, where n e N and {r'p} is a bounded sequence (0 < r'p < 2ri), then from (37) we find that x'p = 2npn + r^, where [r'p} is a bounded sequence such that limp^mlr'p -r'p | = 0. It is obvious that kx'p = 2n[knp] + s'p, kx'p = 2n[knp] + s'p, where s'p = 2n{knp} + r'pk, s'p = 2n{kn.p}+r'p k andlim^^Js^-s^| = 0.Here [•] and {•} denote the integer and fractional parts of a real number, respectively. Since sequences {r'} , {r!} , {/} and {/'} will be
bounded, without loss of generality we can assume that these sequences are convergent. Then let
lim r' = lim r'' = x0,
p^rn V p^rn V
Am^ = =^ (39)
Therefore, we can write the equality A cos x'p = cos(kx'p + b)
A cos r'p = cos (s'p + b) . Then by virtue of (38) and (39), from (40) we get
A cos x0 = cos (y0 + b). Similarly we can obtain
A cos r'p = cos (s'p + b) . Further, from (40) and (42), we have
in I a
r + r r - r
A ■ P P ■ P P
A sin —-— sin —-— = sin
SP +SP . ~p
—-- + b ) sin —
s" -SP
Let us write this equality as
I ii I ii
r + r r - r i p p . p p A sin —-— sin —-—
SP + SP 2
. k(r'p -r'l )
Now dividing both sides of equality (44) by (r'p - r' )/2 = 0
■p rP)
and taking limit as p ^ <x>, by virtue of (3) and (39), we obtain
A sin xo = k sin (yo + b).
Finally, from (41) and (45), we conclude that A2 = 1 - (1 -fc2)sin2(y0 + b) < 1 which is a contradiction. Hence roots of
(36) equation À0(A) = 0 are separate. The lemma is proved.
Denote
On the other hand, since
rn = + n= 0,1,.
Gs := {A:\A- X0n\ > S} (S>0),
where S is sufficiently small positive number (S < fi/2). Lemma 11. For sufficiently large values of n, one has
X e Tn. (47)
\A(X)-A0 (X)\<^ellmMn,
Proof. As it is shown in [38],\A0(A)| > Cse1 ImMn for all X e Gs, where Cs > 0 is some constant. On the other hand, since
lim e-'Im(A(X)-A 0 (X))
|A| — to
= lim e 'lm (J A(n,t) cos Xt dt
\X\ — to
\ B(n,t) si Jo
sin Xt dt
for sufficiently large values of n (see [1]) we get (47). The lemma is proved. □
Lemma 12. The problem L(a, a) has countable set of eigenvalues. If one denotes by X1,X2,... the positive eigenvalues arranged in increasing order and by X_1, X_2,... the negative eigenvalues arranged in decreasing order, then eigenvalues of the problem L(a, a) have the asymptotic behavior
i i0 dn Sn
= K +K + X>,
where Sn e l2 and dn is a bounded sequence, X°n (\/n)ß+(n) + hn, supn\hn\ < rn.
Proof. According to Lemma 11, if n is a sufficiently large natural number and X e Tn, we have \A0(A)\ > Cse'lmXiin > (Cs/2)e1 ImA|" > \A(A) - A0(A)\. Applying Rouche's theorem we conclude that for sufficiently large n inside the contour Tn the functions A0(X) and A0(X) + {A(A) - A0(A)} = A(X) have the same number of zeros counting their multiplicities. That is, there are exactly (n + 1) zeros X0, X1,..., Xn. in Tn. Analogously, it is shown by Rouche's theorem that, for sufficiently large values of n, the function A(X) has a unique zero inside each circle \X - Xn\ < S. Since S > 0 is arbitrary, it follows that Xn = X0n + en, where limn — m£n = 0. Further according to A(Xn) = 0, we have
o A(n,t) cos (X° + £n)t
B (n, t) sin (X0n + en) tdt = 0.
A o (X)
= a+ cos (Xn - ß+ (n)) + oT cos (X (2a - n) + (n)),
A o (X0n + en) = A0 (X0n)en + o(en),
n —> rn,
(50) takes the form of
o A(n,t) cos (X0n + £n)tc
B (n, t) sin (x0n + £n)tdt + 0 (en) = 0,
n —> rn. (52)
It is easy to see that the function A 0(X) is type of "Sine" [39], so there exists ys > 0 such that \A0(A°)\ > ys > 0 is satisfied for all n. We also have
X0n = n+-ß+ (n) + hn, n
where sup n\hn\ < M for some constant M > 0 [40] (see also [41]). Further, substituting (53) into (52) after certain transformations [1, page 67], we reach £n e l2. We can obtain more precisely
£„ =
n 2A0 (X0n) X0n
A o (K)K
[a+ sin (X°nn- ßl (n)) + aT sin (X0n (2a -n) + ß2 (ft))]
(q(x) + p2 (x))dx
- [a+ cos (X0nn- ßl (n))
+ aT cos (X0n (2a - n) + ß2 (ft))]
x(p(n)-p(0))
Bt (n, t) cos Xt dt
- \ At (n,t) sin Xnt dt Jo
o(£n)
+--—, n —> rn.
Since J" B[(n,t) cos X°ntdt e l2, J" A't(n,t) sin X0ntdt e l2, we have
2A0 (K)Xon
a* sin (X°nft - ßi (ft)) + a- sin (Xon (2a -ft) + ß2 (ft))]
(q(x) + p2 (x))dx
- [a* cos (X°nft- ß1 (ft))
+ a- cos (X°n (2a - n) + ß2 (ft))]
x(p(ft)-p(0))} + ^,
where Sn el2. Hence we obtain
i io , dn Sn
~=x "+ K '
2À0 (K)
[a* sin (X°nft - ß+ (ft)) + a- sin (Xon (2a - ft) + ß- (ft))]
(q(x) + p2 (x)) dx
- [a* cos (Xonft - ß+ (ft))
+ a- cos (Xon (2a - ft) + ß- (ft))]
x (p(ft)-p(0))
is a bounded sequence. The proof is completed.
Lemma 13. Normalizing numbers an of the problem L(a, a) arepositive and theformula
[(a*)2 + (a-)2] + §f +d~f
holds, where d11 = -(an/2)p(0), d1n e l2.
Proof. The formula (58) can be easily obtained from the equalities
A (x, x) sin Xnx - B (x, x) cos Xnx
= y{(p(x)-P(0)) cos (Kx-ß* (x))
(q (t) + p2 (t)) dt sin (Xnx - ß* (x))
A(x,2a - x + 0) - A(x,2a - x - 0) sin Xn (2a - x)
- B (x,2a - x + 0) - B (x,2a - x - 0) cos Xn (2a-x)
= t{1 (q(t) + p2 (f))dt sin (Xn (2a-x)+ß- (x))
- (p (x) - p (0)) cos (Xn (2a -x)+ß- (x))
by using the asymptotic formula (49) for Xn.
4. Inverse Problems
Together with L(a, a), we consider the boundary value problem L(a, a) of the same form but with different coefficients (q,p,a,a). It is assumed in what follows that if a certain symbol y denotes an object related to the problem L(a,a), then q will denote the corresponding object related to the problem L(a, a).
In the present section, we investigate some inverse spectral problem of the reconstruction of a boundary value problem L(a,a) of type (1)-(4) from its spectral characteristics. Namely, we consider the inverse problems of reconstruction of the boundary value problem L(a, a) from the Weyl function, from the spectral data {Xn,an}n>0, and from two spectra {Xn, ^n}n>0 and prove that the following two lemmas can be easily obtained from asymptotic behavior (49) of the eigenvalues Xn.
Lemma 14. If Xn = Xn, n = 0,±1,±2,..., then p+(n) =
fi (n), fi-(n) = fi (n), that is, the sequence {Xn} uniquely determines p±(n).
Lemma 15. If Xn = Xn, n = 0, ±1, ±2,..., then a = a, a = a, that is, the sequence {Xn} uniquely determines numbers a and
Let <&(x,X) be the solution of (1) under the conditions U(O) = 1, V(®) = 0 and under the jump conditions (3). One sets M(X) ■= 0(0, A). The functions 0(x, X) and M(X) are called the Weyl solution and Weyl function for the boundary value problem L(a, a), respectively. Using the solution <p(x, X) defined in the previous sections one has
®(x,X) = = S(x,X) + M(X)cp(x,X),
M(X) = -
f(0,X) A (X) '
where f(x, X) is a solution of (1) satisfying the conditions y(n, X) = 0, y'(n, X) = -1, and the jump conditions (3), and S(x, X) is defined from the equality
f (x, X) = f (0, X) (f (x, X)- A (X) S (x, X).
Note that, by virtue of equalities (f(x,X),S(x,X)) = 1 and (60), one has
(O (x,X),<p(x,X)) = 1, {f(x,X),f(x,X)) = -A(X) for x = a.
The following theorem shows that the Weyl function uniquely determines the potentials and the coefficients of the boundary value problem L(a, a).
Theorem 16. IfM(X) = M(X), then L(a, a) = L(a, a). Thus, the boundary value problem L(a, a) is uniquely defined by the Weyl function.
Proof. Since
y(v) (x,X) = o(|XГ1 exp (|Im A| (n - x))), X e Gs,
|A(X)| > Cs exp (\lmX\n), X e Gs, Cs < 0, v = 0,1,
it is easy to observe that
\®(v) (x,X)\ < CS\X\V-1 exp (- |lmX| x), XeGä. (65) Let us define the matrix P(x, X) = [Pjk(x, X)]jk=12, where
p.1 (x, X) = f(j-1) (x, X) G' (x, X) - O(J-1) (x, X) cp' (x, X),
Pj2 (x, X) = O(j-1) (x, X) cp' (x, X) - f(j-1) (x, X) G (x, X).
Then we have
cp (x, X) = P11 (x, X) ((x, X) + P12 (x, X) (' (x, X),
O (x, X) = P11 (x, X) G (x, X) + P12 (x, X)G' (x, X).
According to (60) and (65), for each fixed x, the functions Pjk(x, X) are meromorphic in X with poles at points Xn and
Xn. Denote G°s = Gs n Gs. Byvirtue of (65), (66), and
f(v) (x, X) = 0 dX^ exp (|lm X| x)), Xe G°s,
we get
\Pn (x,X)\ <C5|A|-1, |Pn (x,X)\ < Cs, XeG°s. (69)
It follows from (60) and (66) that if M(X) = M(X), then for each fixed x the functions P1k(x, X) are entire in k. Together with (69) this yields P12(x,X) = 0, P12(x,X) = A(x). Now using (67), we obtain
cp (x, X) = A (x) cp (x, X), O(x,X) = A (x) G (x, X).
Therefore, for |A| ^ >x>, arg X e [e,ri - e] (e > 0), we have cp(x,X)=h- exp (-i(Xx-£i (*)))(! +o(1)), (71)
where b = 1 for x < a and b = a+ for x > a. Similarly, one can calculate
O (x, X) = (iXb)-1 exp (i (Xx-^1 (x)))(1+o(j)), |X —> rn, arg X e [e,n - e],
Finally, taking into account the relations (O(x, X), <p(x, X)) = 1 and (65), we have a+ = a+, A(x) = 1, that is, <p(x,X) = cp(x,X), O(x,X) = O(x,X) for all x and A. Consequently, L(a,a) = L(a, a). Thetheoremisproved. □
The following two theorems show that two spectra and spectral data also uniquely determine the potentials and the coefficients of the boundary value problem L(a, a).
Theorem 17. If Xn = Xn, = Jin, n = 0,±1,±2,..., then L(a, a) = L(a, a).
Proof. It is obvious that characteristic functions A(X) and y(0, X) are uniquely determined by the sequences {X2n} and {^1} (n = 0,±1,±2,...), respectively. If A„ = Xn, = ¡¡n,n = 0,±1,±2,..., then A(X) = A(X), f(0,X) = f(0,X). It follows from (60) that M(X) = M(X). Therefore, applying Theorem 16, we conclude that L(a, a) = L(a, a). The proof is completed. □
Theorem 18. If Xn = Xn, an = an, n = 0,±1,±2,..., then L(a,a) = L(a,a), that is, spectral data {Xn,an} uniquely determines the problem L(a, a).
Proof. It is obvious that the Weyl function M(X) is meromorphic with simple poles at points X2n. Using the expression
A(A) = A 0 (X) + \ A(n,X) cos Xtdt+ \ B(n,X) sin Xtdt
and equalities 2Xnpnan = -A(Xn), y(x, Xn) = pnf(x, Xn), we have
Re sM(X) = -^Xl == 1 ^ A(XJ A.(Xn) 2Xnan
Since the Weyl function M(X) is regular for X e Tn, applying the Rouche theorem, we conclude that
1 ( M(u) (X) = — I —^dp, Xe int rn. 2ni Jr "
Taking (60) and (63) into account, we arrive at M(X) < CS|A|- , X e Gs. Therefore
1 ( M(u) M (X) = lim - I —^du
2ni Jr u-X
where Tn1 = [X : ^ = ^J}, n = 0,±1,±2,.... Hence, by the residue theorem, we have
M(X)= £
n=-m 2Xnan (X-Xn)
Finally, from the equality M(X) = M (A), applying Theorem 16, we conclude that L(a,a) = L(a,a). The theorem is
proved. □
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