Vo lu m e 6 6 , N u m b e r 1 , P a g e s 1 6 5 –1 7 1 (2 0 1 7 ) D O I: 1 0 .1 5 0 1 / C o m m u a 1 _ 0 0 0 0 0 0 0 7 8 5 IS S N 1 3 0 3 –5 9 9 1
INVERSE NODAL PROBLEM FOR A STURM-LIOUVILLE OPERATOR WITH DISCONTINUOUS COEFFICIENT
A. SINAN OZKAN
Abstract. Inverse nodal problem for Sturm–Liouville equation with discon-tinuity coe¢ cient is studied. A uniqueness theorem and an algorithm for recovering the coe¢ cients of the problem from a known sequence related to the nodal points are given.
1. Introduction
Inverse nodal problems consist in recovering the coe¢ cients of operators from the zeros (nodes) of the eigenfunctions. McLaughlin (1988) seems to have been the …rst to consider this kind of inverse problem for the regular Sturm–Liouville equations with Dirichlet boundary conditions[17]. She showed that the potential of the problem can be determined by a given dense subset of nodal points. In 1989, Hald and McLaughlin generalized this result to more general boundary conditions and provide some numerical schemes for the reconstruction of the potential [13]. From then on, their results have been generalized to various problems. Inverse nodal problems for Sturm–Liouville operators without discontinuities have been studied in the several papers ([8], [10], [12], [13], [14], [19], [21], [22] and [24]). The …rst result on inverse nodal problems for the Sturm-Liouville operators with a discontinuity condition was obtained by Shieh and Yurko[20]. This study includes discontinuity conditions at the middle of interval. Inverse nodal problem for Sturm-Liouville operator with boundary conditions dependent on the spectral parameter were investigated in [4], [23] and [18]. Additionally, the studies [5] and [6] include inverse nodal problems for di¤erential pencils.
In the present paper, we consider the boundary value problem L = L (q; h; H) generated by the Sturm–Liouville equation
`y := y[2]+ q(x)y = y; x 2 (0; 1) (1)
Received by the editors: June 28, 2016, Accepted: Sep, 17, 2016. 2000 Mathematics Subject Classi…cation. 34A55, 34B07, 34B24, 34B37.
Key words and phrases. Sturm-Liouville equation; inverse nodal problem; discontinuity condition.
c 2 0 1 7 A n ka ra U n ive rsity C o m m u n ic a tio n s d e la Fa c u lté d e s S c ie n c e s d e l’U n ive rs ité d ’A n ka ra . S é rie s A 1 . M a th e m a t ic s a n d S t a tis tic s .
subject to the boundary conditions
U (y) := y[1](0) hy(0) = 0 (2)
V (y) := y[1](1) + Hy(1) = 0 (3)
and transfer conditions
y(d + 0) = y(d 0)
y[1](d + 0) = y[1](d 0) y(d 0) (4)
where y[1]= py0; y[2]= p (py0)0; q(x) and p(x) are real valued functions in L 2(0; 1);
h, H and are real numbers and is the spectral parameter. We assume that p(x) > 0 and (d)(1) is a rational number in (0; 1)
The equation (1) appears in some physical applications. A Sturm–Liouville equa-tion with the coe¢ cients which are piecewise constant funcequa-tions can be regard as special form of (1). Spectral problems for di¤erential equations with discontinuous coe¢ cients were investigated in several works (see [1], [2], [3], [7], [9], [11], [15] and [16]). These works contain inverse problems according to the various spectral data.
2. Preliminaries
Let a function '(x; ) be the solution of (1) under the initial conditions
'(0; ) = 1; '[1](0; ) = h (5)
and the jump conditions (4). It can be calculated that '(x; ) = '1(x; ); x < d '2(x; ); x > d satis…es the following integral equations:
'1(x; ) = cos (x) + hsin (x) (6) +1 Z x 0 sin [ (x) (t)]q p(t)'(t; )dt '2(x; ) = cos (x) + hsin (x) 2 [sin (x) sin (2 (d) (x))] (7) + h 2 2[cos (x) cos (2 (d) (x))] + 2 2 Z d 0 [cos ( (x) (t)) cos (2 (d) (x) (t))]q p(t)'(t; )dt +1 Z x 0 sin [ (x) (t)]q p(t)'(t; )dt (8) where (x) =R0x dt p(t) and = p .
Using above integral equation we can obtain the following asymptotic relations for j j ! 1: '1(x; ) = cos (x) + 0 @h +1 2 x Z 0 q(u) p(u)du 1 Asin (x)+ o 1exp x ; (9) '2(x; ) = cos (x) + 0 @h 2 + 1 2 x Z 0 q(u) p(u)du 1 Asin (x) (10) + 2 sin (2 (d) (x)) + o 1exp x ;
where = jIm j. By substituting (u) = (1)t to the integrals in (9) and (10) we obtain
'1(x; ) = cos (x) + f (x)sin (x) + o 1exp x ; (11) and '2(x; ) = cos (x) + f (x) 2 sin (x) + (12) + 2 sin (2 (d) (x)) + o 1exp x ; where f (x) = h + (1)2 (x)= (1)R 0 q1(t)dt, q1(t) = qo 1 ( (1)t)
Let f ngn 0be the set of eigenvalues of (1)-(4) and '(x; n) be the eigenfunction
corresponding to the eigenvalue n: It can be proven easily that the numbers n
are real, simple and satisfy the following asymptotic relation for n ! 1:
n = p n= n (1)+ A n 2n cos 2n (d) (1) + o 1 n (13) 1 p n = (1) n 1 A (1) n2 2 (1) 2n2 2cos 2n (d) (1) + o 1 n2 (14) where A = h + H 2 + (1) 2 1 R 0 q1(t)dt: 3. Main results Let X = xj
n : n 2 N be the set of nodal points of the eigenfunctions. Consider
the set Y = ( yj n: ynj = xj n (1) ; x j n2 X )
and the problem eL together with L: It is assumed in what follows that if a certain symbol s denotes an object related to the problem L thenes denotes the corresponding object related to the problem eL.
Theorem 1. Let Y0 Y be dense set on (0; 1): If 1 R 0 q(u) p(u)du = 0; p(x) =p(x) ande Y0 = eY0 then q(x) = q(x) a.e. in (0; 1) ; h = ee h; H = eH and = e: Thus, the
coe¢ cients q(x); h; H and are uniquely determined by Y0.
First, it must be given the following lemma, related to the asymptotic formulae for the elements of Y:
Lemma 1. The elements of Y satisfy the following asymptotic formulae for su¢ -ciently large n, yjn= j + 1 2 n (1) n2 2 A + 2 cos 2n (d) (1) j +12 n + + (1) n2 2f (x j n) + o 1 n2 ; x j n 2 (0; d) (15) yjn= j + 1 2 n (1) n2 2 A + 2 cos 2n (d) (1) j +1 2 n + + (1) n2 2 f (x j n) 2 2cos 2n (d) (1) (16) + o 1 n2 ; x j n 2 (d; 1)
Proof. Use the asymptotic formulae (11) and (12) to get '1(xj n; n) = cos n (xjn) + f (xjn) sin n xj n n + o 1 n ; (17) and '2(xjn; n) = cos n (xjn) + f (xjn) 2 sin n xj n n + (18) + 2 sin n 2 (d) (xjn) n + o 1 n ; Let us consider the second case: '2(xj
n; n) = 0: The …rst case is similar. It is
calculated that, tan n xj n 2 = f (xjn) 2 n 2 n cos 2 n (d) + o 1 n This yields
xjn = 1 n j +1 2 f (xjn) 2 2 n + 2 2 n cos 2 n (d) + o 12 n : We can complete the proof using (13) and (14) .
Proof of Theorem1. Since the set X0:=
j +12
n ; j = 0; n 1; n > 0 is dense on (0; 1); for each …xed x in (0; 1) ; there exist a sequence (j(n)) such that j(n) +
1 2
n converges to x: Thus the set Y is also dense on (0; 1):
Denote Kj n := n2 2 (1) y j n j(n)+1 2
n and m := s:n, with s is denominator of (d) (1):
Therefore, we can show from Lemma 1 that the following limits are exist and …nite: lim m!1K j(m) m = F (x) (19) where F (x) = 8 > > < > > : (h + H) x + h + (1)2 x R 0 q1(t)dt; x 2 [0; d) (h + H) x + h + (1)2 x R 0 q1(t)dt; x 2 (d; 1] (20)
Direct calculation yields
q1(x) = 2 fF0(x) + F (0) F (1) + (1) [F (d + 0) F (d 0)]g ; (21) q(x) = q1 (x) (1) ; (22) h = F (0); H = F (d + 0) F (d 0) F (1) and (23) = F (d 0) F (d + 0): (24)
It is clear that, if p(x) =ep(x) and Y0= eY0 then F (x) = eF (x) and so q(x) = eq(x)
a.e. in (0; 1) ; h = eh; H = eH and = e:
Corollary 1. If Y0is given by (15) and (16), q(x); h; H and can be reconstructed
by the formulae (21)-(24). Corollary 2. If 1 R 0 q(u)
p(u)du = 0; p(x) = ep(x) and X = eX then q(x) = q(x) a.e.e in (0; 1) ; h = eh; H = eH and = e: Thus, the coe¢ cients q(x); h; H and are uniquely determined by the nodal points.
Example 1. Let p(x) = 1; d be a rational number in (0; 1) and Y0 be given by the
ynj = j + 1 2 n + 1 n2 2[1 cos 2n d] j +12 n + + 1 n2 2 1 + 1 2 sin 2 yj n + o 1 n2 ; x j n 2 (0; d) (25) ynj =j + 1 2 n + 1 n2 2[1 cos 2n d] j +12 n + + 1 n2 2 1 2 sin 2 yj n cos 2n d (26) + o 1 n2 ; x j n2 (d; 1)
It can be calculated from (19) and (20) that, F (x) = ( 1 +21 sin2 x; x 2 [0; d) 1 2 sin 2 x 1; x 2 (d; 1] By (21)-(24), it is obtained that q(x) = sin 2 x; h = 1; H = 1 and = 2: References
[1] R.Kh. Amirov, A.S. Ozkan, Discontinuous Sturm-Liouville Problems with Eigenvalue De-pendent Boundary Condition, Math Phys Anal Geom, 17, 483-491, doi:10.1007/s11040-014-9166-1
[2] L. Andersson, Inverse eigenvalue problems with discontiuous coe¢ cients, Inverse Problems, 4, (1988), 353-397.
[3] M.I. Belishev, Inverse spectral inde…nite problem for the equation y00+ p(x)y = 0 on an
interval, Funkts. Anal. Prilozh., 21(2) (1987), 68-69.
[4] P.J. Browne, B.D. Sleeman, Inverse nodal problem for Sturm–Liouville equation with eigen-parameter depend boundary conditions, Inverse Problems 12 (1996), pp. 377–381.
[5] S.A. Buterin, C.T. Shieh, Inverse nodal problem for di¤erential pencils, Appl. Math. Lett. 22, (2009), 1240–1247.
[6] S.A. Buterin, C.T. Shieh, Incomplete inverse spectral and nodal problems for di¤erential pencil. Results Math. 62, (2012), 167-179
[7] R. Carlson, An inverse spectral problem for Sturm-Liouville operators with discontiuous coe¢ cients, Proceed. Amer. Math. Soc., 120(2), (1994), 475-484.
[8] Y.H. Cheng, C-K. Law and J. Tsay, Remarks on a new inverse nodal problem, J. Math. Anal. Appl. 248, (2000), pp. 145–155.
[9] C.F. Coleman, J.R. McLaughlin, Solution of inverse spectral problems for an impedance with integrable derivative, I, II, Comm. Pure and Appl. Math., 46, (1993), 145-184, 185-212. [10] S. Currie, B.A. Watson, Inverse nodal problems for Sturm–Liouville equations on graphs,
Inv. Probl. 23, (2007), pp. 2029–2040.
[11] G. Freiling, V.A. Yurko, Inverse problems for di¤erential equations with turning points, In-verse Problems 13, (1997), 1247-1263.
[12] Y.X. Guo, G.S. Wei Inverse problems: Dense nodal subset on an interior subinterval, J. Di¤erential Equations, 255(7), (2013), 2002–2017.
[13] O.H. Hald, J.R. McLaughlin, Solutions of inverse nodal problems, Inv. Prob. 5 (1989), pp. 307–347.
[14] C.K. Law, J. Tsay, On the well-posedness of the inverse nodal problem, Inv. Probl. 17 (2001), pp. 1493–1512.
[15] J.R. McLaughlin, Analytical methods for recovering coe¢ cients in di¤erential equations from spectral data, SIAM Rev., 28, (1986), 53-72.
[16] A. McNabb, R. Anderssen and E. Lapwood, Asymptotic behaviour of the eigenvalues of a Sturm–Liouville system with discontiuous coe¢ cients, J. Math. Anal. Appl., 54, (1976), 741-751.
[17] J.R. McLaughlin, Inverse spectral theory using nodal points as data – a uniqueness result, J. Di¤. Eq. 73 (1988), pp. 354–362.
[18] A.S. Ozkan, B. Keskin, Inverse nodal problems for Sturm–Liouville equation with eigenparameter-dependent boundary and jump conditions, Inverse Problems in Science and Engineering, i-…rst, (2014), doi:10.1080/17415977.2014.991730.
[19] C.L. Shen, C.T. Shieh, An inverse nodal problem for vectorial Sturm–Liouville equation, Inv. Probl. 16 (2000), pp. 349–356.
[20] C-T Shieh, V. A. Yurko, Inverse nodal and inverse spectral problems for discontinuous bound-ary value problems, J. Math. Anal. Appl. 347 (2008) 266-272.
[21] X-F Yang, A solution of the nodal problem, Inverse Problems, 13, (1997) 203-213. [22] X-F. Yang, A new inverse nodal problem, J. Di¤er. Eqns. 169 (2001), pp. 633–653.
[23] C-F. Yang, Xiao-Ping Yang Inverse nodal problems for the Sturm-Liouville equation with polynomially dependent on the eigenparameter, Inverse Problems in Science and Engineering, 19(7), (2011), 951-961.
[24] C-F. Yang, Inverse nodal problems of discontinuous Sturm–Liouville operator, J. Di¤erential Equations, 254, (2013) 1992–2014.
Current address : Department of Mathematics, Faculty of Science, Cumhuriyet University 58140 Sivas, TURKEY
