IS S N 1 3 0 3 –5 9 9 1
PRINCIPAL FUNCTIONS OF NON-SELFADJOINT MATRIX STURM – LIOUVILLE OPERATORS WITH BOUNDARY CONDITIONS DEPENDENT ON THE SPECTRAL PARAMETER
CAFER COSKUN, DENIZ KATAR AND MURAT OLGUN
Abstract. Let L denote operator generated in L2(R+; E)by the di¤erential
expression
l(y) = y00+ Q(x)y; x 2 R+:= [0; 1) ;
and the boundary condition Y0(0; ) (
0+ 1 + 2 2)Y (0; ) = 0, where Q
is a non-selfadjoint matrix-valued function and 0; 1; 2are non-selfadjoint
matrices, also 2is invertible: In this paper, we investigate the principal
func-tions correspending to the eigenvalues and the spectral singularities of L:
1. Introduction Let us consider the boundary value problem (BVP)
y00+ q (x) y = 2y ; x 2 R+ ; (1.1)
y (0) = 0 ; (1.2)
in L2(R
+) ; where q is a complex-valued function. The spectral theory of the BVP
(1:1)–(1:2) with continuous and point spectrum was investigated by Naimark [1]. He showed the existence of the spectral singularities in the continuous spectrum of the BVP (1:1)–(1:2) : Note that the eigenfunctions and the associated functions (principal functions) correspending to the spectral singularities are not the elements of L2(R+). Also, the spectral singularities belong to the continuous spectrum and
are the poles of the resolvent’s kernel, but are not the eigenvalues of the BVP (1:1)–(1:2). The spectral singularities in the spectral expansion of the BVP (1:1)– (1:2) in terms of the principal functions have been investigated in [2]. The spectral analysis of the quadratic pencil of Schrödinger, Dirac and Klein-Gordon operators with spectral singularities were studied in [3, 4, 5, 6, 7, 8, 9]. The spectral analysis
Received by the editors Feb. 12, 2014; Accepted: May 12, 2014. 2000 Mathematics Subject Classi…cation. 53C15, 53B05(53B20), 53C40.
Key words and phrases. Eigenvalues, spectral singularities, spectral analysis, Sturm - Liouville Operator, non-selfadjoint matrix operator.
c 2 0 1 4 A n ka ra U n ive rsity
of the non-selfadjoint operator, generated in L2(R
+) by (1:1) and the boundary
condition
y0(0)
y (0) = 0+ 1 + 2
2;
where i 2 C, i = 0; 1; 2 with 2 6= 0 was investigated by Bairamov et al.
[20]. The properties of the principal functions corresponding to the eigenvalues and the spectral singularities were studied in [14, 20, 21, 22]. Spectral analysis of the selfadjoint di¤erential and di¤erence equations with matrix coe¢ cients are studied in [10, 11, 12, 13].
Let E be an n-dimensional (n < 1) Euclidian space with the norm k:k and let us introduce the Hilbert space L2(R
+; E) consisting of vector-valued functions with
the values in E: We will consider the BVP
y00+ Q (x) y = 2y ; x 2 R+ ; (1.3)
y (0) = 0; (1.4)
in L2(R
+; E) ; where Q is a non-selfadjoint matrix-valued function (i. e., Q 6= Q ).
It is clear that, the BVP (1:3)–(1:4) is non-selfadjoint. In [15, 16] discrete spec-trum of the non-selfadjoint matrix Sturm–Liouville operator and properties of the principal functions correspending to the eigenvalues and the spectral singularities was investigated.
Let us consider the BVP in L2(R+; E)
y00+ Q(x)y = 2y; x 2 R+; (1.5)
y0(0; ) ( 0+ 1 + 2 2)y(0; ) = 0, (1.6)
where Q is a non-selfadjoint matrix-valued function and 0; 1; 2are non-selfadjoint matrices also 2is invertible. In this paper, which is an extention of [23], we aim to investigate the properties of the principal functions corresponding to the eigenvalues and the spectral singularities of the BVP (1:5)-(1:6) :
2. Jost Solution of (1:5)
We will denote the solution of (1:5) satisfying the condition lim
x!1y(x; )e
i x= I;
2 C+:= f : 2 C; Im 0g ; (2.1)
by E(x; ): The solution E(x; ) is called the Jost solution of (1:5). Under the condition
1
Z
0
x kQ(x)k dx < 1; (2.2)
the Jost solution has a representation E(x; ) = ei xI +
1
Z
x
for 2 C+, where the kernel matrix function K(x; t) satis…es K(x; t) = 1 2 1 Z x + t 2 Q(s)ds +1 2 x + t 2 Z x t+s xZ t+x s Q(s)K(s; v)dvds +1 2 1 Z x + t 2 t+s xZ s Q(s)K(s; v)dvds (2.4)
Moreover, K(x; t) is continuously di¤erentiable with respect to its arguments and kK(x; t)k c (x + t 2 ); (2.5) kKx(x; t)k 1 4 Q( x + t 2 ) + c ( x + t 2 ); (2.6) kKt(x; t)k 1 4 Q( x + t 2 ) + c ( x + t 2 ); (2.7) where (x) = 1 Z x
kQ(s)k ds and c > 0 is a constant. Therefore, E (x; ) is analytic with respect to in C+ := f : 2 C; Im > 0g and continuous on the real axis
([17; chp.1]).
Let ^E (x; ) denote the solutions of (1.5) subject to the conditions lim x!1 ^ E (x; )e i x= I; lim x!1 ^ Ex(x; )e i x= i I; 2 C : (2.8) Then W h E(x; ); ^E (x; ) i = 2i I; 2 C ; (2.9) W [E(x; ); E(x; )] = 2i I; 2 R; (2.10)
where W [f1; f2] is the Wronskian of f1 and f2:
Let '(x; ) denote the solution of (1.5) subject to the initial conditions '(0; ) = I; '0(0; ) = 0+ 1 + 2 2: Therefore '(x; ) is an entire function of .
Let us de…ne the following functions:
A ( ) = '(0; )Ex(0; ) '0(0; )E(0; ) 2 C ; (2.11)
where C = f : 2 C; Im 0g : It is obvious that the functions A+( ) and
A ( ) are analytic in C+and C respectively and continuous on the real axis. The
3. Eigenvalues and Spectral Singularities of L The resolvent of L de…ned by
R (L)f = 1 Z 0 G(x; t; )g(t)dt; g 2 L2(R+; E); (3.1) where G(x; t; ) = G+(x; t; ); 2 C+ G (x; t; ); 2 C : (3.2) and G (x; t; ) = E(x; )A 1( )'T(t; ); 0 t x '(x; ) AT( ) 1ET(t; ); x t < 1 (3.3)
We will show the set of eigenvalues and the set of spectral singularities of the operator L by d and ss respectively.
Let us suppose that
H ( ) = det A ( ): (3.4)
From (2.3) and (3.1)–(3.4)
d= f : 2 C+; H+( ) = 0g [ f : 2 C ; H ( ) = 0g
ss = f : 2 R ; H+( ) = 0g [ f : 2 R ; H ( ) = 0g ; (3.5)
where R = Rn f0g :
We see from that, the functions K+( ) = A^+( ) 2i E(x; ) A+( ) 2i E^ +(x; ); 2 C+; (3.6) K ( ) = A ( )^ 2i E(x; ) A ( ) 2i E (x; );^ 2 C ; (3.7) K( ) = A+( ) 2i E(x; ) A ( ) 2i E(x; ); 2 R ; (3.8)
are the solutions of the boundary problem (1.5)–(1.6) where ^
A ( ) = ^Ex (0; ) ( 0+ 1 + 2 2) ^E (0; ): (3.9) Now let us assume that
Q 2 AC(R+) ; lim x!1Q(x) = 0; supx2R+ h e"pxkQ0(x)k i < 1; " > 0: (3.10) Theorem 3.1. Under the condition (3.10), the operator L has a …nite number of eigenvalues and spectral singularities, and each of them is of …nite multiplicity.
4. Principal Functions of L
Under the condition (3.10), let 1; :::; j and j+1; :::; k denote the zeros H+ in
C+and H in C (which are the eigenvalues of L) with multiplicities m1;:::; mjand
mj+1;:::; mk; respectively. It is obvious that from the de…niton of the Wronskian
dn d nW K +(x; ); E(x; ) = p = d n d nA+( ) = p = 0 (4.1) for n = 0; 1; :::; mp 1; p = 1; 2; :::; j; and dn d nW K (x; ); E(x; ) = p = d n d nA ( ) = p = 0 (4.2) for n = 0; 1; :::; mp 1; p = j + 1; :::; k:
Theorem 4.1. The following formulae: @n @ nK +(x; ) = p = n X m=0 Fm( p) @m @ mE(x; ) = p ; (4.3) n = 0; 1; :::; mp 1; p = 1; 2; :::; j; where Fm( p) = n m @n m @ n m ^ A+( ) = p ; (4.4) @n @ nK (x; ) = p = n X m=0 Nm( p) @m @ mE(x; ) = p ; (4.5) n = 0; 1; :::; mp 1; p = j + 1; :::; k; where Nm( p) = n m @n m @ n m ^ A ( ) = p (4.6) hold.
Proof. We will prove only (4.3) using the method induction, because the case of (4.5) is similar. Let be n = 0: Since K+(x; ) and E(x; ) are linearly dependent
from (4.1), we get
K+(x; p) = f0( p)E(x; p) (4.7)
where f0( p) 6= 0: Let us assume that 1 n0 mp 2; (4.3) holds; that is,
@n0 @ n0K +(x; ) = p = n0 X m=0 Fm( p) @m @ mE(x; ) = p : (4.8)
We will prove that (4.3) holds for n0+ 1: If Y (x; ) is a solution of (1.5), then @n @ nY (x; ) satis…es h d2 dx2 + Q(x) 2i @n @ nY (x; ) = 2 n@ n 1 @ n 1Y (x; )+n(n 1) @n 2 @ n 2Y (x; ): (4.9)
Writing for (4.9) K+(x; ) and E(x; ), and using (4.8), we …nd d2 dx2 + Q(x) 2 g n0+1(x; p) = 0; (4.10) where gn0+1(x; p) = n @n0+1 @ n0+1K +(x; )o = p nX0+1 m=0 Fm( p) @ m @ mE(x; ) = p: (4.11) From (4.1), we have W [gn0+1(x; p); E(x; p)] = dn0+1 d n0+1W K +(x; ); E(x; ) = p = 0: (4.12) Hence there exists a constant fn0+1( p) such that
g
n0+1(x; p) = fn0+1( p)E(x; p): (4.13)
This shows that (4.3) holds for n = n0+ 1:
Using (4.3) and (4.5), de…ne the functions Un;p(x) = @n @ nK +(x; ) = p = n X m=0 Fm( p) @m @ mE(x; ) = p ; (4.14) n = 0; 1; :::; mp 1; p = 1; 2; :::; j and Un;p(x) = @n @ nK (x; ) = p = n X m=0 Nm( p) @m @ mE(x; ) = p ; (4.15) n = 0; 1; :::; mp 1; p = j + 1; :::; k: Then for = p; p = 1; 2; :::; j; j + 1; :::; k; l(U0;p) = 0; l(U1;p) + 1 1! @ @ l(U0;p) = 0; (4.16) l(Un;p) + 1 1! @ @ l(Un 1;p) + 1 2! @2 @ 2l(Un 2;p) = 0; n = 2; 3; :::; mp 1;
hold, where l(u) = u00+ Q(x)u 2u and @m
@ ml(u) denote the di¤erential
ex-pressions whose coe¢ cients are the m-th derivatives with respect to of the cor-responding coe¢ cients of the di¤erential expression l(u): (4.16) shows that U0;p is
the eigenfunction corresponding to the eigenvalue = p; U1;p; U2;p; :::Ump 1;pare
the associated functions of U0;p [18, 19].
U0;p; U1;p; :::Ump 1;p; p = 1; 2; :::; j; j + 1; :::; k are called the principal functions
Theorem 4.2.
Un;p2 L2(R+; E); n = 0; 1; :::mp 1; p = 1; 2; :::; j; j + 1; :::; k: (4.17)
Proof. Let be 0 n mp 1 and 1 p j: Using (2.2), (3.10) and (4.14) we
obtain that kK(x; t)k ce px+t2 : (4.18) From (2.3) we get @n @ nE(x; ) = p x ne x Im p+ c 1 Z x tne px+t2 e t Im pdt; (4.19)
where c > 0 is a constant. Since Im p> 0 for the eigenvalues p; p = 1; 2; :::; j; of
L; (4.19) implies that @n
@ nE(x; ) = p2 L2(R+; E); n = 0; 1; :::mp 1; p = 1; 2; :::; j: (4.20) So we get Un;p 2 L2(R+; E): Similarly we prove the results for 0 n mp 1;
j + 1 p k: This completes the proof.
Let 1; :::; vand v+1; :::; lbe the zeros of A+and A in R with multiplicities
n1; :::; nv and nv+1; :::; nl; respectively. We can show @n @ nK(x; ) = p = n X m=0 Cm( p) @m @ mE(x; ) = p (4.21) n = 0; 1; :::; np 1; p = 1; 2; :::; v; where Cm( p) = n m @n m @ n mA ( ) = p; (4.22) @n @ nK(x; ) = p = n X m=0 Rm( p) @m @ mE(x; ) = p ; n = 0; 1; :::; np 1; p = v + 1; :::; l; where Rm( p) = n m @n m @ n mA+( ) = p: (4.23)
Now de…ne the generalized eigenfunctions and generalized associated functions corresponding to the spectral singularities of L by the following :
Vn;p(x) = @n @ nK(x; ) = p = n X m=0 Cm( p) @m @ mE(x; ) = p (4.24)
n = 0; 1; :::; np 1; p = 1; 2; :::; v; Vn;p(x) = @n @ nK(x; ) = p = n X m=0 Rm( p) @m @ mE(x; ) = p ; n = 0; 1; :::; np 1; p = v + 1; :::; l:
Then Vn;p; n = 0; 1; :::; np 1; p = 1; 2; :::; v; v + 1; :::; l; also satisfy the equations
analogous to (4.16).
V0;p; V1;p; :::; Vnp 1;p; p = 1; 2; :::; v; v + 1; :::; l are called the principal functions
corresponding to the spectral singularities = p; p = 1; 2; :::; v; v + 1; :::; l of L: Theorem 4.3.
Vn;p2 L= 2(R+; E); n = 0; 1; :::np 1; p = 1; 2; :::; v; v + 1; :::; l:
Proof. For 0 n np 1 and 1 p v using (2.3), we obtain
@n @ nE(x; ) = p (ix)nei pxI + 1 Z x (it)nK(x; t)ei ptdt ; since Im p= 0; p = 1; 2; :::; v; we …nd that 1 Z 0 (ix)nei pxI 2dx = 1 Z 0 x2ndx = 1:
So we obtain Vn;p 2 L= 2(R+; E); n = 0; 1; :::np 1; p = 1; 2; :::; v: Using the similar
way, we may also prove the results for 0 n np 1; v + 1 p l:
Now de…ne the Hilbert spaces of vector-valued functions with values in E by
Hn : = 8 < :f : 1 Z 0 (1 + jxj)2nkf(x)k2dx < 1 9 = ;; n = 1; 2; :::; (4.25) H n : = 8 < :g : 1 Z 0 (1 + jxj) 2nkg(x)k2dx < 1 9 = ;; n = 1; 2; :::; (4.26) with the norms
kfk2n:= 1 Z 0 (1 + jxj)2nkf(x)k2dx and kgk2n := 1 Z 0 (1 + jxj) 2nkg(x)k2dx respectively. Then Hn+1$ Hn $ L2(R+; E) $ H n$ H (n+1); n = 1; 2; :::;
and H n is isomorphic to the dual of Hn:
Theorem 4.4.
Vn;p2 H (n+1); n = 0; 1; :::np 1; p = 1; 2; :::; v; v + 1; :::; l:
Proof. For 0 n np 1 and 1 p v using (2.3) and (4.24), we get 1 Z 0 (1 + jxj) 2(n+1)kVn;pk2dx M 1 Z 0 (1 + jxj) 2(n+1) 8 > > < > > : n kE(x; )k2o = p + ::: +n @n @ nE(x; ) 2o = p 9 > > = > > ; dx;
where M > 0 is a constant: Using (2.3), we have
1 Z 0 (1 + jxj) 2(n+1) (ix)nei pxI 2dx < 1 and 1 Z 0 (1 + jxj) 2(n+1) 1 Z x (it)nK(x; t)ei ptdt 2 dx < 1:
Consequently Vn;p 2 H (n+1) for 0 n np 1 and 1 p v: Similarly, we
obtain Vn;p2 H (n+1) for 0 n np 1 and v + 1 p l.
Theorem 4.5.
Vn;p2 H n0; n = 0; 1; :::np 1; p = 1; 2; :::; v; v + 1; :::; l:
References
[1] M. A. Naimark, Investigation of the spectrum and the expansion in eigenfunctions of a non-selfadjoint operators of second order on a semi-axis, AMS Translations 2(16) 1960, 103-193. [2] V. E. Lyance, A di¤erential operator with spectral singularities, I, II, AMS Translations 2
(60)1967, 185-225, 227-283.
[3] E. Bairamov, O. Cakar and A. O. Çelebi, Quadratic pencil of Schröndinger operators with spectral singularities: Discrete spectrum and principal functions, J. Math. Anal. Appl. 216 (1997), 303-320.
[4] E. Bairamov, O. Cakar and A. M. Krall, Spectrum and spectral singularities of a quadratic pencil of a Schrödinger operator with a general boundary condition, J. Di¤erential Equations 151 (1999), no. 2, 252–267.
[5] E. Bairamov, O. Cakar and A. M. Krall, An eigenfunction expansion for a quadratic pencil of a Schrödinger operator with spectral singularities, J. Di¤erential Equations 151 (1999), 268-289.
[6] E. Bairamov and A. O. Celebi, Spectrum and spectral expansion for the non-selfadjoint discrete Dirac operators, Quart. J. Math. Oxford Ser. (2) 50 (1999), no. 200, 371–384. [7] E. Bairamov and A. O. Çelebi, Spectral properties of the Klein-Gordon s-wave equation with
complex potential, Indian J. Pure Appl. Math. 28 (1997), 813-824.
[8] E. Bairamov and G. B. Tunca, Discrete spectrum and principal functions of non-selfadjoint di¤erential operators, Czechoslovak Math. J. 49 (1999), 689-700.
[9] A. M. Krall, E. Bairamov and O. Cakar, Spectral analysis of a non-selfadjoint discrete Schrödinger operators with spectral singularities, Math. Nachr. 231 (2001), 89-104.
[10] R. Carlson, An inverse problem for the matrix Schrödinger equation, J. Math. Anal. Appl. 267(2002), 564-575.
[11] S. Clark and F. Gesztesy, Weyl-Titchmarsh M-function asymptotics, local uniqueness re-sults, trace formulas and Borg-type theorems for Dirac operators ,Trans Amer. Math. Soc. 354(2002), 3475-3534.
[12] S. Clark, F. Gesztesy and W. Renger Trace formulas and Borg-type theorems for matrix-valued Jacobi and Dirac …nite di¤erence operators, J. Di¤erential Equations 219 (2005), 144-182.
[13] F. Gesztesy, A. Kiselev and K. A. Makarov, Uniqueness results for matrix-valued Schrödinger, Jacobi and Dirac-type operators, Math. Nachr. 239 (2002), 103-145.
[14] N. Yokus, Principal functions of non-selfadjoint Sturm-Liouville problems with eigenvalue-dependent boundary conditions,Abstract and Applied Analysis, (2011) 1-12.
[15] M. Olgun, C. Coskun, Non-selfadjoint matrix Sturm-Liouville operators with spectral sin-gularities Applied Mathematics and Computations, 216 (8) (2010)
[16] C. Coskun, M. Olgun, Principal functions of non-selfadjoint matrix Sturm-Liouville equations, Journal of Computational and Applied Mathematics, 235(2011).
[17] Z. S. Agranovich, V. A. Marchenko, The Inverse Problem of Scattering Theory, Gordon and Breach, 1965.
[18] M. V. Keldysh, " On the completeness of the eigenfunctions of some classes of non-selfadjoint linear operators," Soviet Mathematics-Doklady, vol. 77, no. 4, pp. 11-14, 1951.
[19] M. V. Keldysh, " On the completeness of the eigenfunctions of some classes of non-selfadjoint linear operators," Russian Mathematical Surveys, vol. 26, no. 4, pp. 15-41, 1971.
[20] E. Bairamov, and N. Yokus, Spectral Singularities of Sturm-Liouville Problems with Eigenvalue-Dependent Boundary Conditions, Abstract and Applied Analysis, 2009, 1-8. [21] M. Olgun, T. Koprubasi and Y. Aygar, Principal Functions of Non-Selfadjoint Di¤erence
Operator with Spectral Parameter in Boundary Conditions, Abstract and Applied Analysis, pp.1-11, 2011.
[22] Y. Aygar, ;M. Olgun and T. Koprubasi, Principal Functions of Nonselfadjoint Discrete Dirac Equations with Spectral Parameter in Boundary Conditions,Abstract and Applied Analysis, pp.1-15,2012.
[23] M. Olgun, Non-Selfadjoint Matrix Sturm-Liouville Operators with Eigenvalue-Dependent Boundary Conditions, Hacettepe J Math & Stat., (Accepted).
Current address : Ankara University, Faculty of Sciences Department of Mathematics Ankara, TURKEY
E-mail address : ccoskun@ankara.edu.tr, deniz.ktr@hotmail.com, olgun@ankara.edu.tr URL: http://communications.science.ankara.edu.tr/index.php?series=A1
