SPECTRAL EXPANSION OF STURM-LIOUVILLE PROBLEMS WITH EIGENVALUE-DEPENDENT BOUNDARY CONDITIONS

In this paper, we consider the operator L generated in L2(R+) by the differential expression l(y) = −y′′ + q(x)y, x∈R+ := [0,∞) and the boundary condition y′(0) y(0) = α0 + α1λ+ α2λ , where q is a complex valued function and αi ∈ C, i = 0, 1, 2 with α2 6= 0. We have proved that spectral expansion of L in terms of the principal functions under the condition q ∈ AC(R+), lim x→∞ q(x) = 0, sup x∈R+ [e √ x|q′(x)|] <∞, ε > 0 taking into account the spectral singularities. We have also proved the convergence of the spectral expansion.


SPECTRAL EXPANSION OF STURM-LIOUVILLE PROBLEMS WITH EIGENVALUE-DEPENDENT BOUNDARY CONDITIONS
NIHAL YOKUŞ AND ESRA KIR ARPAT Abstract.In this paper, we consider the operator L generated in L 2 (R + ) by the di¤erential expression l(y) = y 00 + q(x)y; x2R + := [0; 1) and the boundary condition where q is a complex valued function and i 2 C; i = 0; 1; 2 with 2 6 = 0. We have proved that spectral expansion of L in terms of the principal functions under the condition

INTRODUCTION
The spectral analysis of a non-selfadjoint di¤erential operators with continuous and discrete spectrum was investigated by Naimark [1].He showed the existence of spectral singularities in the continuous spectrum of the non-selfadjoint di¤erential operator L 0 , generated in L 2 (R + ), by the di¤erential expression l 0 (y) = y 00 + q(x)y; x 2 R + := [0; 1) (1.1) with the boundary condition y 0 (0) hy(0) = 0, where q is a complex valued function and h 2 C. If the following condition Z e "x jq(x)jdx < 1; " > 0 satis…es, then L 0 has a …nite number of eigenvalues and spectral singularities with …nite multiplicities.Lyance investigated the e¤ect of the spectral singularities in the spectral expansion in terms of the principal functions of L 0 [2].The Laurent expansion of the resolvents of non-selfadjoint operators in neigbourhood of spectral singularities was investigated by Gasymov-Maksudov [3] and Maksudov-Allakhverdiev [4].They also studied the e¤ect of spectral singularities in the spectral analysis of these operators.
Using the boundary uniqueness theorems of analytic functions, the structure of the eigenvalues and the spectral singularities of a quadratic pencil of Schrödinger, Klein-Gordon, discrete Dirac and discrete Schrödinger operators was investigated in [5]- [10].The e¤ect of the spectral singularities in the spectral expansion of a quadratic pencil of Schrödinger operators was obtained in [9].In [10] the spectral expansion of the discrete Dirac and Schrödinger operators with spectral singularities was derived using the generalized spectral function (in the sense of Marchenko [11]) and the analytical properties of the Weyl function.
Spectral analysis of the quadratic pencil of Schrödinger operators was done in [9].Spectral expansion of a non-selfadjoint di¤erential operator on the whole axis was studied in [12].The other expansion of the non-selfadjoint Sturm-Liouville Operator with a singular potential was studied in [13].
Let us consider the operator L generated in L 2 (R + )by the di¤erential expression l(y) = y 00 + q(x)y; x2R and the eigenvalue-dependent boundary condition where q is a complex-valued function and i 2 C; i = 0; 1; 2 with 2 6 = 0.In ( [14]) it has been proved that the operator L has of a …nite number and spectral singularities, each of them is of …nite multiplicity under the conditions In this paper, which is a continuation of ( [15]), we …nd a spectral expansion of L in terms of the principal functions under the conditions (1.4) taking into account the spectral singularities using a contour integral method, and the regularization of divergent integrals, using summability factors.We also investigate the convergence of the spectral expansion.

SPECIAL SOLUTIONS
Let us consider the equation We have previously considered in [15] that the only complex valued function, q is almost everywhere continuous in R + and satis…es the following condition ) and e(x, ) denote the solutions of (2.1) satisfying the conditions where W [f 1 ; f 2 ] is the Wronskian of f 1 and f 2 ([14]).

SPECTRAL EXPANSION
Let C 1 0 (R + ) denote the set of in…nitely di¤erentiable functions in R + with compact support.Evidently, where Let r denote the contour with center at the origin having radius r; let @ r be the boundary of r .r will be chosen so that all eigenvalues and spectral singularities of L are in r .P r denotes the part of r lying in the strip jIm j and r = + r [ r , where + r and r are the parts of r nP r in the upper and the lower half-planes, respectively (see Figure 4.1).We chose so small that P r does not contain any eigenvalues of L.
So we easily see that @ r = @ r [ @P r (4.2) From (4.1) we get Using (2.12), (2.13), (3.2) and Jordan's lemma, we see that the …rst term of the right hand side of (4.3) vanishes as r ! 1.The same result holds for the second term.Then considering (4.2) we …nd We easily obtain that the …rst integral in (4.4) gives where Let be the contour which isolates the real zeros of E + by semicircles with centers at i , i = 1; 2; : : : ; having the same radius 0 in the upper-half plane.Similarly, let be the corresponding contour for the real zeros of E in the lower half-plane.The radius of semicircles being chosen so small that their diameters are mutually disjoint and do not contain the point = 0 (see Figure 4.2).
From Figure 4.1, we obtain Therefore (4.4) can be written as where Since the contour + and in (4.6) and (4.7) do not coincide with the continuous spectrum of L, these formulae contains non-spectral objects.The aim of this article is to transform (4.6) and (4.7) into two-fold spectral expansion with respect to the principal functions of L.
Theorem 2. For any 2 C 1 0 (R + ) there exists a constant c > 0 so that Proof.From (3.7) we get where e ( ; ) = Z 1 0 (x)e (x; ) dx: Changing the order of integration, we get e (x; ) = Z 1 0 f(I + K) (t)g e i t dt (4.13) in which the operator I is the unit operator, and K is the operator de…ned by From (2.6) we understand K is a compact operator in L 2 (R + ).Thus (I + K) is a continuous and one-to-one on L 2 (R + ).Using the Parseval's equality for the Fourier transforms and (4.13) we get where c > 0 is a constant.The proof of the theorem is completed by (2.12),(2.13)and (4.14).
By the preceding theorem, for every function 2 L 2 (R + ) the limit exists in the sense of convergence in the mean square, relative to the measure 2 d on the real axis; that is, , the estimate (4.11) may be extended onto where U ( ; ) must be understood in the sense of (4.15).We shall need a generalization of this estimate.
Theorem 3. If 2 H m ; then U ( ; ) has a derivative of order (m 1) which is absolutely continuous of every …nite subinterval of the real axis and satis…es where c n > 0 are constants, n = 1; : : : ; m: The proof is similar to that of Theorem 2.
To transform (4.6) and (4.7) into the spectral expansion of L, we have to reform the integrals over + and onto the real axis.Since the spectral singularities of L are the zeros of E , the integrals over the real axis are divergent in the norm of L 2 (R + ).Now we will investigate the convergence of these integrals in a norm which is weaker than the norm of L 2 (R + ).For this purpose we will use the technique of regularization of divergent integrals.So we de…ne the following summability factor: ; j p j < ; p = 1; : : : ; n 0 ; j p j > ; p = 1; : : : ; n (4.18) ; j p j < ; p = n + 1; :::; k 0 ; j p j > ; p = n + 1; :::; k (4.19) with > 0 : We can choose > 0 so small that the neighborhoods of p ; p = 1; :::; n; n + 1; :::; k have no common points and do not contain the point = 0. De…ne the functions where g 1 and g 2 is chosen so that the right hand side of the above formulae is meaningful.It is evident from (4.18)-(4.19) that 1 ; :::; n are the roots of F + fg 1 ( )g = 0 and n+1 ,..., k are the roots of F fg 2 ( )g = 0 at least of orders m 1 ; :::; m n and m n+1 ; :::; m k , respectively.
Next we want to show the continuity of I + p ; p = 1; :::; n from H (m0+1) into H (m0+1) : From (4.26) we see that Interchanging the order of integration, we get (4.29) can be written as We see that B sp < 1; by (3.6), (4.30) and (4.31).Since where c p are constants.We consider the operator I + 0 which is de…ned by where { + 0 is the characteristic function of the interval + 0 : From (4.33), similar to the proof of Theorem 4.2, we get where c 0 > 0 is a constant.Since In a similar way it follows that Then for every 2 H (m0+1) ; we shall also use the following integral operator (see (4.7)):