C om mun. Fac. Sci. U niv. A nk. Ser. A 1 M ath. Stat. Volum e 68, N umb er 2, Pages 1316–1334 (2019) D O I: 10.31801/cfsuasm as.526270
ISSN 1303–5991 E-ISSN 2618-6470
http://com munications.science.ankara.edu.tr/index.php?series= A 1
SPECTRAL EXPANSION OF STURM-LIOUVILLE PROBLEMS WITH EIGENVALUE-DEPENDENT BOUNDARY CONDITIONS
NIHAL YOKU¸S AND ESRA KIR ARPAT
Abstract. In this paper, we consider the operator L generated in L2(R+)by
the di¤erential expression
l(y) = y00+ q(x)y; x2R+:= [0; 1)
and the boundary condition y0(0)
y(0) = 0+ 1 + 2
2;
where q is a complex valued function and i2 C; i = 0; 1; 2 with 26= 0. We
have proved that spectral expansion of L in terms of the principal functions under the condition
q 2 AC(R+); lim
x!1q(x) = 0; xsup2R+
[e"pxjq0(x)j] < 1; " > 0 taking into account the spectral singularities. We have also proved the con-vergence of the spectral expansion.
1. 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 L0, generated in L2(R+), by the di¤erential expression
l0(y) = y00+ q(x)y; x 2 R+:= [0; 1) (1.1)
with the boundary condition y0(0) hy(0) = 0, where q is a complex valued function
and h 2 C. If the following condition
Received by the editors: November 14, 2017; Accepted: August, 06, 2018. 2010 Mathematics Subject Classi…cation. 47E05, 34B05, 34L05, 47A10.
Key words and phrases. Eigenvalues, spectral singularities, principal functions, resolvent, spec-tral expansion.
c 2 0 1 9 A n ka ra U n ive rsity C o m m u n ic a tio n s Fa c u lty o f S c ie n c e s U n ive rs ity o f A n ka ra -S e rie s A 1 M a th e m a t ic s a n d S ta t is t ic s
Z
e"xjq(x)jdx < 1; " > 0
satis…es, then L0 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 L0[2]. The Laurent
expan-sion of the resolvents of non-selfadjoint operators in neigbourhood of spectral sin-gularities 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 L2(R+)by the di¤erential expression
l(y) = y00+ q(x)y; x2R+ (1.2)
and the eigenvalue-dependent boundary condition
y0(0)
y(0) = 0+ 1 + 2 2 (1.3)
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
q 2 AC(R+); lim
x!1
q(x) = 0; sup
x2R+
[e"pxjq0(x)j] < 1; " > 0 (1.4) 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.
2. SPECIAL SOLUTIONS Let us consider the equation
y00+ q(x)y = 2y; x 2 R+ (2.1)
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
Z 1
0 xjq(x)jdx < 1
(2.2)
Let '(x; ) and e(x, ) denote the solutions of (2.1) satisfying the conditions '(x; ) = 1; '0(x; ) = 0+ 1 + 2 2; lim
x!1
e(x; )e i x= 1; 2 C (2.3) respectively. The solution e(x; ) is called Jost Solution of (2.1). Note that, under the condition (2.2), the solution '(x; ) is an entire function of and the Jost Solution is an analytic function of in C+:= f : 2 C; Im > 0g and continuous
in C+:= f : 2 C; Im > 0g ([14]).
Moreover, Jost Solution has a representation ([11])
e(x; ) = ei x+ Z 1
x
K(x; t)ei tdt; 2 C+ (2.4)
where the kernel K(x; t) satis…es
K(x; t) = 1 2 Z 1 x+t 2 q(s) ds +1 2 Z x+t 2 x Z t+s x t+x s q(s)K(s; u) duds +1 2 Z 1 x+t 2 Z t+s x s q(s)K(s; u) duds (2.5)
and K(x; t) is continuously di¤erentiable with respect to x and t.
jK(x; t)j 6 cw x + t 2 (2.6) jKx(x; t)j; jKt(x; t)j 6 1 4 q x + t 2 + cw x + t 2 (2.7)
where w(x) =Rx1jq(s)j ds and c > 0 is a constant.
Let e (x; ) denote the solutions of (2.1) subject to the conditions lim x!1e i xe (x; ) = 1; lim x!1e i xe x(x; ) = i ; 2 C (2.8) Then W [e(x; ); e (x; )] = 2i ; 2 C W [e(x; ); e (x; )] = 2i ; 2 R = ( 1; 1) (2.9)
We will denote the Wronskian of the solutions with e(x; ) and e(x; ) by E+( ) and E ( ), respectively, where
E+( ) := e0(0; ) ( 0+ 1 + 2 2)e(0; ); 2 C+ (2.10)
E ( ) := e0(0; ) ( 0+ 1 + 2 2)e(0; ); 2 C (2.11)
Therefore, E+and E are analytic with respect to in C
+= f : 2 C; Im >0g
and C = f : 2 C; Im <0g, respectively, and continuous up to real axis, and E+( ) = 2 2+ + + ++ o(1); 2 C+; j j ! 1 (2.12) E ( ) = 2 2+ + + o(1); 2 C ; j j ! 1 (2.13) where += i 1 i 2K(0; 0) + = K(0; 0) 0 i 1K(0; 0) + 2Kt(0; 0) (2.14) f+(t) = Kx(0; t) 0K(0; t) i 1Kt(0; t) + 2Ktt(0; t) hold [14]. 3. THE SPECTRUM OF L We have previously shown ([14]) that
d(L) = f : 2 C+; E+( ) = 0g [ f : 2 C+; E ( ) = 0g ss(L) = f : 2 R ; E+( ) = 0g [ f : 2 R ; E ( ) = 0g
(3.1)
where R = R f0g, by d(L) and ss(L) we denote the eigenvalues and spectral
singularities of L, respectively. Let G(x; t; ) = ( G+(x; t; ); 2 C + G (x; t; ); 2 C (3.2)
be the Green Function of L, where
G+(x; t; ) = ( '(t; )e(x; ) E+( ) ; 06 t 6 x '(x; )e(t; ) E+( ) ; x6 t < 1 ) (3.3) G (x; t; ) = ( '(t; )e(x; ) E ( ) ; 06 t 6 x '(x; )e(t; ) E ( ) ; x6 t < 1 ) (3.4)
Under the conditions (1.4), we know that L has a …nite number of eigenval-ues and spectral singularities, and each of them is …nite multiplicity ([14]). Let
1; : : : ; j and j+1; : : : ; k denote the zeros of E+in C+and E in C (which are
We will also need the Hilbert Spaces Hm= f : Z 1 0 (1 + x)2mjf(x)j2dx < 1 ; m = 0; 1; : : : H m= g : Z 1 0 (1 + x) 2mjg(x)j2dx < 1 ; m = 0; 1; : : : with kfk2m= Z 1 0 (1 + x)2mjf(x)j2dx; kgk2m= Z 1 0 (1 + x) 2mjg(x)j2dx respectively. It is obvious that H0= L2(R+) and
Hm+1$ Hm$ L2(R+) $ H m$ H (m+1); m = 1; 2; : : :
and H mis isomorphic to the dual of Hm: Hm0 H m
We have previously shown that ([15]):
Un;p2 L2(R+); n = 0; 1; :::; mp l; p = 1; 2; :::; (3.5) Un;p2 H (n+1); n = 0; 1; :::; mp l; p = + 1; :::; k (3.6) where Un;p(x) = @n @ n' +(x; ) = p = n X =0 An ( ) @ @ e(x; ) = p Un;p(x) = @n @ n' (x; ) = p = n X =0 An ( ) @ @ e(x; ) = p (3.7)
The functions Un;p(x); n = 0; 1; : : : ; mp 1; p = 1; 2; : : : ; and p = + 1; : : : ; k
are the principal functions corresponding to the eigenvalues and the spectral sin-gularities of L, respectively.
4. SPECTRAL EXPANSION Let C1
0 (R+) denote the set of in…nitely di¤erentiable functions in R+with
com-pact support. Evidently,
(x) = R(L)R 1(L) (x) = R(L)(L 2I) (x) (x) = Z 1 0 G(x; t; ) 00+ q(t) (t) 2 (t) dt for each 2 C1 0 (R+). We obtain (x) = 1 Z 1 0 G(x; t; ) (t) dt D(x; ) (4.1) where (t) = 00+ q(t) (t); D(x; ) = Z 1 0 G(x; t; ) (t) dt
Figure 4.1
Let rdenote the contour with center at the origin having radius r; let @ rbe the boundary of r. r will be chosen so that all eigenvalues and spectral singularities of L are in r. Pr denotes the part of r lying in the strip jIm j and r = +r [ r , where +r and r are the parts of rnPr in the upper and the
lower half-planes, respectively (see Figure 4.1). We chose so small that Pr does
not contain any eigenvalues of L.
So we easily see that
@ r= @ r [ @Pr (4.2) From (4.1) we get (x) = 1 2 i Z @ r 1Z 1 0 G(x; t; ) (t) dt d 1 2 i Z @ r D(x; ) d (4.3)
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
(x) = lim r!1 !0 1 2 i Z @ r D(x; ) d lim r!1 !0 1 2 i Z @Pr D(x; ) d (4.4)
Figure 4.2
We easily obtain that the …rst integral in (4.4) gives
lim r!1 !0 Z D(x; ) d =X i=1 Res = +i D+(x; ) +X i=1 Res = i D (x; ) where D (x; ) = Z 1 0 G (x; t; ) (t) dt:
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
lim r!1 !0 1 2 i Z @Pr D(x; ) d = 1 2 i Z D (x; ) d 1 2 i Z + D+(x; ) d
Therefore (4.4) can be written as
(x) = X i=1 Res = +i D+(x; ) X i=1 Res = i [D (x; )] + 1 2 i Z + D+(x; ) d 1 2 i Z D (x; ) d (4.5)
Theorem 1. For every 2 C1 0 (R+) (x) =X i=1 ( @ @ m+i 1h a+i ( )U (x; ) U ( ; ) i) = + i +X i=1 ( @ @ mi 1h ai ( )U (x; ) U ( ; )i ) = +i + 1 2 i Z + e+ x(0; ) E+( ) U (x; ) U ( ; ) d 1 2 i Z ex(0; ) E ( ) U (x; ) U ( ; ) d (4.6)
0 =X i=1 ( @ @ m+i 1h b+i ( )U (x; ) U ( ; ) i) = + i +X i=1 ( @ @ mi 1h bi ( )U (x; ) U ( ; )i ) = i + 1 2 i Z + e+ x(0; ) E+( ) U (x; ) U ( ; ) d 1 2 i Z ex(0; ) E ( ) U (x; ) U ( ; ) d (4.7) where a+i ( ) = ( + i )mie+x(0; ) (mi 1)!E+( ) ; i = 1; :::; (4.8) ai ( ) = ( i ) mie x(0; ) (mi 1)!E ( ) ; i = 1; :::; k b+i( ) = ( + i )mie+x(0; ) (mi 1)!E+( ) ; i = 1; :::; (4.9) bi ( ) = ( i ) mie x(0; ) (mi 1)!E ( ) ; i = 1; :::; k and U ( ; ) = Z 1 0 (t)U (x; ) dt:
Proof. Let B(x; ) be the solution of (2.1) subject to the initial conditions B(0; ) = 1; B0(0; ) = 0+ 1 + 2 2: Then G (x; t; ) = ex(0; ) E ( ) U (x; ) U (t; ) + a(x; t; ) (4.10) where a (x; t; ) = B(x; )U (t; ) ; 0 < t6 x B(t; )U (x; ) ; x t < 1
and a (x; t; ) is an entire function of . From (4.5) and (4.10) we obtain (4.6). Writing (4.1) as (x) 2 = 1 2 Z 1 0 G(x; t; ) (t) dt D(x; )
Since the contour + and in (4.6) and (4.7) do not coincide with the
contin-uous 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 C01(R+) there exists a constant c > 0 so that
Z 1 1j U ( ; )j 2 d c Z 1 0 j (x)j 2 dx (4.11)
Proof. From (3.7) we get
U ( ; ) = n X =0 Mn ( p) 1 ! @ @ e ( ; ) = p (4.12) where e ( ; ) = Z 1 0 (x)e (x; ) dx: Using (2.4), we obtain e ( ; ) = Z 1 0 ei x+ Z 1 x K(x; t)ei tdt (x) dx = Z 1 0 (x)ei xdx + Z 1 0 Z 1 x (x)K(x; t)ei tdtdx
Changing the order of integration, we get
e (x; ) = Z 1
0 f(I + K) (t)g e
i tdt (4.13)
in which the operator I is the unit operator, and K is the operator de…ned by
K (t) = Z 1
0
K(x; t) (x) dt:
From (2.6) we understand K is a compact operator in L2(R+). Thus (I + K) is a
continuous and one-to-one on L2(R+). Using the Parseval’s equality for the Fourier
transforms and (4.13) we get Z 1 1 e ( ; )2 d c Z 1 0 j (x)j 2 dx (4.14) 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 L2(R+) the limit
U ( ; ) = lim
N!1
Z N 0
exists in the sense of convergence in the mean square, relative to the measure 2d on the real axis; that is,
lim N!1 Z 1 1 U ( ; ) Z N 0 (x)U (x; ) dx 2 2d = 0 (4.15) Since C1
0 (R+) is dense in L2(R+), the estimate (4.11) may be extended onto
L2(R+) for any 2 L2(R+) as Z 1 1j U( ; )j 2 d c Z 1 0 j (x)j 2 dx (4.16)
where U ( ; ) must be understood in the sense of (4.15). We shall need a general-ization of this estimate.
Theorem 3. If 2 Hm; then U ( ; ) has a derivative of order (m 1) which is
absolutely continuous of every …nite subinterval of the real axis and satis…es Z 1 1 d d n [U ( ; )] d 2 cn Z 1 0 (1 + x)2nj (x)j2 dx (4.17)
where cn> 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 L2(R+). Now we will investigate the convergence
of these integrals in a norm which is weaker than the norm of L2(R+). For this
purpose we will use the technique of regularization of divergent integrals. So we de…ne the following summability factor:
Fp+( ) = ( ( p) ! ; j pj < ; p = 1; : : : ; n 0 ; j pj > ; p = 1; : : : ; n (4.18) Fp ( ) = ( ( p) ! ; j pj < ; p = n + 1; :::; k 0 ; j pj > ; 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
F+fg1( )g =g1( ) n X p=1 mXp 1 =0 ( d d g1( ) ) = p Fp+( ) (4.20) F fg2( )g =g2( ) k X p=n+1 mXp 1 =0 ( d d g2( ) ) = p Fp ( ) (4.21)
where g1and g2is chosen so that the right hand side of the above formulae is
mean-ingful. It is evident from (4.18)-(4.19) that 1; :::; n are the roots of F+fg1( )g =
0 and n+1,..., kare the roots of F fg2( )g = 0 at least of orders m1; :::; mnand
mn+1; :::; mk, respectively.
In the following, we will use the operators
P+ (x) = 1 2 i Z + e+x(0; ) E+( ) U (x; ) U ( ; ) d (4.22) P (x) = 1 2 i Z ex(0; ) E ( ) U (x; ) U ( ; ) d (4.23) and I+ (x) = 1 2 i n X p=1 mXp 1 =0 ( @ @ [U (x; ) U ( ; )] ) = p Z + e+ x(0; ) E+( ) F + p ( )d + 1 2 i Z 1 1 e+ x(0; ) E+( ) F + fU (x; ) U ( ; )g d I (x) = 1 2 i k X p=n+1 mXp 1 =0 ( @ @ [U (x; ) U ( ; )] ) = p Z ex(0; ) E ( ) Fp ( )d + 1 2 i Z 1 1 ex(0; ) E ( ) F fU (x; ) U ( ; )g d
Since under the condition (1.4) e+(x; ) and e (x; ) have an analytic continuation to the half-planes Imk > 2and Imk < "2, respectively, we get
P = I for 2 C1
Theorem 4. For each 2 H(m0+1), there exist a constant c > 0 such that I (m 0+1) c1k k(m0+1) (4.24) where m0= max fm1; :::; mn; mn+1; :::; mkg : Proof. De…ne + p = ( p ; p+ ); p = 1; :::; n (4.25) Then 0 =2 +
p, p = 1; :::; n. Using the integral form of remainder in the Taylor
formula, we get F+fU(x; )U( ; )g = ( U (x; )U ( ; ) ; 2 +0 1 (mp 1)! R p( ) mp 1 n @ @ mph U (x; )U ( ; ) io d ; 2 +p (4.26) where +0 = R ( n S p=1 + p )
: If we use the notation
Ip+ (x) = 1 2 i Z + p e+x(0; ) E+( ) n U (x; )U ( ; ) o d ; p = 1; :::; k ~ I+ (x) = 1 2 i k X p=1 mXp 1 =0 @ @ h U (x; )U ( ; ) i = p Z e+ x(0; ) E+( ) F + p ( ) d we obtain I+= I0++ ::: + Ik++ ~I+ (4.27) from (4.25) and (4.26). We now show that each of the operators I0+; :::; I+
p and ~I+
is continuous from H(m0+1) into H (m0+1). We start from with ~I
+. From (4.18)
we …nd the absolute convergence of Z + e+ x(0; ) E+( ) F + p ( ) d
Using (3.6) and the isomorphism H m0 Hm0 0 we see that ~I
+ is continuous from
Hminto H m0or from H(m0+1)into H (m0+1): Hence there exists a constant ~c > 0
such that
~
I+ (x) (m0+1)6 ~c (m0+1) (4.28)
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
Ip+ (x) = 1 2 i(mp 1)! Z + p e+ x(0; ) E+( ) Z p ( )mp 1 @ @ mp [U (x; )U ( ; )] d d (4.29)
Interchanging the order of integration, we get
Ip+ (x) = 1 2 i(mp 1)! 8 > < > : p+ Z p p+ Z @ @ mp [U (x; )U ( ; )] ( )mp 1 e + x(0; ) E+( ) d d p Z p Z p @ @ mp [U (x; )U ( ; )] ( )mp 1 e + x(0; ) E+( ) d d :
Since p is a zero of E+( ) order mp; there exists a continuous function Ep+( )
such that E+
p( p) 6= 0 and E+( ) = ( p)mpE+p( p): On the other hand,
p+ Z ( )mp 1 e + x(0; ) E+( ) d h (1) p ( ) [ln ln( p)] (4.30) if > p; and Z p ( )mp 1 e + x(0; ) E+( ) d h (2) p ( ) [ln( p ) ln ] (4.31) if < p; where h(1)p ( ) = max 2[ ; p+ ] e+x(0; ) Ep+( ) ; h(2)p ( ) = max 2[ p ; ] e+x(0; ) Ep+( ) :
(4.30) and (4.31) show that I+
p; p = 1; :::; n are integral operators with kernels
having logarithmic singularities. (4.29) can be written as Ip+ (x) = Z p mp X s=0 b+sp(x; ) d d s U ( ; ) d :
De…ne Bsp= 1 Z 0 Z p b+sp(x; ) (1 + x)m0+1 2 d dx:
We see that Bsp< 1; by (3.6), (4.30) and (4.31). Since
Ip+ 2 (m0+1)= 1 Z 0 I+ p (x) (1 + x)m0+1 2 dx mp X s=0 1 Z 0 Z + p b+ sp(x; ) (1 + x)m0+1 2 d dx Z + p d d s U ( ; ) 2 d = mp X k=0 Bsp Z + p d d s U ( ; ) 2 d
Utilizing (4.16) and (4.17) we obtain Ip+
(m0+1) cpk km0 cpk k(m0+1); p = 1; :::; n (4.32)
where cp are constants.
We consider the operator I0+ which is de…ned by I0+ = 1 2 i Z 1 1 {0+( ) e+ x(0; ) E+( ) [U (x; )U ( ; )] d (4.33)
where {0+ is the characteristic function of the interval +
0: From (4.33), similar to
the proof of Theorem 4.2, we get
1 Z 0 Ip+ (x)2dx c0 1 Z 0 j (x)j2dx;
where c0> 0 is a constant. Since
H(m0+1)$ L2(R+) $ H (m0+1);
we …nd
kI0 k (m0+1) c0k k(m0+1) (4.34)
From (4.27), (4.28),(4.32) and (4.34) we have
I+ (m
In a similar way it follows that I
(m0+1) c k k(m0+1)
Then for every 2 H(m0+1);
I+ (x) = 1 2 i n X p=1 mXp 1 =0 ( @ @ [U (x; )U ( ; )] ) = p (4.35) Z e+x(0; ) E+( ) F + p ( ) d + 1 2 i 1 Z 1 e+ x(0; ) E+( ) F + fU (x; ) U ( ; )g d and I (x) = 1 2 i k X p=n+1 mXp 1 =0 ( @ @ [U (x; )U ( ; )] ) = p (4.36) Z ex(0; ) E ( ) Fp ( ) d + 1 2 i 1 Z 1 ex(0; ) E ( ) F fU (x; ) U ( ; )g d
Let ap( ) denote any function which is de…ned and di¤erentiable in a neighbourhood
of p, and which satis…es the condition
( d d mp 1 ap( ) ) = p = 8 > > < > > : 1 2 i mp 1 R + e+ x(0; ) E+( ) Fp+( ) d ; p = 1; :::; n 1 2 i mp 1 R ex(0; ) E ( ) Fp ( ) d ; p = n + 1; :::; k (4.37) Then (4.35) and (4.36) can be written as
I+ (x) = n X p=1 ( @ @ mp 1 [ap( )U (x; ) U ( ; )] ) = p + 1 2 i 1 Z 1 e+ x(0; ) E+( ) F + fU (x; ) U ( ; )g d (4.38)
I (x) = k X p=n+1 ( @ @ mp 1 [ap( )U (x; ) U ( ; )] ) = p (4.39) + 1 2 i 1 Z 1 ex(0; ) E ( ) F fU (x; ) U ( ; )g d we shall also use the following integral operator (see (4.7)):
Q+ (x) = 1 2 i Z + e+x(0; ) E+( ) F +[U (x; )U ( ; )] d (4.40) Q (x) = 1 2 i Z ex(0; ) E ( ) F [U (x; )U ( ; )] d (4.41) J+ (x) = 1 2 i n X p=1 mXp 1 =0 ( @ @ [U (x; )U ( ; )] ) = p Z e+ x(0; ) E+( ) F + p ( )d + 1 2 i 1 Z 1 e+ x(0; ) E+( ) F +[U (x; )U ( ; )] d J (x) = 1 2 i n X p=1 mXp 1 =0 ( @ @ [U (x; )U ( ; )] ) = p Z ex(0; ) E ( ) Fp ( )d + 1 2 i 1 Z 1 ex(0; ) E ( ) F [U (x; )U ( ; )] d It is evident that Q = J ; for 2 C1 0 (R+):
Theorem 5. For every each 2 H(m0+1); there exist a constant c > 0 such that
J (m
It is evident that, for every 2 H(m0+1) J+ (x) = n X p=1 ( @ @ mp 1 [bp( ) [U (x; )U ( ; )]] ) = p (4.42) + 1 2 i 1 Z 1 e+ x(0; ) E+( ) F + fU (x; ) U ( ; )g d where ( d d mp 1 bp( ) ) = p = 8 > > < > > : 1 2 i mp 1 R + e+ x(0; ) E+( ) Fp+( ) d p = 1; ::; n 1 2 i mp 1 R ex(0; ) E ( ) Fp ( ) d p = n + 1; ::; k (4.43)
Theorem 6. Under the condition (1.4) the following two-fold spectral expansion in terms of the principal functions of L holds,
(x) = p X i=1 ( @ @ m+i 1 [ai( ) [U (x; )U ( ; )]] ) = i + n X i=p+1 ( @ @ mp 1 [ap( ) [U (x; )U ( ; )]] ) = p (4.44) + 1 2 i 1 Z 1 e+ x(0; ) E+( ) F + fU(x; )U( ; )g d 1 2 i 1 Z 1 ex(0; ) E ( ) F fU(x; )U( ; )g d
0 = p X i=1 8 < : @ @ m+i 1 [bi( ) [U (x; )U ( ; )]] 9 = ; = i + n X i=p+1 ( @ @ mp 1 [bp( ) [U (x; )U ( ; )]] ) = p (4.45) + 1 2 i 1 Z 1 e+(0; ) E+( ) F + fU(x; )U( ; )g d 1 2 i 1 Z 1 ex(0; ) E ( ) F fU(x; )U( ; )g d
for every function 2 H(m0+1): The integrals in (4.44) and (4.45) converge in the
norm of H (m0+1) where ai; bi; F , ap and bp de…ned by (4.8), (4.9), (4.20), (4.21),
(4.37), and (4.43) respectively.
Proof. We obtain (4.44) and (4.45) for 2 C1
0 (R+) H(m0+1); by use of (4.6),
(4.7), (4.22), (4.23) and (4.38)-(??). The convergence of the integrals appearing in (4.44) and (4.45) in the norm of H (m0+1); has been given in Theorem 4 and
Theorem 5. As C1
0 (R+) is dense in H(m0+1); the proof is …nished.
Acknowledgements. The authors would like to thank Professor E. Bairamov for his helpful suggestions during the preparation of this paper.
References
[1] Naimark, M.A., Investigation of the spectrum and the expansion in eigenfunctions of a non-selfadjoint operator of the second order on a semi-axis, American Mathematical Society Trans-lations Series 2, 16, (1960), 103–193.
[2] Lyance, V.E., A di¤erential operator with spectral singularities I, II, American Mathematical Society Transactions Series 2, 60, (1967), 185–225, 227–283.
[3] Gasymov, M.G. and Maksudov, F.G., The principal part of the resolvent of non-selfadjoint opeerators in neighbourhood of spectral singularities, Func. Anal. Appl, 6, (1972), 185–192. [4] Maksudov, F.G. and Allakhverdiev, B.P., Spectral analysis of a new class of non-selfadjoint
operators with continuous and point spectrum, Soviet Math. Dokl., 30, (1984), 566–569. [5] Ad¬var, M. and Bairamov, E., Spectral properties of non-selfadjoint di¤erence operators,
Journal of Mathematical Analysis and Applications, 261(2), (2001), 461–478.
[6] Bairamov, E., Çakar, Ö. and Yan¬k, C., Spectral singularities of the Klein-Gordon s-wave equation, Indian Journal of Pure and Applied Mathematics, 32(6), (2001), 851–857. [7] Bairamov, E. and Çelebi, A.O., Spectrum and spectral expansion for the non-selfadjoint
dis-crete Dirac operators, The Quarterly Journal of Mathematics. Oxford Second Series, 50(200), (1999), 371–384.
[8] Bairamov, E. and Karaman, Ö., Spectral singularities of the Klein-Gordon s-wave equations with and integral boundary conditions, Acta Mathematica Hungarica, 97(1–2), (2002), 121– 131.
[9] Krall, A.M., Bairamov, E. and Çakar, Ö., Spectrum and spectral singularities of a quadratic pencil of a Schrödinger operator with a general boundary condition, Journal of Di¤ erential Equations, 151(2), (1999), 252–267.
[10] Krall, A.M., Bairamov, E. and Çakar, Ö., Spectral analysis of non-selfadjoint discrete Schrödinger operators with spectral singularities, Mathematische Nachrichten, 231, (2001), 89–104.
[11] Marchenko, V.A., Expansion in eigenfunctions of non-selfadjoint singular second-order di¤er-ential operators, American Mathematical Society Transactions Series 2, 25, (1963), 99.77– 130.
[12] Ba¸scanbaz-Tunca, G, Spectral expasion of a non-selfadjoint di¤erential operator on the whole axis, J.Math.Anal.Appl., 252(1), (2000), 278–297.
[13] K¬r Arpat, E., An eingenfunction expansion of the non-selfadjoint Sturm-Liouville operator with a singular potential, Journal of Mathematical Chemistry, 51(8), (2013), 2196–2213. [14] Bairamov, E. and Yoku¸s, N., Spectral singularities of Sturm-Liouville problems with
eigenvalue-dependent boundary conditions, Abstract and Applied Analysis, 2009, Article ID 289596, (2009), 8 pages.
[15] Yoku¸s, N., Principal functions of non-selfadjoint sturm-liouville problems with eigenvalue-dependent boundary conditions, Abstract and Applied Analysis, 2011, Article ID 358912, (2011), 12 pages.
Current address : Nihal Yoku¸s: Karamano¼glu Mehmetbey University, Department of Mathe-matics, Karaman, Turkey.
E-mail address : nyokus@kmu.edu.tr
ORCID Address: https://orcid.org/0000-0002-5327-2312
Current address : Esra K¬r Arpat (Corresponding author): Gazi University, Faculty of Sciences, Department of Mathematics, Teknikokullar-06500, Ankara, Turkey.
E-mail address : esrakir@gazi.edu.tr
