Abstract

We consider the operator 𝐿 generated in 𝐿2(+) by the differential expression 𝑙(𝑦)=𝑦+𝑞(𝑥)𝑦, 𝑥+=[0,) and the boundary condition 𝑦(0)/𝑦(0)=𝛼0+𝛼1𝜆+𝛼2𝜆2, where 𝑞 is a complex-valued function and 𝛼𝑖, 𝑖=0,1,2 with 𝛼20. In this paper we obtain the properties of the principal functions corresponding to the spectral singularities of 𝐿.

1. Introduction

Let 𝑇 be a nonselfadjoint, closed operator in a Hilbert space 𝐻. We will denote the continuous spectrum and the set of all eigenvalues of 𝑇 by 𝜎𝑐(𝑇) and 𝜎𝑑(𝑇), respectively. Let us assume that 𝜎𝑐(𝑇).

Definition 1.1. If 𝜆=𝜆0 is a pole of the resolvent of 𝑇 and 𝜆0𝜎𝑐(𝑇), but 𝜆0𝜎𝑑(𝑇), then 𝜆0 is called a spectral singularity of 𝑇.

Let us consider the nonselfadjoint operator 𝐿0 generated in 𝐿2(+) by the differential expression𝑙0(𝑦)=𝑦+𝑞(𝑥)𝑦,𝑥+,(1.1) and the boundary condition 𝑦(0)=0, where 𝑞 is a complex-valued function. The spectrum and spectral expansion of 𝐿0 were investigated by Naĭmark [1]. He proved that the spectrum of 𝐿0 is composed of continuous spectrum, eigenvalues, and spectral singularities. He showed that spectral singularities are on the continuous spectrum and are the poles of the resolvent kernel, which are not eigenvalues.

Lyance investigated the effect of the spectral singularities in the spectral expansion in terms of the principal functions of 𝐿0 [2, 3]. He also showed that the spectral singularities play an important role in the spectral analysis of 𝐿0.

The spectral analysis of the non-self-adjoint operator 𝐿1 generated in 𝐿2(+) by (1.1) and the boundary condition0𝐾(𝑥)𝑦(𝑥)𝑑𝑥+𝛼𝑦(0)𝛽𝑦(0)=0,(1.2) in which 𝐾𝐿2(+) is a complex valued function and 𝛼,𝛽, was investigated in detail by Krall [48] In [4] he obtained the adjoint 𝐿1 of the operator 𝐿1. Note that 𝐿1 deserves special interest, since it is not a purely differential operator. The eigenfunction expansions of 𝐿1 and 𝐿1 were investigated in [5].

In [9] the results of Naimark were extended to the three-dimensional Schrödinger operators.

The Laurent expansion of the resolvents of the abstract non-self-adjoint operators in the neighborhood of the spectral singularities was studied in [10].

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 [1120]. By regularization of a divergent integral, the effect of the spectral singularities in the spectral expansion of a quadratic pencil of Schrödinger operators was obtained in [13]. In [19, 20] 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 [21]) and the analytical properties of the Weyl function.

Let 𝐿 denote the operator generated in 𝐿2(+) by the differential expression𝑙(𝑦)=𝑦+𝑞(𝑥)𝑦,𝑥+(1.3) and the boundary condition𝑦(0)𝑦(0)=𝛼0+𝛼1𝜆+𝛼2𝜆2,(1.4) where 𝑞 is a complex-valued function and 𝛼𝑖, 𝑖=0,1,2 with 𝛼20. In this work we obtain the properties of the principal functions corresponding to the spectral singularities of 𝐿.

2. The Jost Solution and Jost Function

We consider the equation𝑦+𝑞(𝑥)𝑦=𝜆2𝑦,𝑥+(2.1) related to the operator 𝐿.

Now we will assume that the complex valued function 𝑞 is almost everywhere continuous in + and satisfies the following: 0𝑥||||𝑞(𝑥)𝑑𝑥<.(2.2) Let 𝜑(𝑥,𝜆) and 𝑒(𝑥,𝜆) denote the solutions of (2.1) satisfying the conditions𝜑(0,𝜆)=1,𝜑(0,𝜆)=𝛼0+𝛼1𝜆+𝛼2𝜆2,lim𝑥𝑒(𝑥,𝜆)𝑒𝑖𝜆𝑥=1,𝜆+,(2.3) respectively. The solution 𝑒(𝑥,𝜆) is called the Jost solution of (2.1). Note that, under the condition (2.2), the solution 𝜑(𝑥,𝜆) is an entire function of 𝜆 and the Jost solution is an analytic function of 𝜆 in +={𝜆𝜆,Im𝜆>0} and continuous in +={𝜆𝜆,Im𝜆0}.

In addition, Jost solution has a representation ([22])𝑒(𝑥,𝜆)=𝑒𝑖𝜆𝑥+𝑥𝐾(𝑥,𝑡)𝑒𝑖𝜆𝑡𝑑𝑡,𝜆+,(2.4) where the kernel 𝐾(𝑥,𝑡) satisfies1𝐾(𝑥,𝑡)=2(𝑥+𝑡)/21𝑞(𝑠)𝑑𝑠+2𝑥(𝑥+𝑡)/2𝑡+𝑠𝑥𝑡+𝑥𝑠+1𝑞(𝑠)𝐾(𝑠,𝑢)𝑑𝑢𝑑𝑠2(𝑥+𝑡)/2𝑠𝑡+𝑠𝑥𝑞(𝑠)𝐾(𝑠,𝑢)𝑑𝑢𝑑𝑠(2.5) and 𝐾(𝑥,𝑡) is continuously differentiable with respect to its arguments. We also have||||𝐾(𝑥,𝑡)𝑐𝑤𝑥+𝑡2,(2.6)||𝐾𝑥||,||𝐾(𝑥,𝑡)𝑡||1(𝑥,𝑡)4|||𝑞𝑥+𝑡2|||+𝑐𝑤𝑥+𝑡2,(2.7) where 𝑤(𝑥)=𝑥|𝑞(𝑠)|𝑑𝑠 and 𝑐>0 is a constant.

Let ̂𝑒±(𝑥,𝜆) denote the solutions of (2.1) subject to the conditionslim𝑥𝑒±𝑖𝜆𝑥̂𝑒±(𝑥,𝜆)=1,lim𝑥𝑒±𝑖𝜆𝑥̂𝑒±𝑥(𝑥,𝜆)=±𝑖𝜆,𝜆±.(2.8) Then𝑊𝑒(𝑥,𝜆),̂𝑒±(𝑥,𝜆)=2𝑖𝜆,𝜆±,𝑊[]𝑒(𝑥,𝜆),𝑒(𝑥,𝜆)=2𝑖𝜆,𝜆=(,),(2.9) where 𝑊[𝑓1,𝑓2] is the Wronskian of 𝑓1 and 𝑓2, ([23]).

We will denote the Wronskian of the solutions 𝜑(𝑥,𝜆) with 𝑒(𝑥,𝜆) and 𝑒(𝑥,𝜆) by 𝐸+(𝜆) and 𝐸(𝜆), respectively, where𝐸+(𝜆)=𝑒𝛼(0,𝜆)0+𝛼1𝜆+𝛼2𝜆2𝑒(0,𝜆),𝜆+,𝐸(𝜆)=𝑒𝛼(0,𝜆)0+𝛼1𝜆+𝛼2𝜆2𝑒(0,𝜆),𝜆,(2.10) and ={𝜆𝜆,Im𝜆0}. Therefore 𝐸+ and 𝐸 are analytic in + and ={𝜆𝜆,Im𝜆<0}, respectively, and continuous up to real axis.

The functions 𝐸+ and 𝐸 are called Jost functions of 𝐿.

3. Eigenvalues and Spectral Singularities of 𝐿

Let𝐺𝐺(𝑥,𝑡;𝜆)=+(𝑥,𝑡;𝜆),𝜆+,𝐺(𝑥,𝑡;𝜆),𝜆(3.1) be the Green function of 𝐿 (obtained by the standard techniques), where𝐺+(𝑥,𝑡;𝜆)=𝜑(𝑡,𝜆)𝑒(𝑥,𝜆)𝐸+(𝜆),0𝑡𝑥𝜑(𝑥,𝜆)𝑒(𝑡,𝜆)𝐸+𝐺(𝜆),𝑥𝑡<(𝑥,𝑡;𝜆)=𝜑(𝑡,𝜆)𝑒(𝑥,𝜆)𝐸(𝜆),0𝑡𝑥𝜑(𝑥,𝜆)𝑒(𝑡,𝜆)𝐸(𝜆),𝑥𝑡<.(3.2) We will denote the set of eigenvalues and spectral singularities of 𝐿 by 𝜎𝑑(𝐿) and 𝜎ss(𝐿), respectively. From (3.1)–(3.2)𝜎𝑑(𝐿)=𝜆𝜆+,𝐸+(𝜆)=0𝜆𝜆,𝐸,𝜎(𝜆)=0ss(𝐿)=𝜆𝜆,𝐸+(𝜆)=0𝜆𝜆,𝐸(,𝜆)=0(3.3) where ={0}.

From (3.3) we obtain that to investigate the structure of the eigenvalues and the spectral singularities of 𝐿, we need to discuss the structure of the zeros of the functions 𝐸+ and 𝐸 in + and , respectively.

Definition 3.1. The multiplicity of zero of the function 𝐸+(or 𝐸) in +(or ) is called the multiplicity of the corresponding eigenvalue and spectral singularity of 𝐿.
We see from (2.9) that the functions𝜓+𝐸(𝑥,𝜆)=+(𝜆)𝐸2𝑖𝜆𝑒(𝑥,𝜆)+(𝜆)2𝑖𝜆̂𝑒+(𝑥,𝜆),𝜆+,𝜓𝐸(𝑥,𝜆)=(𝜆)𝐸2𝑖𝜆𝑒(𝑥,𝜆)(𝜆)2𝑖𝜆̂𝑒(𝑥,𝜆),𝜆,𝐸𝜓(𝑥,𝜆)=+(𝜆)𝐸2𝑖𝜆𝑒(𝑥,𝜆)(𝜆)2𝑖𝜆𝑒(𝑥,𝜆),𝜆(3.4) are the solutions of the boundary value problem 𝑦+𝑞(𝑥)𝑦=𝜆2𝑦,𝑥+,𝑦(0)𝑦(0)=𝛼0+𝛼1𝜆+𝛼2𝜆2,(3.5) where 𝐸±(𝜆)=̂𝑒±𝛼(0,𝜆)0+𝛼1𝜆+𝛼2𝜆2̂𝑒±(0,𝜆).(3.6) Now let us assume that 𝑞𝐴𝐶+,lim𝑥𝑞(𝑥)=0,sup𝑥+𝑒𝜀𝑥||𝑞||(𝑥)<,𝜀>0.(3.7)

Theorem 3.2 (see [24]). Under the condition (3.7) the operator 𝐿 has a finite number of eigenvalues and spectral singularities, and each of them is of a finite multiplicity.

4. Principal Functions of 𝐿

In this section we assume that (3.7) holds. Let 𝜆1,,𝜆𝑗 and 𝜆𝑗+1,,𝜆𝑘 denote the zeros of 𝐸+ in + and 𝐸 in (which are the eigenvalues of 𝐿) with multiplicities 𝑚1,,𝑚𝑗 and 𝑚𝑗+1,,𝑚𝑘, respectively. It is obvious that from definition of the Wronskian𝑑𝑛𝑑𝜆𝑛𝑊𝜓+(𝑥,𝜆),𝑒(𝑥,𝜆)𝜆=𝜆𝑝=𝑑𝑛𝑑𝜆𝑛𝐸+(𝜆)𝜆=𝜆𝑝=0(4.1) for 𝑛=0,1,,𝑚𝑝1,𝑝=1,2,,𝑗, and𝑑𝑛𝑑𝜆𝑛𝑊[𝜓](𝑥,𝜆),𝑒(𝑥,𝜆)𝜆=𝜆𝑝=𝑑𝑛𝑑𝜆𝑛𝐸(𝜆)𝜆=𝜆𝑝=0(4.2) for 𝑛=0,1,...,𝑚𝑝1,𝑝=𝑗+1,...,𝑘.

Theorem 4.1. The fallowing formulae: 𝜕𝑛𝜕𝜆𝑛𝜓+(𝑥,𝜆)𝜆=𝜆𝑝=𝑛𝑚=0𝐴𝑚𝜆𝑝𝜕𝑚𝜕𝜆𝑚𝑒(𝑥,𝜆)𝜆=𝜆𝑝,(4.3)𝑛=0,1,,𝑚𝑝1,𝑝=1,2,,𝑗, where 𝐴𝑚𝜆𝑝=𝑛𝑚𝜕𝑛𝑚𝜕𝜆𝑛𝑚𝐸+(𝜆)𝜆=𝜆𝑝,(4.4)𝜕𝑛𝜕𝜆𝑛𝜓(𝑥,𝜆)𝜆=𝜆𝑝=𝑛𝑚=0𝐵𝑚𝜆𝑝𝜕𝑚𝜕𝜆𝑚𝑒(𝑥,𝜆)𝜆=𝜆𝑝,(4.5)𝑛=0,1,,𝑚𝑝1,𝑝=𝑗+1,,𝑘, where 𝐵𝑚𝜆𝑝=𝑛𝑚𝜕𝑛𝑚𝜕𝜆𝑛𝑚𝐸(𝜆)𝜆=𝜆𝑝(4.6) holds.

Proof. We will proceed by mathematical induction, we prove first (4.3). Let 𝑛=0. From (4.1) we get 𝜓+𝑥,𝜆𝑝=𝑎0𝜆𝑝𝑒𝑥,𝜆𝑝,(4.7) where 𝑎0(𝜆𝑝)0. Let us assume that for 1𝑛0𝑚𝑝2, (4.3) holds; that is, 𝜕𝑛0𝜕𝜆𝑛0𝜓+(𝑥,𝜆)𝜆=𝜆𝑝=𝑛0𝑚=0𝐴𝑚𝜆𝑝𝜕𝑚𝜕𝜆𝑚𝑒(𝑥,𝜆)𝜆=𝜆𝑝.(4.8) Now we will prove that (4.3) holds for 𝑛0+1. If 𝑦(𝑥,𝜆) is a solution of (2.1), then (𝜕𝑛/𝜕𝜆𝑛)𝑦(𝑥,𝜆) satisfies 𝑑2𝑑𝑥2+𝑞(𝑥)𝜆2𝜕𝑛𝜕𝜆𝑛𝜕𝑦(𝑥,𝜆)=2𝜆𝑛𝑛1𝜕𝜆𝑛1𝜕𝑦(𝑥,𝜆)+𝑛(𝑛1)𝑛2𝜕𝜆𝑛2𝑦(𝑥,𝜆).(4.9) Writing (4.9) for 𝜓+(𝑥,𝜆) and 𝑒(𝑥,𝜆), and using (4.8), we find 𝑑2𝑑𝑥2+𝑞(𝑥)𝜆2𝑓𝑛0+1𝑥,𝜆𝑝=0,(4.10) where 𝑓𝑛0+1𝑥,𝜆𝑝=𝜕𝑛0+1𝜕𝜆𝑛0+1𝜓+(𝑥,𝜆)𝜆=𝜆𝑝𝑛0+1𝑚=0𝐴𝑚𝜆𝑝𝜕𝑚𝜕𝜆𝑚𝑒(𝑥,𝜆)𝜆=𝜆𝑝.(4.11) From (4.1) we have 𝑊𝑓𝑛0+1𝑥,𝜆𝑝,𝑒𝑥,𝜆𝑝=𝑑𝑛0+1𝑑𝜆𝑛0+1𝑊𝜓+(𝑥,𝜆),𝑒(𝑥,𝜆)𝜆=𝜆𝑝=0.(4.12) Hence there exists a constant 𝑎𝑛0+1(𝜆𝑝) such that 𝑓𝑛0+1𝑥,𝜆𝑝=𝑎𝑛0+1𝜆𝑝𝑒𝑥,𝜆𝑝.(4.13) This shows that (4.3) holds for 𝑛=𝑛0+1.
Similarly we can prove that (4.5) holds.

Definition 4.2. Let 𝜆=𝜆0 be an eigenvalue of L. If the functions 𝑦0𝑥,𝜆0,𝑦1𝑥,𝜆0,,𝑦𝑠𝑥,𝜆0(4.14) satisfy the equations 𝑙𝑦0𝜆0𝑦0𝑦=0,𝑙𝑗𝜆0𝑦𝑗𝑦𝑗1=0,𝑗=1,2,,𝑠,(4.15) then the function 𝑦0(𝑥,𝜆0) is called the eigenfunction corresponding to the eigenvalue 𝜆=𝜆0 of 𝐿. The functions 𝑦1(𝑥,𝜆0),,𝑦𝑠(𝑥,𝜆0) are called the associated functions corresponding 𝜆=𝜆0. The eigenfunctions and the associated functions corresponding to 𝜆=𝜆0 are called the principal functions of the eigenvalue 𝜆=𝜆0.
The principal functions of the spectral singularities of 𝐿 are defined similarly.

Now using (4.3) and (4.5) define the functions𝑈𝑛,𝑝𝜕(𝑥)=𝑛𝜕𝜆𝑛𝜓+(𝑥,𝜆)𝜆=𝜆𝑝=𝑛𝑚=0𝐴𝑚𝜆𝑝𝜕𝑚𝜕𝜆𝑚𝑒(𝑥,𝜆)𝜆=𝜆𝑝,(4.16)𝑛=0,1,,𝑚𝑝1,𝑝=1,2,,𝑗

and𝑈𝑛,𝑝𝜕(𝑥)=𝑛𝜕𝜆𝑛𝜓(𝑥,𝜆)𝜆=𝜆𝑝=𝑛𝑚=0𝐵𝑚𝜆𝑝𝜕𝑚𝜕𝜆𝑚𝑒(𝑥,𝜆)𝜆=𝜆𝑝,(4.17)𝑛=0,1,,𝑚𝑝1,𝑝=𝑗+1,,𝑘.

Then for 𝜆=𝜆𝑝,𝑝=1,2,,𝑗,𝑗+1,,𝑘,𝑙𝑈0,𝑝𝑙𝑈=0,1,𝑝+1𝜕1!𝑙𝑈𝜕𝜆0,𝑝𝑙𝑈=0,𝑛,𝑝+1𝜕1!𝑙𝑈𝜕𝜆𝑛1,𝑝+1𝜕2!2𝜕𝜆2𝑙𝑈𝑛2,𝑝=0,(4.18)𝑛=2,3,,𝑚𝑝1,

hold, where 𝑙(𝑢)=𝑢+𝑞(𝑥)𝑢𝜆2𝑢 and (𝜕𝑚/𝜕𝜆𝑚)𝑙(𝑢) denotes the differential expressions whose coefficients are the m-th derivatives with respect to 𝜆 of the corresponding coefficients of the differential expression 𝑙(𝑢). Equation (4.18) shows that 𝑈0,𝑝 is the eigenfunction corresponding to the eigenvalue 𝜆=𝜆𝑝;𝑈1,𝑝,𝑈2,𝑝,,𝑈𝑚𝑝1,𝑝 are the associated functions of 𝑈0,𝑝 ([25, 26]).

𝑈0,𝑝,𝑈1,𝑝,,𝑈𝑚𝑝1,𝑝,𝑝=1,2,,𝑗,𝑗+1,,𝑘 are called the principal functions corresponding to the eigenvalue 𝜆=𝜆𝑝,𝑝=1,2,,𝑗,𝑗+1,,𝑘 of 𝐿.

Theorem 4.3. One has 𝑈𝑛,𝑝𝐿2+,𝑛=0,1,,𝑚𝑝1,𝑝=1,2,,𝑗,𝑗+1,,𝑘.(4.19)

Proof. Let 0𝑛𝑚𝑝1 and 1𝑝𝑗. Using (2.6) and (3.7) we obtain that ||||𝐾(𝑥,𝑡)𝑐𝑒𝜀(𝑥+𝑡)/2.(4.20) From (2.4) we get |||||𝜕𝑛𝜕𝜆𝑛𝑒(𝑥,𝜆)𝜆=𝜆𝑝|||||𝑥𝑛𝑒𝑥Im𝜆𝑝+𝑐𝑥𝑡𝑛𝑒𝜀(𝑥+𝑡)/2𝑒𝑡Im𝜆𝑝𝑑𝑡,(4.21) where 𝑐>0 is a constant. Since Im𝜆𝑝>0 for the eigenvalues 𝜆𝑝,𝑝=1,,𝑗, of 𝐿, (4.21) implies that 𝜕𝑛𝜕𝜆𝑛𝑒(𝑥,𝜆)𝜆=𝜆𝑝𝐿2+,𝑛=0,1,,𝑚𝑝1,𝑝=1,2,,𝑗.(4.22) The proof of theorem is obtained from (4.16) and (4.22). In a similar way using (4.17) we may also prove the results for 0𝑛𝑚𝑝1 and 𝑗+1𝑝𝑘.

Let 𝜇1,,𝜇𝑣, and 𝜇𝑣+1,,𝜇𝑙 be the zeros of 𝐸+ and 𝐸 in ={0} (which are the spectral singularities of 𝐿) with multiplicities 𝑛1,,𝑛𝑣 and 𝑛𝑣+1,,𝑛𝑙, respectively.

Similar to (4.3) and (4.5) we can show the following:𝜕𝑛𝜕𝜆𝑛𝜓(𝑥,𝜆)𝜆=𝜇𝑝=𝑛𝑚=0𝐶𝑚𝜇𝑝𝜕𝑚𝜕𝜆𝑚𝑒(𝑥,𝜆)𝜆=𝜇𝑝,(4.23)

𝑛=0,1,,𝑛𝑝1,𝑝=1,2,,𝑣,

where𝐶𝑚𝜇𝑝𝑛𝑚𝜕=𝑛𝑚𝜕𝜆𝑛𝑚𝐸(𝜆)𝜆=𝜇𝑝,𝜕𝑛𝜕𝜆𝑛𝜓(𝑥,𝜆)𝜆=𝜇𝑝=𝑛𝑚=0𝐷𝑚𝜇𝑝𝜕𝑚𝜕𝜆𝑚𝑒(𝑥,𝜆)𝜆=𝜇𝑝,(4.24)

𝑛=0,1,,𝑛𝑝1,𝑝=𝑣+1,,𝑙,

where𝐷𝑚𝜇𝑝=𝑛𝑚𝜕𝑛𝑚𝜕𝜆𝑛𝑚𝐸+(𝜆)𝜆=𝜇𝑝.(4.25) Now define the generalized eigenfunctions and generalized associated functions corresponding to the spectral singularities of 𝐿 by the following:𝜐𝑛,𝑝𝜕(𝑥)=𝑛𝜕𝜆𝑛𝜓(𝑥,𝜆)𝜆=𝜇𝑝=𝑛𝑚=0𝐶𝑚𝜇𝑝𝜕𝑚𝜕𝜆𝑚𝑒(𝑥,𝜆)𝜆=𝜇𝑝,(4.26)𝑛=0,1,,𝑛𝑝1,𝑝=1,2,,𝑣, 𝜐𝑛,𝑝𝜕(𝑥)=𝑛𝜕𝜆𝑛𝜓(𝑥,𝜆)𝜆=𝜇𝑝=𝑛𝑚=0𝐷𝑚𝜇𝑝𝜕𝑚𝜕𝜆𝑚𝑒(𝑥,𝜆)𝜆=𝜇𝑝,(4.27)𝑛=0,1,,𝑛𝑝1,𝑝=𝑣+1,,𝑙.

Then 𝜐𝑛,𝑝,𝑛=0,1,,𝑛𝑝1,𝑝=1,2,,𝑣,𝑣+1,,𝑙, also satisfy the equations analogous to (4.18).

𝜐0,𝑝,𝜐1,𝑝,,𝜐𝑛𝑝1,𝑝,𝑝=1,2,,𝑣,𝑣+1,,𝑙 are called the principal functions corresponding to the spectral singularities 𝜆=𝜇𝑝,𝑝=1,2,,𝑣,𝑣+1,,𝑙 of 𝐿.

Theorem 4.4. One has 𝜐𝑛,𝑝𝐿2+,𝑛=0,1,,𝑛𝑝1,𝑝=1,2,,𝑣,𝑣+1,,𝑙.(4.28)

Proof. If we consider (4.21) for the principal functions corresponding to the spectral singularities 𝜆=𝜇𝑝,𝑝=1,2,,𝑣,𝑣+1,,𝑙, of 𝐿 and consider that Im𝜆𝑝=0 for the spectral singularities, then we have (4.28), by (4.26) and (4.27).

Now introduce the Hilbert spaces𝐻𝑛=𝑓0(1+𝑥)2𝑛||||𝑓(𝑥)2𝐻𝑑𝑥<,𝑛=1,2,,𝑛=𝑔0(1+𝑥)2𝑛||||𝑔(𝑥)2𝑑𝑥<,𝑛=1,2,,(4.29) with𝑓2𝑛=0(1+𝑥)2𝑛||||𝑓(𝑥)2𝑑𝑥;𝑔2𝑛=0(1+𝑥)2𝑛||||𝑔(𝑥)2𝑑𝑥,(4.30) respectively. Then𝐻𝑛+1𝐻𝑛𝐿2+𝐻𝑛𝐻(𝑛+1),𝑛=1,2,,(4.31) and 𝐻𝑛 is isomorphic to the dual of 𝐻𝑛.

Theorem 4.5. One has 𝜐𝑛,𝑝𝐻(𝑛+1),𝑛=0,1,,𝑛𝑝1,𝑝=1,2,,𝑣,𝑣+1,,𝑙.(4.32)

Proof. From (2.4) we have 0(1+𝑥)2(𝑛+1)||(𝑖𝑥)𝑛𝑒𝑖𝜇𝑝𝑥||2𝑑𝑥<,0(1+𝑥)2(𝑛+1)||||𝑥(𝑖𝑡)𝑛𝐾(𝑥,𝑡)𝑒𝑖𝜇𝑝𝑡||||𝑑𝑡2𝑑𝑥<.(4.33) Using (4.26), (4.33) we obtain 𝜐𝑛,𝑝𝐻(𝑛+1),𝑛=0,1,,𝑛𝑝1,𝑝=1,2,,𝑣.(4.34) In a similar way, we find 𝜐𝑛,𝑝𝐻(𝑛+1),𝑛=0,1,,𝑛𝑝1,𝑝=𝑣+1,,𝑙.(4.35)

Let us choose 𝑛0 so that𝑛0𝑛=max1,𝑛2,𝑛𝑣,𝑛𝑣+1,,𝑛𝑙.(4.36)

By Theorem 4.5 and (4.31) we get following theorem

Theorem 4.6. One has 𝜐𝑛,𝑝𝐻𝑛0,𝑛=0,1,,𝑛𝑝1,𝑝=1,2,,𝑣,𝑣+1,,𝑙.(4.37)

Acknowledgment

The author would like to thank Professor E. Bairamov for his helpful suggestions during the preparation of this work.