Abstract

We investigate the principal functions corresponding to the eigenvalues and the spectral singularities of the boundary value problem (BVP) 𝑎 𝑛 − 1 𝑦 𝑛 − 1 + 𝑏 𝑛 𝑦 𝑛 + 𝑎 𝑛 𝑦 𝑛 + 1 = 𝜆 𝑦 𝑛 , 𝑛 ∈ ℕ and ( 𝛾 0 + 𝛾 1 𝜆 ) 𝑦 1 + ( 𝛽 0 + 𝛽 1 𝜆 ) 𝑦 0 = 0 , where ( 𝑎 𝑛 ) and ( 𝑏 𝑛 ) are complex sequences, 𝜆 is an eigenparameter, and 𝛾 𝑖 , 𝛽 𝑖 ∈ ℂ for 𝑖 = 0 , 1.

1. Introduction

Let us consider the (BVP) − 𝑦 + 𝑞 ( 𝑥 ) 𝑦 = 𝜆 2 𝑦 𝑦 , 0 ≤ 𝑥 < ∞ , ( 0 ) − ℎ 𝑦 ( 0 ) = 0 ( 1 . 1 ) in 𝐿 2 ( ℝ + ) , where 𝑞 is a complex-valued function and 𝜆 ∈ ℂ is a spectral parameter and ℎ ∈ ℂ . The spectral theory of the above BVP with continuous and point spectrum was investigated by Naĭmark [1]. He showed that the existence of the spectral singularities in the continuous spectrum of the BVP. He noted that the spectral singularities that belong to the continuous spectrum are the poles of the resolvents kernel but they are not the eigenvalues of the BVP. Also he showed that eigenfunctions and the associated functions (principal functions) corresponding to the spectral singularities are not the element of 𝐿 2 ( ℝ + ) . The spectral singularities in the spectral expansion of the BVP in terms of principal functions have been investigated in [2]. The spectral analysis of the quadratic pencil of Schrödinger, Dirac, and Klein-Gordon operators with spectral singularities was studied in [3–8]. The spectral analysis of a non-selfadjoint difference equation with spectral parameter has been studied in [9]. In this paper, it is proved that the BVP 𝑎 𝑛 − 1 𝑦 𝑛 − 1 + 𝑏 𝑛 𝑦 𝑛 + 𝑎 𝑛 𝑦 𝑛 + 1 = 𝜆 𝑦 𝑛 𝛾 , 𝑛 ∈ ℕ , ( 1 . 2 ) 0 + 𝛾 1 𝜆 𝑦 1 + 𝛽 0 + 𝛽 1 𝜆 𝑦 0 = 0 ( 1 . 3 ) has a finite number of eigenvalues and spectral singularities with a finite multiplicities if s u p 𝑛 ∈ ℕ e x p 𝜀 𝑛 𝛿 | | 1 − 𝑎 𝑛 | | + | | 𝑏 𝑛 | | < ∞ ( 1 . 4 ) for some 𝜀 > 0 and 1 / 2 ≤ 𝛿 ≤ 1 .

Let 𝐿 denote difference operator of second order generated in ℓ 2 ( ℕ ) by ( ℓ 𝑦 ) 𝑛 = 𝑎 𝑛 − 1 𝑦 𝑛 − 1 + 𝑏 𝑛 𝑦 𝑛 + 𝑎 𝑛 𝑦 𝑛 + 1 , 𝑛 ∈ ℕ ( 1 . 5 ) and with boundary condition 𝛾 0 + 𝛾 1 𝜆 𝑦 1 + 𝛽 0 + 𝛽 1 𝜆 𝑦 0 = 0 , 𝛾 0 𝛽 1 − 𝛾 1 𝛽 0 ≠ 0 , 𝛾 1 ≠ 𝑎 0 − 1 𝛽 0 , ( 1 . 6 ) where { 𝑎 𝑛 } 𝑛 ∈ ℕ , { 𝑏 𝑛 } 𝑛 ∈ ℕ are complex sequences and 𝑎 𝑛 ≠ 0 for all 𝑛 ∈ ℕ ∪ { 0 } and 𝛾 𝑖 , 𝛽 𝑖 ∈ ℂ for 𝑖 = 0 , 1 .

In this paper, which is extension of [9], we aim to investigate the properties of the principal functions corresponding to the eigenvalues and spectral singularities of the BVP (1.2)-(1.3).

2. Discrete Spectrum of (1.2)-(1.3)

Let s u p 𝑛 ∈ ℕ e x p 𝜀 𝑛 𝛿 | | 1 − 𝑎 𝑛 | | + | | 𝑏 𝑛 | | < ∞ ( 2 . 1 ) for some 𝜀 > 0 and 1 / 2 ≤ 𝛿 ≤ 1 . The following result is obtained in [10, 11]: under the condition (2.1), equation (1.2) has the solution 𝑒 𝑛 ( 𝑧 ) = 𝛼 𝑛 𝑒 𝑖 𝑛 𝑧 1 + ∞ 𝑚 = 1 𝐴 𝑛 𝑚 𝑒 𝑖 𝑚 𝑧 , 𝑛 ∈ ℕ ∪ { 0 } ( 2 . 2 ) for 𝜆 = 2 c o s 𝑧 , where 𝑧 ∈ ℂ + ∶ = { 𝑧 ∶ 𝑧 ∈ ℂ , I m 𝑧 ≥ 0 } and 𝛼 𝑛 , 𝐴 𝑛 𝑚 are expressed in terms of ( 𝑎 𝑛 ) and ( 𝑏 𝑛 ) as 𝛼 𝑛 = ∞ 𝑘 = 𝑛 𝑎 𝑘 − 1 , 𝐴 𝑛 , 1 = − ∞ 𝑘 = 𝑛 + 1 𝑏 𝑘 , 𝐴 𝑛 , 2 = − ∞ 𝑘 = 𝑛 + 1 1 − 𝑎 2 𝑘 + ∞ 𝑘 = 𝑛 + 1 𝑏 𝑘 ∞ 𝑝 = 𝑘 + 1 𝑏 𝑝 , 𝐴 𝑛 , 𝑚 + 2 = ∞ 𝑘 = 𝑛 + 1 1 − 𝑎 2 𝑘 𝐴 ∞ 𝑘 + 1 , 𝑚 𝑘 = 𝑛 + 1 𝑏 𝑘 𝐴 𝑘 , 𝑚 + 1 + 𝐴 𝑛 + 1 , 𝑚 . ( 2 . 3 ) Moreover, 𝐴 𝑛 𝑚 satisfies | | 𝐴 𝑛 𝑚 | | ≤ 𝐶 ∞ 𝑘 = 𝑛 + ⌈ 𝑚 / 2 ⌉ | | 1 − 𝑎 𝑘 | | + | | 𝑏 𝑘 | | , ( 2 . 4 ) where ⌈ 𝑚 / 2 ⌉ is the integer part of 𝑚 / 2 and 𝐶 > 0 is a constant. So 𝑒 ( 𝑧 ) = { 𝑒 𝑛 ( 𝑧 ) } is continuous in I m 𝑧 = 0 and analytic in ℂ + ∶ = { 𝑧 ∶ 𝑧 ∈ ℂ , I m 𝑧 > 0 } with respect to 𝑧 .

Let us define 𝑓 ( 𝑧 ) using (2.2) and the boundary condition (1.3) as 𝑓 𝛾 ( 𝑧 ) = 0 + 2 𝛾 1 𝑒 c o s 𝑧 1 𝛽 ( 𝑧 ) + 0 + 2 𝛽 1 𝑒 c o s 𝑧 0 ( 𝑧 ) . ( 2 . 5 ) The function 𝑓 is analytic in ℂ + , continuous in ℂ + , and 𝑓 ( 𝑧 ) = 𝑓 ( 𝑧 + 2 𝜋 ) .

We denote the set of eigenvalues and spectral singularities of 𝐿 by 𝜎 𝑑 ( 𝐿 ) and 𝜎 𝑠 𝑠 ( 𝐿 ) , respectively. From the definition of the eigenvalues and spectral singularities, we have [12] 𝜎 𝑑 ( 𝐿 ) = 𝜆 ∶ 𝜆 = 2 c o s 𝑧 , 𝑧 ∈ 𝑃 0 , 𝜎 , 𝐹 ( 𝑧 ) = 0 𝑠 𝑠 − 𝜋 ( 𝐿 ) = 𝜆 ∶ 𝜆 = 2 c o s 𝑧 , 𝑧 ∈ 2 , 3 𝜋 2 , 𝐹 ( 𝑧 ) = 0 ⧵ { 0 } . ( 2 . 6 ) From (2.2) and (2.5), we get 𝛾 𝑓 ( 𝑧 ) = 0 + 𝛾 1 𝑒 𝑖 𝑧 + 𝑒 − 𝑖 𝑧 𝛼 1 𝑒 𝑖 𝑧 1 + ∞ 𝑚 = 1 𝐴 1 𝑚 𝑒 𝑖 𝑚 𝑧 + 𝛽 0 + 𝛽 1 𝑒 𝑖 𝑧 + 𝑒 − 𝑖 𝑧 𝛼 0 1 + ∞ 𝑚 = 1 𝐴 0 𝑚 𝑒 𝑖 𝑚 𝑧 = 𝛼 0 𝛽 1 𝑒 − 𝑖 𝑧 + 𝛾 1 𝛼 1 + 𝛼 0 𝛽 0 + 𝛾 0 𝛼 1 + 𝛼 0 𝛽 1 𝑒 𝑖 𝑧 + 𝛾 1 𝛼 1 𝑒 𝑖 2 𝑧 + ∞ 𝑚 = 1 𝛼 0 𝛽 1 𝐴 0 𝑚 𝑒 𝑖 ( 𝑚 − 1 ) 𝑧 + ∞ 𝑚 = 1 𝛾 1 𝛼 1 𝐴 1 𝑚 + 𝛼 0 𝛽 0 𝐴 0 𝑚 𝑒 𝑖 𝑚 𝑧 + ∞ 𝑚 = 1 𝛾 0 𝛼 1 𝐴 1 𝑚 + 𝛼 0 𝛽 1 𝐴 0 𝑚 𝑒 𝑖 ( 𝑚 + 1 ) 𝑧 + ∞ 𝑚 = 1 𝛾 1 𝛼 1 𝐴 1 𝑚 𝑒 𝑖 ( 𝑚 + 2 ) 𝑧 . ( 2 . 7 ) Let 𝐹 ( 𝑧 ) = 𝑓 ( 𝑧 ) 𝑒 𝑖 𝑧 = 𝛼 0 𝛽 1 + 𝛾 1 𝛼 1 + 𝛼 0 𝛽 0 𝑒 𝑖 𝑧 + 𝛾 0 𝛼 1 + 𝛼 0 𝛽 1 𝑒 2 𝑖 𝑧 + 𝛾 1 𝛼 1 𝑒 3 𝑖 𝑧 + ∞ 𝑚 = 1 𝛼 0 𝛽 1 𝐴 0 𝑚 𝑒 𝑖 𝑚 𝑧 + ∞ 𝑚 = 1 𝛾 1 𝛼 1 𝐴 1 𝑚 + 𝛼 0 𝛽 0 𝐴 0 𝑚 𝑒 𝑖 ( 𝑚 + 1 ) 𝑧 + ∞ 𝑚 = 1 𝛾 0 𝛼 1 𝐴 1 𝑚 + 𝛼 0 𝛽 1 𝐴 0 𝑚 𝑒 𝑖 ( 𝑚 + 2 ) 𝑧 + ∞ 𝑚 = 1 𝛾 1 𝛼 1 𝐴 1 𝑚 𝑒 𝑖 ( 𝑚 + 3 ) 𝑧 ; ( 2 . 8 ) then the function 𝐹 is analytic in ℂ + , continuous in ℂ + , and 𝐹 ( 𝑧 ) = 𝐹 ( 𝑧 + 2 𝜋 ) . It follows from (2.6) and (2.8) that 𝜎 𝑑 ( 𝐿 ) = 𝜆 ∶ 𝜆 = 2 c o s 𝑧 , 𝑧 ∈ 𝑃 0 , 𝜎 , 𝐹 ( 𝑧 ) = 0 𝑠 𝑠 − 𝜋 ( 𝐿 ) = 𝜆 ∶ 𝜆 = 2 c o s 𝑧 , 𝑧 ∈ 2 , 3 𝜋 2 , 𝐹 ( 𝑧 ) = 0 ⧵ { 0 } . ( 2 . 9 )

Definition 2.1. The multiplicity of a zero of 𝐹 in 𝑃 is called the multiplicity of the corresponding eigenvalue or spectral singularity of the BVP (1.2) and (1.3).

3. Principal Functions

Let 𝜆 1 , 𝜆 2 , … , 𝜆 𝑝 and 𝜆 𝑝 + 1 , 𝜆 𝑝 + 2 , … , 𝜆 𝑞 denote the zeros of 𝐹 in 𝑃 0 ∶ = { 𝑧 ∶ 𝑧 ∈ ℂ , 𝑧 = 𝑥 + 𝑖 𝑦 , − 𝜋 / 2 ≤ 𝑥 ≤ 3 𝜋 / 2 , 𝑦 > 0 } and [ − 𝜋 / 2 , 3 𝜋 / 2 ] with multiplicities 𝑚 1 , 𝑚 2 , … , 𝑚 𝑝 and 𝑚 𝑝 + 1 , 𝑚 𝑝 + 2 , … , 𝑚 𝑞 , respectively.

Definition 3.1. Let 𝜆 = 𝜆 0 be an eigenvalue of 𝐿 . If the vectors 𝑦 ( 0 ) , 𝑦 ( 1 ) , … , 𝑦 ( 𝑠 ) ; 𝑦 ( 𝑘 ) = { 𝑦 𝑛 ( 𝑘 ) } 𝑛 ∈ ℕ 𝑘 = 0 , 1 , … , 𝑠 satisfy the equations 𝑙 𝑦 ( 0 ) 𝑛 − 𝜆 0 𝑦 𝑛 ( 0 ) = 0 , 𝑙 𝑦 ( 𝑘 ) 𝑛 − 𝜆 0 𝑦 𝑛 ( 𝑘 ) − 𝑦 𝑛 ( 𝑘 − 1 ) = 0 , 𝑘 = 1 , 2 , . . . , 𝑠 ; 𝑛 ∈ ℕ , ( 3 . 1 ) then vector 𝑦 ( 0 ) is called the eigenvector corresponding to the eigenvalue 𝜆 = 𝜆 0 of 𝐿 . The vectors 𝑦 ( 1 ) , … , 𝑦 ( 𝑠 ) are called the associated vectors corresponding to 𝜆 = 𝜆 0 . The eigenvector and the associated vectors corresponding to 𝜆 = 𝜆 0 are called the principal vectors of the eigenvalue 𝜆 = 𝜆 0 .

The principal vectors of the spectral singularities of 𝐿 are defined similarly.

We define the vectors 𝑉 𝑛 ( 𝑘 ) 𝜆 𝑗 = 1 𝑑 𝑘 ! 𝑘 𝑑 𝜆 𝑘 𝐸 𝑛 | | | | ( 𝜆 ) 𝜆 = 𝜆 𝑗 , 𝑘 = 0 , 1 , … , 𝑚 𝑗 𝑉 − 1 ; 𝑗 = 1 , 2 , … , 𝑝 , 𝑛 ( 𝑘 ) 𝜆 𝑗 = 1 𝑑 𝑘 ! 𝑘 𝑑 𝜆 𝑘 𝐸 𝑛 | | | | ( 𝜆 ) 𝜆 = 𝜆 𝑗 , 𝑘 = 0 , 1 , … , 𝑚 𝑗 − 1 ; 𝑗 = 𝑝 + 1 , 𝑝 + 2 , … , 𝑞 , ( 3 . 2 ) where 𝜆 = 2 c o s 𝑧 , 𝑧 ∈ 𝑃 0 , and 𝐸 𝑛 𝑒 ( 𝜆 ) ∶ = 𝑛 𝜆 a r c c o s 2 , 𝑛 ∈ ℕ . ( 3 . 3 ) Moreover, if 𝑦 ( 𝜆 ) = { 𝑦 𝑛 ( 𝜆 ) } 𝑛 ∈ ℕ is a solution of (1.2), then ( 𝑑 𝑘 / 𝑑 𝜆 𝑘 ) 𝑦 ( 𝜆 ) = { ( 𝑑 𝑘 / 𝑑 𝜆 𝑘 ) 𝑦 𝑛 ( 𝜆 ) } 𝑛 ∈ ℕ satisfies 𝑎 𝑛 − 1 𝑑 𝑘 𝑑 𝜆 𝑘 𝑦 𝑛 − 1 ( 𝜆 ) + 𝑏 𝑛 𝑑 𝑘 𝑑 𝜆 𝑘 𝑦 𝑛 ( 𝜆 ) + 𝑎 𝑛 𝑑 𝑘 𝑑 𝜆 𝑘 𝑦 𝑛 + 1 𝑑 ( 𝜆 ) = 𝜆 𝑘 𝑑 𝜆 𝑘 𝑦 𝑛 𝑑 ( 𝜆 ) + 𝑘 𝑘 − 1 𝑑 𝜆 𝑘 − 1 𝑦 𝑛 ( 𝜆 ) . ( 3 . 4 ) From (3.2) and (3.4), we get that ℓ 𝑉 ( 0 ) 𝜆 𝑗 𝑛 − 𝜆 𝑗 𝑉 𝑛 ( 0 ) 𝜆 𝑗 = 0 , ℓ 𝑉 ( 𝑘 ) 𝜆 𝑗 𝑛 − 𝜆 𝑗 𝑉 𝑛 ( 𝑘 ) 𝜆 𝑗 − 𝑉 𝑛 ( 𝑘 − 1 ) 𝜆 𝑗 = 0 , 𝑘 = 1 , 2 , … , 𝑚 𝑗 − 1 ; 𝑗 = 1 , 2 , … , 𝑞 . ( 3 . 5 ) Consequently, the vectors 𝑉 𝑛 ( 𝑘 ) ( 𝜆 𝑗 ) ; 𝑘 = 0 , 1 , … , 𝑚 𝑗 − 1 , 𝑗 = 1 , 2 , … , 𝑝 and 𝑉 𝑛 ( 𝑘 ) ( 𝜆 𝑗 ) ; 𝑘 = 0 , 1 , … , 𝑚 𝑗 − 1 , 𝑗 = 𝑝 + 1 , 𝑝 + 2 , … , 𝑞 are the principal vectors of eigenvalues and spectral singularities of 𝐿 , respectively.

Theorem 3.2. 𝑉 𝑛 ( 𝑘 ) 𝜆 𝑗 ∈ ℓ 2 ( ℕ ) ; 𝑘 = 0 , 1 , … , 𝑚 𝑗 𝑉 − 1 , 𝑗 = 1 , 2 , … , 𝑝 , 𝑛 ( 𝑘 ) 𝜆 𝑗 ∉ ℓ 2 ( ℕ ) ; 𝑘 = 0 , 1 , … , 𝑚 𝑗 − 1 , 𝑗 = 𝑝 + 1 , … , 𝑞 . ( 3 . 6 )

Proof. Using 𝐸 𝑛 ( 𝜆 ) = 𝑒 𝑛 ( a r c c o s ( 𝜆 / 2 ) ) , we obtain that 𝑑 𝑘 𝑑 𝜆 𝑘 𝐸 𝑛 | | | | ( 𝜆 ) 𝜆 = 𝜆 𝑗 = 𝑘 𝜈 = 0 𝐶 𝜈 𝑑 𝜈 𝑑 𝜆 𝜈 𝑒 𝑛 ( 𝑧 ) 𝑧 = 𝑧 𝑗 , 𝑛 ∈ ℕ , ( 3 . 7 ) where 𝜆 𝑗 = 2 c o s 𝑧 𝑗 ; 𝑧 𝑗 ∈ 𝑃 = 𝑃 0 ∪ [ − 𝜋 / 2 , 3 𝜋 / 2 ] , 𝑗 = 1 , 2 , … , 𝑞 ; 𝐶 𝜈 is a constant depending on 𝜆 𝑗 .
From (2.2), we find that 𝑑 𝜈 𝑑 𝑧 𝜈 𝑒 𝑛 ( 𝑧 ) 𝑧 = 𝑧 𝑗 = 𝛼 𝑛 𝑒 𝑖 𝑛 𝑧 𝑗 ( 𝑖 𝑛 ) 𝜈 + ∞ 𝑚 = 1 [ ] 𝑖 ( 𝑛 + 𝑚 ) 𝜈 𝐴 𝑛 𝑚 𝑒 𝑖 𝑚 𝑧 𝑗 = 𝛼 𝑛 𝑒 𝑖 𝑛 𝑧 𝑗 ( 𝑖 𝑛 ) 𝜈 + 𝛼 𝑛 𝑒 𝑖 𝑛 𝑧 𝑗 ∞ 𝑚 = 1 [ ] 𝑖 ( 𝑛 + 𝑚 ) 𝜈 𝐴 𝑛 𝑚 𝑒 𝑖 𝑚 𝑧 𝑗 . ( 3 . 8 ) For the principal vectors 𝑉 𝑛 ( 𝑘 ) ( 𝜆 𝑗 ) = { 𝑉 ( 𝑘 ) ( 𝜆 𝑗 ) } 𝑛 ∈ ℕ , 𝑘 = 0 , 1 , … , 𝑚 𝑗 − 1 , 𝑗 = 1 , 2 , … , 𝑝 , corresponding to the eigenvalues 𝜆 𝑗 = 2 c o s 𝑧 𝑗 , 𝑗 = 1 , 2 , … , 𝑝 , of 𝐿 , we get 𝑑 𝑘 𝑑 𝜆 𝑘 𝐸 𝑛 | | | | ( 𝜆 ) 𝜆 = 𝜆 𝑗 = 𝑘 𝜈 = 0 𝐶 𝜈 𝛼 𝑛 𝑒 𝑖 𝑛 𝑧 𝑗 ( 𝑖 𝑛 ) 𝜈 + 𝛼 𝑛 𝑒 𝑖 𝑛 𝑧 𝑗 ∞ 𝑚 = 1 [ ] 𝑖 ( 𝑛 + 𝑚 ) 𝜈 𝐴 𝑛 𝑚 𝑒 𝑖 𝑚 𝑧 𝑗 ; ( 3 . 9 ) then 𝑉 𝑛 ( 𝑘 ) 𝜆 𝑗 = 1 𝑘 ! 𝑘 𝜈 = 0 𝐶 𝜈 𝛼 𝑛 𝑒 𝑖 𝑛 𝑧 𝑗 ( 𝑖 𝑛 ) 𝜈 + 𝛼 𝑛 𝑒 𝑖 𝑛 𝑧 𝑗 ∞ 𝑚 = 1 [ ] 𝑖 ( 𝑛 + 𝑚 ) 𝜈 𝐴 𝑛 𝑚 𝑒 𝑖 𝑚 𝑧 𝑗 ( 3 . 1 0 ) for 𝑘 = 0 , 1 , … , 𝑚 𝑗 − 1 , 𝑗 = 1 , 2 , … , 𝑝 .
Since I m 𝜆 𝑗 > 0 , 𝑗 = 1 , 2 , … , 𝑝 from (3.10) we obtain that ∞ 𝑛 = 1 | | | | | 1 𝑘 ! 𝑘 𝜈 = 0 𝐶 𝜈 𝛼 𝑛 𝑒 𝑖 𝑛 𝑧 𝑗 ( 𝑖 𝑛 ) 𝜈 | | | | | 2 ≤ 1 ( 𝑘 ! ) 2 ∞ 𝑘 𝑛 = 1 𝜈 = 0 | | 𝐶 𝜈 | | | | 𝛼 𝑛 | | 𝑒 − 𝑛 I m 𝑧 𝑗 | | 𝑛 𝜈 | | 2 ≤ 𝐴 ( 𝑘 ! ) 2 ∞ 𝑛 = 1 𝑒 − 𝑛 I m 𝑧 𝑗 1 + 𝑛 + 𝑛 2 + ⋯ + 𝑛 𝑘 2 ≤ 𝐴 ( 𝑘 ! ) 2 ( 𝑘 + 1 ) 2 ∞ 𝑛 = 1 𝑒 − 𝑛 I m 𝑧 𝑗 𝑛 𝑘 2 < ∞ , ( 3 . 1 1 ) where 𝐴 is a constant. Now we define the function 𝑔 𝑛 1 ( 𝑧 ) = 𝑘 ! 𝑘 𝜈 = 0 𝛼 𝑛 𝑒 𝑖 𝑛 𝑧 𝑗 ∞ 𝑚 = 1 [ ] 𝑖 ( 𝑛 + 𝑚 ) 𝜈 𝐴 𝑛 𝑚 𝑒 𝑖 𝑚 𝑧 𝑗 , 𝑗 = 1 , 2 , … , 𝑝 . ( 3 . 1 2 ) From (2.4), we obtain that | | 𝑔 𝑛 | | ≤ ( 𝑧 ) 𝑘 𝜈 = 0 | | 𝛼 𝑛 | | 𝑒 − 𝑛 I m 𝑧 𝑗 ∞ 𝑚 = 1 | 𝑛 + 𝑚 | 𝜈 | | 𝐴 𝑛 𝑚 | | 𝑒 − 𝑚 I m 𝑧 𝑗 ≤ | | 𝛼 𝑛 | | 𝑒 − 𝑛 I m 𝑧 𝑗 ∞ 𝑚 = 1 | | 𝐴 𝑛 𝑚 | | 𝑒 − 𝑚 I m 𝑧 𝑗 + ∞ 𝑚 = 1 | | 𝐴 ( 𝑛 + 𝑚 ) 𝑛 𝑚 | | 𝑒 − 𝑚 I m 𝑧 𝑗 + ⋯ + ∞ 𝑚 = 1 ( 𝑛 + 𝑚 ) 𝑘 | | 𝐴 𝑛 𝑚 | | 𝑒 − 𝑚 I m 𝑧 𝑗 < 𝐵 𝑒 − 𝑛 I m 𝑧 𝑗 , ( 3 . 1 3 ) where 𝐵 = | 𝛼 𝑛 | ∑ ∞ 𝑚 = 1 ∑ 𝑘 𝜈 = 0 | 𝐴 𝑛 𝑚 | 𝑒 − 𝑚 I m 𝑧 𝑗 ( 𝑛 + 𝑚 ) 𝜈 . Therefore, we have ∞ 𝑛 = 1 | | 𝑔 𝑛 ( | | 𝑧 ) 2 ≤ ∞ 𝑛 = 1 𝐵 2 𝑒 − 2 𝑛 I m 𝑧 𝑗 , 𝑗 = 1 , 2 , … , 𝑝 < ∞ . ( 3 . 1 4 ) It follows from (3.11) and (3.14) that 𝑉 𝑛 ( 𝑘 ) ( 𝜆 𝑗 ) ∈ ℓ 2 ( ℕ ) , 𝑘 = 0 , 1 , … , 𝑚 𝑗 − 1 , 𝑗 = 1 , 2 , … , 𝑝 .
If we consider (3.10) for the principal vectors corresponding to the spectral singularities 𝜆 𝑗 = 2 c o s 𝑧 𝑗 , 𝑗 = 𝑝 + 1 , 𝑝 + 2 , … , 𝑞 , of 𝐿 and consider that I m 𝑧 𝑗 = 0 for the spectral singularities, then we have 𝑉 𝑛 ( 𝑘 ) 𝜆 𝑗 = 1 𝑘 ! 𝑘 𝜈 = 0 𝐶 𝜈 𝛼 𝑛 𝑒 𝑖 𝑛 𝑧 𝑗 ( 𝑖 𝑛 ) 𝜈 + 𝛼 𝑛 𝑒 𝑖 𝑛 𝑧 𝑗 𝑘 ∞ 𝜈 = 0 𝑚 = 1 [ ] 𝑖 ( 𝑛 + 𝑚 ) 𝜈 𝐴 𝑛 𝑚 𝑒 𝑖 𝑚 𝑧 𝑗 ( 3 . 1 5 ) for 𝑘 = 0 , 1 , … , 𝑚 𝑗 − 1 , 𝑗 = 𝑝 + 1 , 𝑝 + 2 , … , 𝑞 .
Since I m 𝜆 𝑗 = 0 , 𝑗 = 𝑝 + 1 , … , 𝑞 from (3.15) we find that 1 𝑘 ! ∞ 𝑛 = 1 | | | | | 𝑘 𝜈 = 0 𝐶 𝜈 𝛼 𝑛 𝑒 𝑖 𝑛 𝑧 𝑗 ( 𝑖 𝑛 ) 𝜈 | | | | | 2 = ∞ . ( 3 . 1 6 ) Now we define 𝑡 𝑛 ∑ ( 𝑧 ) = 𝑘 𝜈 = 0 ∑ ∞ 𝑚 = 1 [ 𝑖 ( 𝑛 + 𝑚 ) ] 𝜈 𝐴 𝑛 𝑚 𝑒 𝑖 𝑚 𝑧 𝑗 , and using (2.4) we get | | 𝑡 𝑛 | | ≤ ( 𝑧 ) 𝑘 ∞ 𝜈 = 0 𝑚 = 1 | | ( 𝑛 + 𝑚 ) 𝜈 | | | | 𝐴 𝑛 𝑚 | | ≤ 𝑘 ∞ 𝜈 = 0 𝑚 = 1 ( 𝑛 + 𝑚 ) 𝜈 𝐶 ∞ 𝑘 = 𝑛 + ⌈ 𝑚 / 2 ⌉ | | 1 − 𝑎 𝑘 | | + | | 𝑏 𝑘 | | ≤ 𝐶 𝑘 ∞ 𝜈 = 0 𝑚 = 1 ( 𝑛 + 𝑚 ) 𝜈 ∞ 𝑘 = 𝑛 + ⌈ 𝑚 / 2 ⌉ | | e x p ( − 𝜀 𝑘 ) e x p ( 𝜀 𝑘 ) 1 − 𝑎 𝑘 | | + | | 𝑏 𝑘 | | ≤ 𝐶 𝑘 ∞ 𝜈 = 0 𝑚 = 1 ( 𝑛 + 𝑚 ) 𝜈 e x p − 𝜀 4 ( 𝑛 + 𝑚 ) ∞ 𝑘 = 𝑛 + ⌈ 𝑚 / 2 ⌉ | | e x p ( 𝜀 𝑘 ) 1 − 𝑎 𝑘 | | + | | 𝑏 𝑘 | | ≤ 𝐶 1 𝑘 ∞ 𝜈 = 0 𝑚 = 1 ( 𝑛 + 𝑚 ) 𝜈 e x p − 𝜀 4 ( 𝑛 + 𝑚 ) = 𝐶 1 𝑒 ∞ ( − 𝜀 / 4 ) 𝑛 𝑘 𝑚 = 1 𝜈 = 0 ( 𝑛 + 𝑚 ) 𝜈 e x p − 𝜀 4 𝑚 = 𝐴 𝑒 ( − 𝜀 / 4 ) 𝑛 , ( 3 . 1 7 ) where 𝐴 = 𝐶 1 ∞ 𝑘 𝑚 = 1 𝜈 = 0 ( 𝑛 + 𝑚 ) 𝜈 e x p − 𝜀 4 𝑚 . ( 3 . 1 8 )
If we use (3.17), we obtain that 1 𝑘 ! ∞ 𝑛 = 1 | | | | | 𝛼 𝑛 𝑒 𝑖 𝑛 𝑧 𝑗 𝑘 ∞ 𝜈 = 0 𝑚 = 1 [ ] 𝑖 ( 𝑛 + 𝑚 ) 𝜈 𝐴 𝑛 𝑚 𝑒 𝑖 𝑚 𝑧 𝑗 | | | | | 2 ≤ 1 𝑘 ! ∞ 𝑛 = 1 𝛼 2 𝑛 𝐴 2 𝑒 − 𝜀 𝑛 / 2 < ∞ . ( 3 . 1 9 ) So 𝑉 𝑛 ( 𝑘 ) ∉ ℓ 2 ( ℕ ) , 𝑘 = 0 , 1 , … , 𝑚 𝑗 − 1 , 𝑗 = 𝑝 + 1 , 𝑝 + 2 , … 𝑞 .

Let us introduce Hilbert spaces 𝐻 𝑘 𝑦 ( ℕ ) = 𝑦 = 𝑛 𝑛 ∈ ℕ ∶ 𝑛 ∈ ℕ ( 1 + | 𝑛 | ) 2 𝑘 | | 𝑦 𝑛 | | 2 , 𝐻 < ∞ − 𝑘 𝑢 ( ℕ ) = 𝑢 = 𝑛 𝑛 ∈ ℕ ∶ 𝑛 ∈ ℕ ( 1 + | 𝑛 | ) − 2 𝑘 | | 𝑢 𝑛 | | 2 < ∞ , 𝑘 = 0 , 1 , 2 , … , ( 3 . 2 0 ) with ‖ 𝑦 ‖ 2 𝑘 = ∑ 𝑛 ∈ ℕ ( 1 + | 𝑛 | ) 2 𝑘 | 𝑦 𝑛 | 2 , ‖ 𝑢 ‖ 2 − 𝑘 = ∑ 𝑛 ∈ ℕ ( 1 + | 𝑛 | ) − 2 𝑘 | 𝑢 𝑛 | 2 , respectively. It is obvious that 𝐻 0 ( ℕ ) = ℓ 2 ( ℕ ) and 𝐻 𝑘 + 1 ( ℕ ) ⫋ 𝐻 𝑘 ( ℕ ) ⫋ 𝑙 2 ( ℕ ) ⫋ 𝐻 − 𝑘 ( ℕ ) ⫋ 𝐻 − ( 𝑘 + 1 ) ( ℕ ) , 𝑘 = 1 , 2 , … ( 3 . 2 1 )

Theorem 3.3. 𝑉 𝑛 ( 𝑘 ) ( 𝜆 𝑗 ) ∈ 𝐻 − ( 𝑘 + 1 ) ( ℕ ) , 𝑘 = 0 , 1 , … , 𝑚 𝑗 − 1 , 𝑗 = 𝑝 + 1 , … , 𝑞 .

Proof. From (3.15), we have ∞ 𝑛 = 1 ( 1 + | 𝑛 | ) − 2 ( 𝑘 + 1 ) | | | | | 1 𝑘 ! 𝑘 𝜈 = 0 𝐶 𝜈 𝛼 𝑛 𝑒 𝑖 𝑛 𝑧 𝑗 ( 𝑖 𝑛 ) 𝜈 | | | | | 2 < ∞ , ∞ 𝑛 = 1 ( 1 + | 𝑛 | ) − 2 ( 𝑘 + 1 ) | | | | | 1 𝑘 ! 𝑘 𝜈 = 0 𝛼 𝑛 𝑒 𝑖 𝑛 𝑧 𝑗 ∞ 𝑚 = 1 [ ] 𝑖 ( 𝑛 + 𝑚 ) 𝜈 𝐴 𝑛 𝑚 𝑒 𝑖 𝑚 𝑧 𝑗 | | | | | 2 < ∞ ( 3 . 2 2 ) for 𝑘 = 0 , 1 , … , 𝑚 𝑗 − 1 , 𝑗 = 𝑝 + 1 , 𝑝 + 2 , … , 𝑞 . Therefore, we obtain that 𝑉 𝑛 ( 𝑘 ) ( 𝜆 𝑗 ) ∈ 𝐻 − ( 𝑘 + 1 ) ( ℕ ) , 𝑘 = 0 , 1 , … , 𝑚 𝑗 − 1 , 𝑗 = 𝑝 + 1 , 𝑝 + 2 , … , 𝑞 .

Let us choose 𝑚 0 = m a x { 𝑚 𝑝 + 1 , 𝑚 𝑝 + 2 , … , 𝑚 𝑞 } . By Theorem 3.2 and (3.21), we get the following.

Theorem 3.4. 𝑉 𝑛 ( 𝑘 ) ( 𝜆 𝑗 ) ∈ 𝐻 − 𝑚 0 ( ℕ ) , 𝑘 = 0 , 1 , … , 𝑚 𝑗 − 1 , 𝑗 = 𝑝 + 1 , 𝑝 + 2 , … , 𝑞 .

Proof. The proof of theorem is trivial.