Principal Functions of Nonselfadjoint Discrete Dirac Equations with Spectral Parameter in Boundary Conditions

Abstract

Let 𝐿 denote the operator generated in ℓ₂(ℕ,ℂ²) by 𝑎_𝑛+1𝑦⁽²⁾_𝑛+1+𝑏_𝑛𝑦_𝑛⁽²⁾+𝑝_𝑛𝑦_𝑛⁽¹⁾=𝜆𝑦_𝑛⁽¹⁾, 𝑎_𝑛−1𝑦⁽¹⁾_𝑛−1+𝑏_𝑛𝑦_𝑛⁽¹⁾+𝑞_𝑛𝑦_𝑛⁽²⁾=𝜆𝑦_𝑛⁽²⁾, 𝑛∈ℕ, and the boundary condition (𝛾₀+𝛾₁𝜆)𝑦₁⁽²⁾+(𝛽₀+𝛽₁𝜆)𝑦₀⁽¹⁾=𝟎, where (𝑎_𝑛), (𝑏_𝑛), (𝑝_𝑛), and (𝑞_𝑛), 𝑛∈ℕ are complex sequences, 𝛾_𝑖,𝛽_𝑖∈ℂ, 𝑖=0,1, and 𝜆 is an eigenparameter. In this paper we investigated the principal functions corresponding to the eigenvalues and the spectral singularities of 𝐿.

1. Introduction

Consider the boundary value problem (BVP) −𝑦^′′+𝑞(𝑥)𝑦=𝜆²𝑦(0)𝑦,0≤𝑥<∞,=0, (1.1) in 𝐿²(ℝ₊), where 𝑞 is a complex-valued function and 𝜆∈ℂ is a spectral parameter. The spectral theory of the above BVP with continuous and point spectrum was investigated by Naĭmark [1]. He showed the existence of the spectral singularities in the continuous spectrum of (1.1). Note that the eigen and associated functions corresponding to the spectral singularities are not the elements of 𝐿²(ℝ₊).

In [2, 3] the effect of the spectral singularities in the spectral expansion in terms of the principal vectors was considered. Some problems related to the spectral analysis of difference equations with spectral singularities were discussed in [4–7]. The spectral analysis of eigenparameter dependent nonselfadjoint difference equation was studied in [8, 9].

Let us consider the nonselfadjoint BVP for the discrete Dirac equations 𝑎_𝑛+1𝑦⁽²⁾_𝑛+1+𝑏_𝑛𝑦_𝑛⁽²⁾+𝑝_𝑛𝑦_𝑛⁽¹⁾=𝜆𝑦_𝑛⁽¹⁾𝑎,𝑛∈ℕ,_𝑛−1𝑦⁽¹⁾_𝑛−1+𝑏_𝑛𝑦_𝑛⁽¹⁾+𝑞_𝑛𝑦_𝑛⁽²⁾=𝜆𝑦_𝑛⁽²⁾𝛾,𝑛∈ℕ,₀+𝛾₁𝜆𝑦₁⁽²⁾+𝛽₀+𝛽₁𝜆𝑦₀⁽¹⁾=0,𝛾₀𝛽₁−𝛾₁𝛽₀≠0,𝛾₁≠𝑎₀⁻¹𝛽₀, (1.2)(1.3) where ^𝑦𝑛(1)_𝑦𝑛(2),𝑛∈ℕ are vector sequences, 𝑎_𝑛≠0, 𝑏_𝑛≠0 for all 𝑛∈ℕ,  𝛾_𝑖,𝛽_𝑖∈ℂ, and 𝑖=0,1 and, 𝜆 is a spectral parameter.

In [10] the authors proved that eigenvalues and spectral singularities of (1.2)-(1.3) have a finite number with finite multiplicities, if the condition, ^∞_𝑛=1exp𝜀𝑛^𝛿1−𝑎_𝑛+1+𝑏_𝑛+𝑝_𝑛+𝑞_𝑛<∞ (1.4) holds, for some 𝜀>0 and 1/2≤𝛿<1.

In this paper, we aim to investigate the principal functions corresponding to the eigenvalues and the spectral singularities of the BVP (1.2)-(1.3).

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

Let for some 𝜀>0 and 1/2≤𝛿<1, ^∞_𝑛=1exp𝜀𝑛^𝛿1−𝑎_𝑛+1+𝑏_𝑛+𝑝_𝑛+𝑞_𝑛<∞, (2.1) be satisfied. It has been shown that [10] under the condition (2.1), (1.2) has the solution 𝑓_𝑛(𝑧)=𝑓_𝑛⁽¹⁾(𝑧)𝑓_𝑛⁽²⁾(𝑧)=𝛼_𝑛𝐼+^∞_𝑚=1𝐴_𝑛𝑚𝑒^𝑖𝑚𝑧𝑒^𝑖𝑧/2𝑒−𝑖^𝑖𝑛𝑧𝑓,𝑛=1,2,…,₀₍₁₎(𝑧)=𝛼₀¹¹𝑒^𝑖𝑧/21+^∞_𝑚=1𝐴¹¹_0𝑚𝑒^𝑖𝑚𝑧−𝑖^∞_𝑚=1𝐴¹²_0𝑚𝑒^𝑖𝑚𝑧, (2.2)(2.3) for 𝜆=2sin(𝑧/2) and 𝑧∈ℂ₊∶={𝑧∈ℂ∶Im𝑧≥0}, where 𝛼_𝑛=𝛼_𝑛¹¹𝛼_𝑛¹²𝛼_𝑛²¹𝛼_𝑛²²,𝐼=1001,𝐴_𝑛𝑚=𝐴¹¹_𝑛𝑚𝐴¹²_𝑛𝑚𝐴²¹_𝑛𝑚𝐴²²_𝑛𝑚. (2.4) Note that 𝛼_𝑛^𝑖𝑗 and 𝐴^𝑖𝑗_𝑛𝑚 (𝑖,𝑗=1,2) are uniquely expressed in terms of (𝑎_𝑛), (𝑏_𝑛), (𝑝_𝑛), and (𝑞_𝑛), 𝑛∈ℕ as follows 𝛼_𝑛¹¹=^∞_𝑘=𝑛+1(−1)^𝑛−𝑘𝑏_𝑘𝑎_𝑘−1⁻¹,𝛼_𝑛¹²𝛼=0,_𝑛22=𝑏_𝑛^∞_𝑘=𝑛+1(−1)^{𝑛−𝑘+1}𝑏_𝑘𝑎_𝑘−1⁻¹,𝛼_𝑛²¹=𝛼_𝑛²²𝑝_𝑛+^∞_𝑘=𝑛+1𝑝_𝑘+𝑞_𝑘,𝐴¹²_𝑛1=−^∞_𝑘=𝑛+1𝑝_𝑘+𝑞_𝑘,𝐴¹¹_𝑛1=^∞_𝑘=𝑛+1𝑎_𝑘+1𝑎_𝑘+𝑏²_𝑘−𝑝_𝑘𝑞_𝑘+𝑝_𝑘+𝑞_𝑘𝐴¹²_𝑘1,𝐴−2_22𝑛1=−1+𝑎_𝑛+1𝑎_𝑛+𝐴¹²_𝑛1²+𝐴¹¹_𝑛1,𝐴²¹_𝑛1=−^∞_𝑘=𝑛𝑞_𝑘+1+𝐴¹²_𝑘1𝑎_𝑘+1𝑎_𝑘+𝑞_𝑘+1𝑝_𝑘+1+𝑞_𝑘+1+𝑞_𝑘+1𝐴¹²_𝑘1+𝑏²_𝑘+1+𝐴¹¹_𝑘+1,1−1−𝐴¹²_𝑘11+𝐴¹¹_𝑘1+^∞_𝑘=𝑛+1𝑞_𝑘𝐴²²_𝑘1−𝑏²_𝑘𝑝_𝑘,𝐴¹²_𝑛2=−𝑎_𝑛+1𝑎_𝑛𝑞_𝑛+1+𝐴¹²_𝑛1+𝐴¹²_𝑛1𝐴¹¹_𝑛1+𝐴¹²_𝑛1−𝐴²¹_𝑛1,𝐴¹¹_𝑛2=^∞_𝑘=𝑛+1𝑏²_𝑘−1𝐴¹¹_𝑘1−𝑎_𝑘+1𝑎_𝑘𝑞_𝑘+1+𝐴¹²_𝑘1𝐴¹²_𝑘+1,1−𝐴²²_𝑘+1,1−𝑝_𝑘−𝐴¹²_𝑘1𝑞_𝑘𝐴¹¹_𝑘1+𝐴¹²_𝑘1−𝐴¹²_𝑘2−𝑞_𝑘𝐴²¹_𝑘1+𝐴¹²_𝑘1𝐴¹²_𝑘2−𝐴²²_𝑘1,𝐴²²_𝑛2=−𝑎_𝑛+1𝑎_𝑛𝑞_𝑛+1+𝐴¹²_𝑛1𝐴¹²_𝑛+1,1+𝑎_𝑛+1𝑎_𝑛𝐴²²_𝑛+1,1+𝐴¹²_𝑛1𝐴¹²_𝑛2−𝐴¹¹_𝑛1+𝐴¹¹_𝑛2,𝐴²¹_𝑛2=^∞_𝑘=𝑛𝐴¹²_𝑘1𝐴¹¹_𝑘2+A²¹_𝑘2−𝑎_𝑘+1𝑎_𝑘𝑞_𝑘+1+𝐴¹²_𝑘1𝐴¹¹_𝑘+1,1−𝐴²¹_𝑘+1,1−^∞_𝑘=𝑛+1𝑞_𝑘+𝐴¹²_𝑘−1,1𝑞_𝑘𝐴¹²_𝑘2−𝐴¹¹_𝑘1+𝐴¹¹_𝑘2+𝑏²_𝑘𝐴²¹_𝑘2−𝑝_𝑘𝐴²²_𝑘2+𝐴²¹_𝑘1, (2.5) and for 𝑚≥3𝐴¹²_𝑛𝑚=−𝑎_𝑛+1𝑎_𝑛𝑞_𝑛+1+𝐴¹²_𝑛1𝐴¹¹_{𝑛+1,𝑚−2}+𝐴²¹_{𝑛+1,𝑚−2}+𝐴¹²_𝑛1𝐴¹¹_{𝑛,𝑚−1}+𝐴¹²_{𝑛,𝑚−1}−𝐴²¹_{𝑛,𝑚−1},𝐴¹¹_𝑛𝑚=−^∞_𝑘=𝑛+1𝑎_𝑘+1𝑎_𝑘𝑞_𝑘+1+𝐴¹²_𝑘1𝐴¹²_{𝑘+1,𝑚−1}−𝐴²²_{𝑘+1,𝑚−1}−^∞_𝑘=𝑛+1𝑝_𝑘−𝐴¹²_𝑘1𝑞_𝑘𝐴¹¹_{𝑘,𝑚−1}+𝐴¹²_{𝑘,𝑚−1}−𝐴¹²_𝑘𝑚+^∞_𝑘=𝑛+1𝑏²_𝑘−1𝐴¹¹_{𝑘,𝑚−1}−^∞_𝑘=𝑛+1𝑞_𝑘𝐴²¹_{𝑘,𝑚−1}+^∞_𝑘=𝑛+1𝐴¹²_𝑘1𝐴¹²_𝑘𝑚−^∞_𝑘=𝑛+1𝐴²²_{𝑘,𝑚−1},𝐴²²_𝑛𝑚=−𝑎_𝑛+1𝑎_𝑛𝑞_𝑛+1+𝐴¹²_𝑛1𝐴¹¹_{𝑛+1,𝑚−1}−𝐴²²_{𝑛+1,𝑚−1}+𝐴¹²_𝑛1𝐴¹²_𝑛𝑚+𝐴¹¹_𝑛𝑚−𝐴¹¹_{𝑛,𝑚−1},𝐴²¹_𝑛𝑚=−^∞_𝑘=𝑛𝑎_𝑘+1𝑎_𝑘𝑞_𝑘+1+𝐴¹²_𝑘1𝐴¹¹_{𝑘+1,𝑚−1}−𝐴²¹_{𝑘+1,𝑚−1}−^∞_𝑘=𝑛+1𝑞_𝑘−𝐴¹²_𝑘−1,1𝑞_𝑘𝐴²¹_𝑘𝑚+𝐴¹¹_{𝑘,𝑚−1}−𝐴²²_𝑘𝑚−^∞_𝑘=𝑛+1𝑏²_𝑘−1𝐴¹²_𝑘𝑚+^∞_𝑘=𝑛𝐴¹²_𝑘1𝐴²²_𝑘𝑚+^∞_𝑘=𝑛+1𝑞_𝑘𝐴²²_𝑘𝑚+^∞_𝑘=𝑛𝐴¹²_𝑘𝑚−^∞_𝑘=𝑛+1𝐴²¹_{𝑘,𝑚−1}. (2.6) Moreover 𝐴^𝑖𝑗_𝑛𝑚≤𝐶^∞_{𝑘=𝑛+[|𝑚/2|]}1−𝑎_𝑘+1+𝑏_𝑘+𝑝_𝑘+𝑞_𝑘 (2.7) holds, where [|𝑚/2|] is the integer part of 𝑚/2 and 𝐶>0 is a constant. Therefore 𝑓_𝑛 is vector-valued analytic function with respect to 𝑧 in ℂ₊∶={𝑧∈ℂ∶Im𝑧>0} and continuous in ℂ₊ [10]. The solution 𝑓(𝑧)=(𝑓_𝑛(𝑧))=^𝑓𝑛(1)_𝑓^(𝑧)_{𝑛(2)(𝑧)} is called Jost solution of (1.2).

Let us define 𝐹(𝑧)=𝛾₀+2𝛾₁𝑧sin2𝑓₁⁽²⁾(𝑧)+𝛽₀+2𝛽₁𝑧sin2𝑓₀⁽¹⁾(𝑧). (2.8) It follows (2.2) and (2.3) that the function 𝐹 is analytic in ℂ₊, continuous up to the real axis, and 𝐹(𝑧+4𝜋)=𝐹(𝑧). (2.9) We denote the set of eigenvalues and spectral singularities of 𝐿 by 𝜎_𝑑(𝐿) and 𝜎_ss(𝐿), respectively. From the definition of the eigenvalues and spectral singularities we have [10] 𝜎_𝑑=𝑧𝜆∶𝜆=2sin2,𝑧∈𝑃₀(𝑧),𝜎,𝐹=0_ss=𝑧𝜆∶𝜆=2sin2,𝑧∈[0,4𝜋],𝐹(𝑧)=0, (2.10) where 𝑃₀∶={𝑧∶𝑧∈ℂ,𝑧=𝑥+𝑖𝑦,0≤𝑥≤4𝜋,𝑦>0}. The finiteness of the multiplicities of eigenvalues and spectral singularities has been proven in [10]. Using (2.2), (2.3), and (2.8) we obtain 𝐹(𝑧)=𝛾₀+𝛾₁(−𝑖)𝑒^𝑖(𝑧/2)−𝑒^{−𝑖(𝑧/2)}𝑓₁⁽²⁾(𝑧)+𝛽₀+𝛽₁(−𝑖)𝑒^𝑖(𝑧/2)−𝑒^{−𝑖(𝑧/2)}𝑓₀⁽¹⁾(𝑧)=𝑖𝛼₀¹¹𝛽₁+𝛾₁𝛼₁²²+𝛼₀¹¹𝛽₀𝑒^𝑖(𝑧/2)+𝑖−𝛾₀𝛼₁²²+𝛾₁𝛼₁²²−𝛼₀¹¹𝛽₁𝑒^𝑖𝑧+𝛾₀𝛼₁²¹−𝛾₁𝛼₁²²𝑒^{𝑖(3𝑧/2)}−𝑖𝛾₁𝛼₁²¹𝑒^2𝑖𝑧+^∞_𝑚=1𝛼₀¹¹𝛽₁𝐴¹²_0𝑚𝑒^{𝑖(𝑚−(1/2))𝑧}+𝑖^∞_𝑚=1−𝛼₀¹¹𝛽₀𝐴¹²_0𝑚+𝛼₀¹¹𝛽₁𝐴¹¹_0𝑚𝑒^𝑖𝑚𝑧+^∞_𝑚=1𝛾₁𝛼₁²¹𝐴¹²_1𝑚+𝛾₁𝛼₁²²𝐴²²_1𝑚+𝛼₀¹¹𝛽₀𝐴¹¹_0𝑚−𝛼₀¹¹𝛽₁𝐴¹²_0𝑚𝑒^{𝑖(𝑚+(1/2))𝑧}+𝑖^∞_𝑚=1−𝛾₀𝛼₁²¹𝐴¹²_1𝑚−𝛾₀𝛼₁²²𝐴²²_1𝑚+𝛾₁𝛼₁²¹𝐴¹¹_1𝑚+𝛾₁𝛼₁²²𝐴²¹_1𝑚−𝛼₀¹¹𝛽₁𝐴¹¹_0𝑚𝑒^{𝑖(𝑚+1)𝑧}+^∞_𝑚=1𝛾₀𝛼₁²¹𝐴¹¹_1𝑚+𝛾₀𝛼₁²²𝐴²¹_1𝑚−𝛾₁𝛼₁²¹𝐴¹²_1𝑚−𝛾₁𝛼₁²²𝐴²²_1𝑚𝑒^{𝑖(𝑚+(3/2))𝑧}+𝑖^∞_𝑚=1−𝛾₁𝛼₁²¹𝐴¹¹_1𝑚−𝛾₁𝛼₁²²𝐴²¹_1𝑚𝑒^{𝑖(𝑚+2)𝑧}.+𝑖^∞_𝑚=1−𝛾₁𝛼₁²¹𝐴¹¹_1𝑚−𝛾₁𝛼₁²²𝐴²¹_1𝑚𝑒^{𝑖(𝑚+2)𝑧}. (2.11)

Definition 2.1. The multiplicity of a zero of 𝐹 in 𝑃∶=𝑃₀∪[0,4𝜋] is called the multiplicity of the corresponding eigenvalue or spectral singularity of the BVP (1.2), (1.3).

3. Principal Functions

In this section we also assume that (2.1) holds.

Let 𝜆₁,𝜆₂,…,𝜆_𝑘 and 𝜆_𝑘+1,𝜆_𝑘+2,…,𝜆_𝜈 denote the zeros of 𝐹 in 𝑃₀ and [0,4𝜋] with multiplicities 𝑚₁,𝑚₂,…,𝑚_𝑘 and 𝑚_𝑘+1,𝑚_𝑘+2,…,𝑚_𝜈, respectively.

Let us define ℓ̃ℓ∶=ℓ̂ where ℓ𝑦̃_𝑛=𝑎_𝑛+1𝑦⁽²⁾_𝑛+1+𝑏_𝑛𝑦_𝑛⁽²⁾+𝑝_𝑛𝑦_𝑛⁽¹⁾̂,𝑛∈ℕ,ℓ𝑦_𝑛=𝑎_𝑛−1𝑦⁽¹⁾_𝑛−1+𝑏_𝑛𝑦_𝑛⁽¹⁾+𝑞_𝑛𝑦_𝑛⁽²⁾,𝑛∈ℕ. (3.1)

Definition 3.1. Let 𝜆=𝜆₀ be an eigenvalue of 𝐿. If the vectors 𝑦_𝑛, 𝑑/(𝑑𝜆𝑦_𝑛),𝑑²/𝑑𝜆²𝑦_𝑛,…,𝑑^𝜈/𝑑𝜆^𝜈𝑦_𝑛, 𝑑^𝑗𝑑𝜆^𝑗𝑑𝑦∶=^𝑗𝑑𝜆^𝑗𝑦_{𝑛𝑛∈ℕ},𝑗=0,1,…,𝜈;𝑛∈ℕ, (3.2) satisfy the equations ℓ𝑦_𝑛−𝜆₀𝑦_𝑛ℓ𝑑=0,_𝑗𝑑𝜆^𝑗𝑦_𝑛−𝜆₀𝑑^𝑗𝑑𝜆^𝑗𝑦_𝑛−𝑑^𝑗−1𝑑𝜆^𝑗−1𝑦_𝑛=0,𝑗=1,2,…,𝜈,𝑛∈ℕ, (3.3) then the vector 𝑦_𝑛 is called the eigenvector corresponding to the eigenvalue 𝜆=𝜆₀ of 𝐿. The vectors (𝑑/𝑑𝜆)𝑦_𝑛,(𝑑²/𝑑𝜆²)𝑦_𝑛,…,(𝑑^𝜈/𝑑𝜆^𝜈)𝑦_𝑛 are called the associated vectors corresponding to 𝜆=𝜆₀. The eigenvector and the associated vectors corresponding to 𝜆=𝜆₀ are called the principal vectors of the eigenvalue 𝜆=𝜆₀. The principal vectors of the spectral singularities of 𝐿 are defined similarly.

We define the vectors 𝑑^𝑗𝑑𝜆^𝑗𝑉_𝑛𝜆_𝑖=1𝑑𝑗!𝑑𝜆^𝑗𝑗𝐸_𝑛⁽¹⁾(𝜆)_{𝜆=𝜆𝑖}1𝑑𝑗!𝑑𝜆^𝑗𝑗𝐸_𝑛⁽²⁾(𝜆)_{𝜆=𝜆𝑖},𝑛∈ℕ,𝑗=0,1,…,𝑚_𝑖−1,𝑖=1,2,…,𝑘,𝑘+1,…,𝜈, (3.4) where 𝜆=2sin(𝑧/2) and 𝐸_𝑛(𝜆)=𝐸_𝑛⁽¹⁾(𝜆)𝐸_𝑛⁽²⁾(𝜆)∶=𝑓_𝑛𝜆2arcsin2=𝑓_𝑛⁽¹⁾𝜆2arcsin2𝑓_𝑛⁽²⁾𝜆2arcsin2. (3.5) If 𝑦(𝜆)=𝑦_𝑛(𝜆𝑦)∶=^𝑛(1)(𝜆)𝑦_𝑛⁽²⁾(𝜆)_𝑛∈ℕ (3.6) is a solution of (1.2), then 𝑑^𝑗𝑑𝜆^𝑗𝑦(𝜆)=𝑑^𝑗𝑑𝜆^𝑗𝑦_𝑛(𝜆)_𝑛∈ℕ𝑑∶=𝑑𝜆^𝑗𝑗𝑦_𝑛⁽¹⁾(𝜆)𝑑^𝑗𝑑𝜆^𝑗𝑦_𝑛⁽²⁾(𝜆) (3.7) satisfies 𝑎_𝑛−1𝑑^𝑗𝑑𝜆^𝑗𝑦⁽²⁾_𝑛+1(𝜆)+𝑏_𝑛𝑑^𝑗𝑑𝜆^𝑗𝑦_𝑛⁽²⁾(𝜆)+𝑝_𝑛𝑑^𝑗𝑑𝜆^𝑗𝑦_𝑛⁽¹⁾(𝜆)𝑎_𝑛−1𝑑^𝑗𝑑𝜆^𝑗𝑦⁽¹⁾_𝑛−1(𝜆)+𝑏_𝑛𝑑^𝑗𝑑𝜆^𝑗𝑦_𝑛⁽¹⁾(𝜆)+𝑞_𝑛𝑑^𝑗𝑑𝜆^𝑗𝑦_𝑛⁽²⁾(𝜆)=𝜆𝑑^𝑗𝑑𝜆^𝑗𝑦_𝑛⁽¹⁾(𝜆)𝑑+𝑗^𝑗−1𝑑𝜆^𝑗−1𝑦_𝑛⁽¹⁾(𝜆)𝜆𝑑^𝑗𝑑𝜆^𝑗𝑦_𝑛⁽²⁾(𝜆)𝑑+𝑗^𝑗−1𝑑𝜆^𝑗−1𝑦_𝑛⁽²⁾(𝜆). (3.8) From (3.4) and (3.8) we get that ℓ𝑉𝜆_𝑖𝑛−𝜆₀𝑉_𝑛𝜆_𝑖ℓ𝑑=0,_𝑗𝑑𝜆^𝑗𝑉𝜆_𝑖𝑛−𝜆₀𝑑^𝑗𝑑𝜆^𝑗𝑉_𝑛𝜆_𝑖−𝑑^𝑗−1𝑑𝜆^𝑗−1𝑉_𝑛𝜆_𝑖=0,𝑛∈ℕ,𝑗=1,2,…,𝑚_𝑖−1,𝑖=1,2,…,𝜈. (3.9) The vectors 𝑑^𝑗/𝑑𝜆^𝑗𝑉_𝑛(𝜆_𝑖), 𝑗=0,1,2,…,𝑚_𝑖−1, 𝑖=1,2,…,𝑘 and 𝑑^𝑗/𝑑𝜆^𝑗𝑉_𝑛(𝜆_𝑖), 𝑗=0,1,2,…,𝑚_𝑖−1, 𝑖=𝑘+1,𝑘+2,…,𝜈 are the principal vectors of eigenvalues and spectral singularities of 𝐿, respectively.

Theorem 3.2. 𝑑^𝑗𝑑𝜆^𝑗𝑉_𝑛𝜆_𝑖∈ℓ₂ℕ,ℂ²,𝑗=0,1,2,…,𝑚_𝑖𝑑−1,𝑖=1,2,…,𝑘,_𝑗𝑑𝜆^𝑗𝑉_𝑛𝜆_𝑖∉ℓ₂ℕ,ℂ²,𝑗=0,1,2,…,𝑚_𝑖−1,𝑖=𝑘+1,𝑘+2,…,𝜈. (3.10)

Proof. Using (3.5) we get that 𝑑^𝑗𝑑𝜆^𝑗𝐸_𝑛⁽¹⁾(𝜆)_{𝜆=𝜆𝑖}=^𝑗_𝑡=0𝐶_𝑡𝑑^𝑡𝑑𝜆^𝑡𝑓_𝑛⁽¹⁾(𝑧)_{𝑧=𝑧𝑖}𝑑,𝑛∈ℕ,_𝑗𝑑𝜆^𝑗𝐸_𝑛⁽²⁾(𝜆)_{𝜆=𝜆𝑖}=^𝑗_𝑡=0𝐷_𝑡𝑑^𝑡𝑑𝜆^𝑡𝑓_𝑛⁽²⁾(𝑧)_{𝑧=𝑧𝑖},𝑛∈ℕ, (3.11) where 𝜆_𝑖=2sin𝑧_𝑖/2, 𝑧_𝑖∈𝑃=𝑃₀∪[0,4], 𝑖=1,2,…,𝑘 and 𝐶_𝑡, 𝐷_𝑡 are constant depending on 𝜆. From (2.2) we obtain that 𝑑^𝑡𝑑𝜆^𝑡𝑓_𝑛⁽¹⁾(𝑧)_{𝑧=𝑧𝑖}=𝛼_𝑛¹¹𝑖^𝑡1𝑛+2^𝑡𝑒^{𝑖𝑧𝑖(𝑛+(1/2))}+^∞_𝑚=1𝛼_𝑛¹¹𝐴¹¹_𝑛𝑚𝑖^𝑡1𝑚+𝑛+2^𝑡𝑒^{𝑖(𝑚+𝑛+(1/2))𝑧𝑖}−𝐴¹²_𝑛𝑚𝑖^𝑡+1(𝑚+𝑛)^𝑡𝑒^{𝑖(𝑚+𝑛)𝑧𝑖},𝑑^𝑡𝑑𝜆^𝑡𝑓_𝑛⁽²⁾(𝑧)_{𝑧=𝑧𝑖}=𝛼_𝑛²¹𝑖^𝑡1𝑛+2^𝑡𝑒^{𝑖𝑧𝑖(𝑛+(1/2))}−𝑖(𝑖𝑛)^𝑡𝛼_𝑛²²𝑒^{𝑖𝑛𝑧𝑖}+^∞_𝑚=1𝛼_𝑛²¹𝐴¹¹_𝑛𝑚𝑖^𝑡1𝑚+𝑛+2^𝑡𝑒^{𝑖(𝑚+𝑛+(1/2))𝑧𝑖}−𝐴¹²_𝑛𝑚𝑖^𝑡+1(𝑚+𝑛)^𝑡𝑒^{𝑖(𝑚+𝑛)𝑧𝑖}+^∞_𝑚=1𝛼_𝑛²²𝐴²¹_𝑛𝑚𝑖^𝑡1𝑚+𝑛+2^𝑡𝑒^{𝑖(𝑚+𝑛+(1/2))𝑧𝑖}−𝐴²²_𝑛𝑚𝑖^𝑡+1(𝑚+𝑛)^𝑡𝑒^{𝑖(𝑚+𝑛)𝑧𝑖}. (3.12)(3.13) For the principal vectors (𝑑^𝑗/𝑑𝜆^𝑗)𝑉_𝑛(𝜆_𝑖)={(𝑑^𝑗/𝑑𝜆^𝑗)𝑉_𝑛(𝜆_𝑖)}_𝑛∈ℕ, 𝑗=0,1,…,𝑚_𝑖−1, 𝑖=1,2,…,𝑘 corresponding to the eigenvalues of 𝐿 we get 1𝑑𝑗!𝑑𝜆^𝑗𝑗𝐸_𝑛⁽¹⁾(𝜆)_{𝜆=𝜆𝑖}=1^𝑗𝑗!^𝑗_𝑡=0𝐶_𝑡𝑑^𝑡𝑑𝜆^𝑡𝑓_𝑛⁽¹⁾𝑧_𝑖,𝑗=0,1,…,𝑚_𝑖1−1,𝑖=1,2,…,𝑘,𝑑𝑗!𝑑𝜆^𝑗𝑗𝐸_𝑛⁽²⁾(𝜆)_{𝜆=𝜆𝑖}=1^𝑗𝑗!^𝑗_𝑡=0𝐷_𝑡𝑑^𝑡𝑑𝜆^𝑡𝑓_𝑛⁽²⁾𝑧_𝑖,𝑗=0,1,…,𝑚_𝑖−1,𝑖=1,2,…,𝑘. (3.14) Since Im𝜆_𝑖>0 for 𝑖=1,2,…,𝑘 from (3.14) we obtain that 𝑑^𝑗𝑑𝜆^𝑗𝑉_𝑛²=^∞_𝑛=11𝑑𝑗!𝑑𝜆^𝑗𝑗𝐸_𝑛⁽¹⁾(𝜆)_{𝜆=𝜆𝑖}²+1𝑑𝑗!𝑑𝜆^𝑗𝑗𝐸_𝑛⁽²⁾(𝜆)_{𝜆=𝜆𝑖}²=1𝑗!^2∞_𝑛=1^𝑗_𝑡=0𝐶_𝑡𝑑^𝑡𝑑𝜆^𝑡𝑓_𝑛⁽¹⁾𝑧_𝑖²+^𝑗_𝑡=0𝐷_𝑡𝑑^𝑡𝑑𝜆^𝑡𝑓_𝑛⁽²⁾𝑧_𝑖²≤1𝑗!^2∞_𝑛=1^𝑗_𝑡=0𝐶_𝑡𝑑^𝑡𝑑𝜆^𝑡𝑓_𝑛⁽¹⁾𝑧_𝑖²+^𝑗_𝑡=0𝐷_𝑡𝑑^𝑡𝑑𝜆^𝑡𝑓_𝑛⁽²⁾𝑧_𝑖²≤1𝑗!^2∞𝑗_𝑛=1𝑡=0max𝐶_𝑡,𝐷_𝑡𝑑^𝑡𝑑𝜆^𝑡𝑓_𝑛⁽¹⁾𝑧_𝑖+𝑑^𝑡𝑑𝜆^𝑡𝑓_𝑛⁽²⁾𝑧_𝑖², (3.15) or 𝑑^𝑗𝑑𝜆^𝑗𝑉_𝑛²≤1𝑗!^2∞_𝑛=1^𝑗_𝑡=0max𝐶_𝑡,𝐷_𝑡×𝛼_𝑛¹¹+𝛼_𝑛²¹1𝑛+2^𝑡𝑒^{−(𝑛+(1/2))Im𝑧𝑖}+𝛼_𝑛²²|𝑛|^𝑡𝑒^{−𝑛Im𝑧𝑖}+^𝑗_𝑡=0max𝐶_𝑡,𝐷_𝑡^∞_𝑚=1𝛼_𝑛¹¹+𝛼_𝑛²¹𝐴¹¹_𝑛𝑚1𝑚+𝑛+2^𝑡𝑒^{−(𝑚+𝑛+(1/2))Im𝑧𝑖}+𝐴¹²_𝑛𝑚|𝑚+𝑛|^𝑡𝑒^{−(𝑚+𝑛)Im𝑧𝑖}+^𝑗_𝑡=0𝐷_𝑡^∞_𝑚=1𝛼_𝑛²²𝐴²¹_𝑛𝑚1𝑚+𝑛+2^𝑡𝑒^{−(𝑚+𝑛+(1/2))Im𝑧𝑖}+𝐴²²_𝑛𝑚|𝑚+𝑛|^𝑡×𝑒^{−(𝑚+𝑛)Im𝑧𝑖2}. (3.16) From (3.16), 1𝑗!^2∞𝑗_𝑛=1𝑡=0max𝐶_𝑡,𝐷_𝑡𝛼_𝑛¹¹+𝛼_𝑛²¹1𝑛+2^𝑡𝑒^{−(𝑛+(1/2))Im𝑧𝑖}+𝛼_𝑛²²|𝑛|^𝑡𝑒^{−𝑛Im𝑧𝑖}≤𝐴𝑗!^2∞_𝑛=111+𝑛+2+1𝑛+2²1+⋯+𝑛+2^𝑗𝑒^{−(𝑛+(1/2))Im𝑧𝑖}+1+𝑛+𝑛²+⋯+𝑛^𝑗𝑒^{−𝑛Im𝑧𝑖}≤𝐴𝑗+1²𝑗!^2∞_𝑛=11𝑛+2^𝑗𝑒^{−(𝑛+(1/2))Im𝑧𝑖}+𝑛^𝑗𝑒^{−𝑛Im𝑧𝑖}<∞. (3.17) Holds, where 𝐴=max𝐶_𝑡,𝐷_𝑡max𝛼_𝑛¹¹+𝛼_𝑛²¹,𝛼_𝑛²². (3.18) Now we define the function 𝑔_𝑛(𝑧)=^𝑗_𝑡=0max𝐶_𝑡,𝐷_𝑡^∞_𝑚=1𝛼_𝑛¹¹+𝛼_𝑛²¹𝐴¹¹_𝑛𝑚1𝑚+𝑛+2^𝑡𝑒^{−(𝑚+𝑛+(1/2))Im𝑧𝑖}+𝐴¹²_𝑛𝑚|𝑚+𝑛|^𝑡𝑒^{−(𝑚+𝑛)Im𝑧𝑖}+^𝑗_𝑡=0𝐷_𝑡^∞_𝑚=1𝛼_𝑛²²𝐴²¹_𝑛𝑚1𝑚+𝑛+2^𝑡𝑒^{−(𝑚+𝑛+(1/2))Im𝑧𝑖}+𝐴²²_𝑛𝑚|𝑚+𝑛|^𝑡𝑒^{−(𝑚+𝑛)Im𝑧𝑖}. (3.19) So we get 1𝑗!^2∞_𝑛=1^𝑗_𝑡=0max𝐶_𝑡,𝐷_𝑡^∞_𝑚=1𝛼_𝑛¹¹+𝛼_𝑛²¹𝐴¹¹_𝑛𝑚1𝑚+𝑛+2^𝑡𝑒^{−(𝑚+𝑛+(1/2))Im𝑧𝑖}+𝐴¹²_𝑛𝑚|𝑚+𝑛|^𝑡𝑒^{−(𝑚+𝑛)Im𝑧𝑖}+^𝑗_𝑡=0𝐷_𝑡^∞_𝑚=1𝛼_𝑛²²𝐴²¹_𝑛𝑚1𝑚+𝑛+2^𝑡𝑒^{−(𝑚+𝑛+(1/2))Im𝑧𝑖}+𝐴²²_𝑛𝑚|𝑚+𝑛|^𝑡𝑒^{−(𝑚+𝑛)Im𝑧𝑖}=1𝑗!^2∞_𝑛=1𝑔_𝑛(𝑧). (3.20) Using the boundness of 𝐴^𝑖𝑗_𝑛𝑚 and 𝛼_𝑛^𝑖𝑗, 𝑖,𝑗=1,2 we obtain that 𝑔_𝑛(𝑧)≤max𝐶_𝑡,𝐷_𝑡𝑀^𝑗∞_𝑡=0𝑚=11𝑚+𝑛+2^𝑡𝑒^{−(𝑚+𝑛+(1/2))Im𝑧𝑖}+|𝑚+𝑛|^𝑡𝑒^{−(𝑚+𝑛)Im𝑧𝑖}, (3.21) where 𝑀=max𝛼_𝑛¹¹+𝛼_𝑛²¹𝐴¹¹_𝑛𝑚,𝛼_𝑛²²𝐴²¹_𝑛𝑚,𝛼_𝑛¹¹+𝛼_𝑛²¹𝐴¹²_𝑛𝑚,𝛼_𝑛²²𝐴²²_𝑛𝑚. (3.22) If we take max{|𝐶_𝑡|,|𝐷_𝑡|}𝑀=𝑁, we can write 𝑔_𝑛(𝑧)≤𝑁^𝑗_𝑡=0𝑒^{−𝑛Im𝑧𝑖∞}_𝑚=11𝑚+𝑛+2^𝑡𝑒^{−𝑚Im𝑧𝑖}+(𝑚+𝑛)^𝑡𝑒^{−𝑚Im𝑧𝑖}=𝑁𝑒^{−𝑛Im𝑧𝑖∞}_𝑚=12𝑒^{−𝑚Im𝑧𝑖}+^∞_𝑚=1𝑒^{−𝑚Im𝑧𝑖}1𝑚+𝑛+2+(+𝑚+𝑛)+⋯_∞𝑚=1𝑒^{−𝑚Im𝑧𝑖}1𝑚+𝑛+2^𝑗+(𝑚+𝑛)^𝑗≤𝑁𝑒^{−𝑛Im𝑧𝑖∞𝑗}_𝑚=1𝑡=0𝑒^{−𝑚Im𝑧𝑖}1𝑚+𝑛+2^𝑡+(𝑚+𝑛)^𝑡≤𝐵𝑒^{−𝑛Im𝑧𝑖}, (3.23) where 𝐵=𝐴^∞𝑗_𝑚=1𝑡=0𝑒^{−𝑚Im𝑧𝑖}1𝑚+𝑛+2^𝑡+(𝑚+𝑛)^𝑡. (3.24) Therefore we have 1𝑗!^2∞_𝑛=1𝑔_𝑛(𝑧)≤1𝑗!^2∞_𝑛=1𝐵𝑒^{−𝑛Im𝑧𝑖}<∞. (3.25) From (3.17) and (3.25) 𝑑^𝑗/𝑑𝜆^𝑗𝑉_𝑛(𝜆_𝑖)∈ℓ₂(ℕ,ℂ²), 𝑗=0,1,…,𝑚_𝑖−1, 𝑖=1,2,…,𝑘.
On the other hand, since Im𝑧_𝑖=0 for 𝑗=0,1,…,𝑚_𝑖−1; 𝑖=𝑘+1,𝑘+2,…,𝜈 using (3.12) we find that ^∞_𝑛=1𝛼_𝑛¹¹𝑖^𝑡1𝑛+2^𝑡𝑒^{𝑖𝑧𝑖(𝑛+(1/2))2}=∞, (3.26) but the other terms in (3.12) belongs ℓ₂(ℕ,ℂ²), so 𝑑^𝑗/(𝑑𝜆^𝑗)𝐸_𝑛⁽¹⁾(𝜆)∉ℓ₂(ℕ,ℂ²). Similarly from (3.13) we get 𝑑^𝑗/(𝑑𝜆^𝑗)𝐸_𝑛⁽²⁾(𝜆)∉ℓ₂(ℕ,ℂ²), then we obtain that 𝑑^𝑗/(𝑑𝜆^𝑗)𝑉_𝑛(𝜆_𝑖)∉ℓ₂(ℕ,ℂ²), 𝑗=0,1,…,𝑚_𝑖−1; 𝑖=𝑘+1,𝑘+2,…,𝜈.

Let us introduce Hilbert space 𝑗=0,1,2,…𝐻_−𝑗(ℕ)=𝑦𝑦=^𝑛(1)𝑦_𝑛⁽²⁾∶_𝑛∈ℕ(1+|𝑛|)^−2𝑗𝑦_𝑛⁽¹⁾²+𝑦_𝑛⁽²⁾²<∞, (3.27) with 𝑦²_−𝑗=_𝑛∈ℕ(1+|𝑛|)^−2𝑗𝑦_𝑛⁽¹⁾²+𝑦_𝑛⁽²⁾². (3.28)

Theorem 3.3. 𝑑^𝑗/(𝑑𝜆^𝑗)𝑉_𝑛(𝜆_𝑖)∈𝐻_−(𝑗+1)(ℕ), 𝑗=0,1,2,…,𝑚_𝑖−1, 𝑖=𝑘+1, 𝑘+2,…,𝜈.

Proof. Using (3.4), (3.14) we have _𝑛∈ℕ(1+|𝑛|)^−2(𝑗+1)1𝑑𝑗!𝑑𝜆^𝑗𝑗𝐸_𝑛⁽¹⁾(𝜆)_{𝜆=𝜆𝑖}²+1𝑑𝑗!𝑑𝜆^𝑗𝑗𝐸_𝑛⁽²⁾(𝜆)_{𝜆=𝜆𝑖}²=_𝑛∈ℕ(1+|𝑛|)^−2(𝑗+1)𝑗!^2𝑗_𝑡=0𝐶_𝑡𝑑^𝑡𝑑𝜆^𝑡𝑓_𝑛⁽¹⁾𝑧_𝑖²+^𝑗_𝑡=0𝐷_𝑡𝑑^𝑡𝑑𝜆^𝑡𝑓_𝑛⁽²⁾𝑧_𝑖²≤1𝑗!^2∞_𝑛=1(1+|𝑛|)^{−2(𝑗+1)𝑗}_𝑡=0𝐶_𝑡𝑑^𝑡𝑑𝜆^𝑡𝑓_𝑛⁽¹⁾𝑧_𝑖²+^𝑗_𝑡=0𝐷_𝑡𝑑^𝑡𝑑𝜆^𝑡𝑓_𝑛⁽²⁾𝑧_𝑖², (3.29) for 𝑗=0,1,2,…,𝑚_𝑖−1, 𝑖=𝑘+1,𝑘+2,𝜈. Since Im𝑧_𝑖=0, using (3.29) we obtain _𝑛∈ℕ(1+|𝑛|)^−2(𝑗+1)1𝑗!^2𝑗_𝑡=0𝐶_𝑡𝑑^𝑡𝑑𝜆^𝑡𝑓_𝑛⁽¹⁾𝑧_𝑖²=1𝑗!^2∞_𝑛=1^𝑗_𝑡=0(1+|𝑛|)^−(𝑗+1)1𝑛+2^𝑡𝛼_𝑛¹¹𝐶_𝑡+^𝑗_𝑡=0𝐶_𝑡𝛼_𝑛¹¹(1+|𝑛|)^{∞−(𝑗+1)}_𝑚=1𝐴¹¹_𝑛𝑚1𝑚+𝑛+2^𝑡+𝐴¹²_𝑛𝑚(𝑚+𝑛)^𝑡2=1𝑗!^2∞_𝑛=1^𝑗_𝑡=0(1+|𝑛|)^−(𝑗+1)1𝑛+2^𝑡𝛼_𝑛¹¹𝐶_𝑡²+2(1+|𝑛|)^−2(𝑗+1)𝛼_𝑛^112𝑗_𝑡=01𝑛+2^𝑡𝐶_𝑡×^𝑗_𝑡=0𝐶_𝑡^∞_𝑚=1𝐴¹¹_𝑛𝑚1𝑚+𝑛+2^𝑡+𝐴¹²_𝑛𝑚(𝑚+𝑛)^𝑡+^𝑗_𝑡=0𝐶_𝑡(1+|𝑛|)^−(𝑗+1)𝛼_𝑛^11∞_𝑚=1𝐴¹¹_𝑛𝑚1𝑚+𝑛+2^𝑡+𝐴¹²_𝑛𝑚(𝑚+𝑛)^𝑡2. (3.30) Using (3.30), (2.1), and (2.7) we first obtain that ^𝑗_𝑡=0𝐶_𝑡𝛼_𝑛¹¹(1+|𝑛|)^{∞−(𝑗+1)}_𝑚=1𝐴¹¹_𝑛𝑚1𝑚+n+2^𝑡+𝐴¹²_𝑛𝑚(𝑚+𝑛)^𝑡2≤4^𝑗_𝑡=0𝐶_𝑡𝛼_𝑛^11∞_𝑚=1(1+|𝑛|)^−(𝑗+1)1𝑚+𝑛+2^𝑡𝐶×^∞_{⌈⌉𝑗=𝑛+𝑚/2}1−𝑎_𝑗+1+𝑏_𝑗+𝑝_𝑗+𝑞_𝑗𝑒^{−𝜀𝑗𝛿}𝑒^{𝜀𝑗𝛿2}≤4^𝑗_𝑡=0𝛼_𝑛^11∞_𝑚=1(1+|𝑛|)^−(𝑗+1)1𝑚+𝑛+2^𝑡𝐶exp−𝜀𝑛+𝑚4^𝛿×^∞_{⌈⌉𝑗=𝑛+𝑚/2}𝑒^𝜀𝑗𝛿1−𝑎_𝑗+1+𝑏_𝑗+𝑝_𝑗+𝑞_𝑗²≤𝐶₁^𝑗_𝑡=0(1+|𝑛|)^{∞−(𝑗+1)}_𝑚=11𝑚+𝑛+2^𝑡exp−𝜀𝑛+𝑚4^𝛿2≤𝐶₁^𝑗_𝑡=0(1+|𝑛|)^{∞−(𝑗+1)}_𝑚=11𝑚+𝑛+2^𝑡exp−𝜀𝑛+𝑚4^1/22≤𝐶₁^𝑗_𝑡=0(1+|𝑛|)^{∞−(𝑗+1)}_𝑚=11𝑚+𝑛+2^𝑡√exp−𝜀24𝑛121+𝑚2²=𝐶₁(1+|𝑛|)^−2(𝑗+1)√exp−𝜀24𝑛^1/2𝑗∞_𝑡=0𝑚=11𝑚+𝑛+2^𝑡√exp−𝜀24𝑚^1/22√=𝐺exp−𝜀24𝑛12(1+|𝑛|)^−2(𝑗+1), (3.31) where 𝐶₁=2𝐶𝛼_𝑛^11∞_{⌈⌉𝑗=𝑛+𝑚/2}𝑒^𝜀𝑗𝛿1−𝑎_𝑗+1+𝑏_𝑗+𝑝_𝑗+𝑞_𝑗²,𝐺=𝐶₁^𝑗∞_𝑡=0𝑚=11𝑚+𝑛+2^𝑡√exp−𝜀24𝑚^1/22. (3.32) So we get from (3.31) ^∞_𝑛=1^𝑗_𝑡=0𝐶_𝑡(1+|𝑛|)^−(𝑗+1)𝛼_𝑛^11∞_𝑚=1𝐴¹¹_𝑛𝑚1𝑚+𝑛+2^𝑡+𝐴¹²_𝑛𝑚(𝑚+𝑛)^𝑡2≤𝐺^∞_𝑛=1√exp−𝜀24𝑛^1/2(1+|𝑛|)^−2(𝑗+1)<∞. (3.33) Secondly, using (3.30) and (3.31) we obtain that ^∞_𝑛=12^𝑗_𝑡=0𝛼_𝑛¹¹𝐶_𝑡(1+|𝑛|)^−(𝑗+1)1𝑛+2^𝑡×^𝑗_𝑡=0𝐶_𝑡𝛼_𝑛^11∞_𝑚=1(1+|𝑛|)^−(𝑗+1)1𝑚+𝑛+2^𝑡𝐴¹¹_𝑛𝑚+(𝑚+𝑛)^𝑡𝐴¹²_𝑛𝑚≤^∞_𝑛=1^𝑗_𝑡=0(1+|𝑛|)^−2(𝑗+1)1𝑛+2^𝑡√exp−𝜀24𝑛^1/2𝐺^1/2<∞, (3.34) and also the first part of the (3.31) obviously convergent so, we get from (3.33) and (3.34) _𝑛∈ℕ(1+|𝑛|)^−2(𝑗+1)1𝑗!^2𝑗_𝑡=0𝐶_𝑡𝑑^𝑡𝑑𝜆^𝑡𝑓_𝑛⁽¹⁾𝑧_𝑖²<∞, (3.35) and similarly _𝑛∈ℕ(1+|𝑛|)^−2(𝑗+1)1𝑗!^2𝑗_𝑡=0𝐷_𝑡𝑑^𝑡𝑑𝜆^𝑡𝑓_𝑛⁽²⁾𝑧_𝑖²<∞. (3.36) Finally 𝑑^𝑗/(𝑑𝜆^𝑗)𝑉_𝑛(𝜆_𝑖)∈𝐻_−(𝑗+1)(ℕ), 𝑗=0,1,2,…,𝑚_𝑖−1, 𝑖=𝑘+1,𝑘+2,…,𝜈.