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

Authors: Yelda Aygar; Murat Olgun; Turhan Koprubasi
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

LetL denote the operator generated in ℓ2(ℕ,ℂ2) by an+1yn+1(2)+bnyn(2)+pnyn(1)=λyn(1), an-1yn-1(1)+bnyn(1)+qnyn(2)=λyn(2), n∈ℕ, and the boundary condition (γ0+γ1λ)y1(2)+(β0+β1λ)y0(1)=0, where (an), (bn), (pn), and (qn), n∈ℕ are complex sequences, γi,βi∈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.


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## Body

## 1. Introduction

Consider the boundary value problem (BVP)(1.1)-y′′+q(x)y=λ2y,0≤x<∞,y(0)=0,
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(1.2)an+1yn+1(2)+bnyn(2)+pnyn(1)=λyn(1),n∈ℕ,an-1yn-1(1)+bnyn(1)+qnyn(2)=λyn(2),n∈ℕ,(1.3)(γ0+γ1λ)y1(2)+(β0+β1λ)y0(1)=0,γ0β1-γ1β0≠0,γ1≠a0-1β0,
where (yn(1)yn(2)),n∈ℕ are vector sequences, an≠0,  bn≠0 for all n∈ℕ,  γi,βi∈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,
(1.4)∑n=1∞‍exp(εnδ)(|1-an|+|1+bn|+|pn|+|qn|)<∞
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 someε>0 and 1/2≤δ<1,
(2.1)∑n=1∞‍exp(εnδ)(|1-an|+|1+bn|+|pn|+|qn|)<∞,
be satisfied. It has been shown that [10] under the condition (2.1), (1.2) has the solution
(2.2)fn(z)=(fn(1)(z)fn(2)(z))=αn(I+∑m=1∞‍Anmeimz)(eiz/2-i)einz,n=1,2,…,(2.3)f0(1)(z)=α011{eiz/2[1+∑m=1∞‍A0m11eimz]-i∑m=1∞‍A0m12eimz},
for λ=2sin(z/2) and z∈ℂ¯+:={z∈ℂ:Imz≥0}, where
(2.4)αn=(αn11αn12αn21αn22),I=(1001),Anm=(Anm11Anm12Anm21Anm22).
Note that αnij and Anmij (i,j=1,2) are uniquely expressed in terms of (an), (bn), (pn), and (qn), n∈ℕ as follows
(2.5)αn11=[∏k=n+1∞(-1)n-kbkak-1]-1,αn12=0,αn22=[bn∏k=n+1∞(-1)n-k+1bkak-1]-1,αn21=αn22[pn+∑k=n+1∞‍(pk+qk)],An112=-∑k=n+1∞‍(pk+qk),An111=∑k=n+1∞‍[ak+1ak+bk2-pkqk+(pk+qk)Ak112-2],An122=-1+an+1an+(An112)2+An111,An121=-∑k=n∞‍{(qk+1+Ak112)[ak+1ak+qk+1(pk+1+qk+1)+qk+1Ak112+bk+12+Ak+1,111-1]-Ak112(1+Ak111)}+∑k=n+1∞‍(qkAk122-bk2pk),An212=-an+1an(qn+1+An112)+An112An111+An112-An121,An211=∑k=n+1∞‍{(bk2-1)Ak111-ak+1ak[(qk+1+Ak112)Ak+1,112-Ak+1,122]-(pk-Ak112)[qkAk111+Ak112-Ak212]-qkAk121+Ak112Ak212-Ak122},An222=-an+1an(qn+1+An112)An+1,112+an+1anAn+1,122+An112An212-An111+An211,An221=∑k=n∞‍{Ak112Ak211+Ak221-ak+1ak[(qk+1+Ak112)Ak+1,111-Ak+1,121]}-∑k=n+1∞‍[(qk+Ak-1,112)(qkAk212-Ak111+Ak211)+bk2Ak221-pkAk222+Ak121],
and for m≥3(2.6)Anm12=-an+1an[(qn+1+An112)An+1,m-211+An+1,m-221]+An112An,m-111+An,m-112-An,m-121,Anm11=-∑k=n+1∞‍ak+1ak[(qk+1+Ak112)Ak+1,m-112-Ak+1,m-122]-∑k=n+1∞‍(pk-Ak112)(qkAk,m-111+Ak,m-112-Akm12)+∑k=n+1∞‍(bk2-1)Ak,m-111-∑k=n+1∞‍qkAk,m-121+∑k=n+1∞‍Ak112Akm12-∑k=n+1∞‍Ak,m-122,Anm22=-an+1an[(qn+1+An112)An+1,m-111-An+1,m-122]+An112Anm12+Anm11-An,m-111,Anm21=-∑k=n∞‍ak+1ak[(qk+1+Ak112)Ak+1,m-111-Ak+1,m-121]-∑k=n+1∞‍(qk-Ak-1,112)(qkAkm21+Ak,m-111-Akm22)-∑k=n+1∞‍(bk2-1)Akm12+∑k=n∞‍Ak112Akm22+∑k=n+1∞‍qkAkm22+∑k=n∞‍Akm12-∑k=n+1∞‍Ak,m-121.
Moreover
(2.7)|Anmij|≤C∑k=n+[|m/2|]∞‍(|1-ak|+|1+bk|+|pk|+|qk|)
holds, where [|m/2|] is the integer part of m/2 and C>0 is a constant. Therefore fn is vector-valued analytic function with respect to z in ℂ+:={z∈ℂ:Imz>0} and continuous in ℂ¯+ [10]. The solution f(z)=(fn(z))=(fn(1)(z)fn(2)(z)) is called Jost solution of (1.2).Let us define(2.8)F(z)=(γ0+2γ1sinz2)f1(2)(z)+(β0+2β1sinz2)f0(1)(z).
It follows (2.2) and (2.3) that the function F is analytic in ℂ+, continuous up to the real axis, and
(2.9)F(z+4π)=F(z).
We denote the set of eigenvalues and spectral singularities of L by σd(L) and σss(L), respectively. From the definition of the eigenvalues and spectral singularities we have [10]
(2.10)σd={λ:λ=2sinz2,z∈P0,F(z)=0},σss={λ:λ=2sinz2,z∈[0,4π],F(z)=0},
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
(2.11)F(z)={γ0+γ1[(-i)(ei(z/2)-e-i(z/2))]}f1(2)(z)+{β0+β1[(-i)(ei(z/2)-e-i(z/2))]}f0(1)(z)=iα011β1+(γ1α122+α011β0)ei(z/2)+i(-γ0α122+γ1α122-α011β1)eiz+(γ0α121-γ1α122)ei(3z/2)-iγ1α121e2iz+∑m=1∞‍α011β1A0m12ei(m-(1/2))z+i∑m=1∞‍(-α011β0A0m12+α011β1A0m11)eimz+∑m=1∞‍(γ1α121A1m12+γ1α122A1m22+α011β0A0m11-α011β1A0m12)ei(m+(1/2))z+i∑m=1∞‍(-γ0α121A1m12-γ0α122A1m22+γ1α121A1m11+γ1α122A1m21-α011β1A0m11)ei(m+1)z+∑m=1∞‍(γ0α121A1m11+γ0α122A1m21-γ1α121A1m12-γ1α122A1m22)ei(m+(3/2))z+i∑m=1∞‍(-γ1α121A1m11-γ1α122A1m21)ei(m+2)z.+i∑m=1∞‍(-γ1α121A1m11-γ1α122A1m21)ei(m+2)z.Definition 2.1.
The multiplicity of a zero ofF 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.Letλ1,λ2,…,λk and λk+1,λk+2,…,λν denote the zeros of  F in P0 and [0,4π] with multiplicities m1,m2,…,mk and mk+1,mk+2,…,mν, respectively.Let us defineℓ:=(ℓ~ℓ^) where
(3.1)(ℓ~y)n=an+1yn+1(2)+bnyn(2)+pnyn(1),n∈ℕ,(ℓ^y)n=an-1yn-1(1)+bnyn(1)+qnyn(2),n∈ℕ.Definition 3.1.
Letλ=λ0 be an eigenvalue of L. If the vectors yn,  d/(dλyn),d2/dλ2yn,…,dν/dλνyn,
(3.2)djdλjy:={djdλjyn}n∈ℕ,(j=0,1,…,ν;n∈ℕ),
satisfy the equations
(3.3)(ℓy)n-λ0yn=0,(ℓ(djdλjy))n-λ0djdλjyn-dj-1dλj-1yn=0,j=1,2,…,ν,n∈ℕ,
then the vector yn is called the eigenvector corresponding to the eigenvalue λ=λ0 of L. The vectors (d/dλ)yn,(d2/dλ2)yn,…,(dν/dλν)yn 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(3.4)djdλjVn(λi)=(1j!{djdλjEn(1)(λ)}λ=λi1j!{djdλjEn(2)(λ)}λ=λi),n∈ℕ,j=0,1,…,mi-1,i=1,2,…,k,k+1,…,ν,
where λ=2sin(z/2) and
(3.5)En(λ)=(En(1)(λ)En(2)(λ)):=fn(2arcsinλ2)=(fn(1)(2arcsinλ2)fn(2)(2arcsinλ2)).
If
(3.6)y(λ)={yn(λ)}:=(yn(1)(λ)yn(2)(λ))n∈ℕ
is a solution of (1.2), then
(3.7)djdλjy(λ)={(djdλj)yn(λ)}n∈ℕ:={(djdλj)yn(1)(λ)(djdλj)yn(2)(λ)}
satisfies
(3.8)(an-1djdλjyn+1(2)(λ)+bndjdλjyn(2)(λ)+pndjdλjyn(1)(λ)an-1djdλjyn-1(1)(λ)+bndjdλjyn(1)(λ)+qndjdλjyn(2)(λ))=(λdjdλjyn(1)(λ)+jdj-1dλj-1yn(1)(λ)λdjdλjyn(2)(λ)+jdj-1dλj-1yn(2)(λ)).
From (3.4) and (3.8) we get that
(3.9)(ℓV(λi))n-λ0Vn(λi)=0,(ℓ(djdλjV(λi)))n-λ0djdλjVn(λi)-dj-1dλj-1Vn(λi)=0,n∈ℕ,j=1,2,…,mi-1,i=1,2,…,ν.
The vectors dj/dλjVn(λi), j=0,1,2,…,mi-1, i=1,2,…,k and dj/dλjVn(λi), j=0,1,2,…,mi-1, i=k+1,k+2,…,ν are the principal vectors of eigenvalues and spectral singularities of L, respectively.Theorem 3.2.
(3.10)
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.Proof.
Using (3.5) we get that
(3.11){djdλjEn(1)(λ)}λ=λi=∑t=0jCt{dtdλtfn(1)(z)}z=zi,n∈ℕ,{djdλjEn(2)(λ)}λ=λi=∑t=0jDt{dtdλtfn(2)(z)}z=zi,n∈ℕ,
where λi=2sinzi/2, zi∈P=P0∪[0,4], i=1,2,…,k and Ct,  Dt are constant depending on λ. From (2.2)  we obtain that
(3.12){dtdλtfn(1)(z)}z=zi=αn11it(n+12)teizi(n+(1/2))+∑m=1∞αn11{Anm11it(m+n+12)tei(m+n+(1/2))zi-Anm12it+1(m+n)tei(m+n)zi(m+n+12)t},(3.13){dtdλtfn(2)(z)}z=zi=αn21it(n+12)teizi(n+(1/2))-i(in)tαn22einzi+∑m=1∞αn21{Anm11it(m+n+12)tei(m+n+(1/2))zi-Anm12it+1(m+n)tei(m+n)zi(m+n+12)t}+∑m=1∞αn22{Anm21it(m+n+12)tei(m+n+(1/2))zi-Anm22it+1(m+n)tei(m+n)zi(m+n+12)t}.
For the principal vectors (dj/dλj)Vn(λi)={(dj/dλj)Vn(λi)}n∈ℕ, j=0,1,…,mi-1, i=1,2,…,k corresponding to the eigenvalues of L we get
(3.14)1j!{djdλjEn(1)(λ)}λ=λi=1jj!∑t=0jCt{dtdλtfn(1)(zi)},j=0,1,…,mi-1,i=1,2,…,k,1j!{djdλjEn(2)(λ)}λ=λi=1jj!∑t=0jDt{dtdλtfn(2)(zi)},j=0,1,…,mi-1,i=1,2,…,k.
Since Imλi>0 for i=1,2,…,k from (3.14) we obtain that
(3.15)∥djdλjVn∥2=∑n=1∞(|1j!{djdλjEn(1)(λ)}λ=λi|2+|1j!{djdλjEn(2)(λ)}λ=λi|2)=(1j!)2∑n=1∞(|∑t=0jCt{dtdλtfn(1)(zi)}|2+|∑t=0jDt{dtdλtfn(2)(zi)}|2)≤(1j!)2∑n=1∞{(∑t=0j|Ct{dtdλtfn(1)(zi)}|2)+(∑t=0j|Dt{dtdλtfn(2)(zi)}|2)}≤(1j!)2(∑n=1∞∑t=0jmax{|Ct|,|Dt|}(|{dtdλtfn(1)(zi)}|+|{dtdλtfn(2)(zi)}|))2,
or
(3.16)∥djdλjVn∥2≤(1j!)2{∑n=1∞[∑t=0jmax{|Ct|,|Dt|}×{(|αn11|+|αn21|)(|n+12|te-(n+(1/2))Imzi)+|αn22||n|te-nImzi∑t=0jmax{|Ct|,}}+∑t=0jmax{|Ct|,|Dt|}{∑m=1∞(|αn11|+|αn21|)(|Anm11||m+n+12|te-(m+n+(1/2))Imzi)+|Anm12||m+n|te-(m+n)Imzi(|Anm11||m+n+12|te-(m+n+(12))Imzi)}+∑t=0j|Dt|{∑m=1∞|αn22|(|Anm21||m+n+12|te-(m+n+(1/2))Imzi+|Anm22||m+n|t×e-(m+n)Imzi|m+n+12|t)}]}2.
From (3.16),
(3.17)(1j!)2∑n=1∞∑t=0jmax{|Ct|,|Dt|}{(|αn11|+|αn21|)(|n+12|te-(n+(1/2))Imzi)+|αn22||n|te-nImzi}≤A(j!)2∑n=1∞(1+(n+12)+(n+12)2+⋯+(n+12)j)e-(n+(1/2))Imzi+(1+n+n2+⋯+nj)e-nImzi≤A(j+1)2(j!)2∑n=1∞[(n+12)je-(n+(1/2))Imzi+nje-nImzi]<∞.
Holds, where
(3.18)A=max{|Ct|,|Dt|}max{(|αn11|+|αn21|),|αn22|}.
Now we define the function
(3.19)gn(z)=∑t=0jmax{|Ct|,|Dt|}{∑m=1∞(|αn11|+|αn21|)(|Anm11||m+n+12|te-(m+n+(1/2))Imzi+|Anm12||m+n|te-(m+n)Imzi|m+n+12|t)}+∑t=0j|Dt|{∑m=1∞|αn22|(|Anm21||m+n+12|te-(m+n+(1/2))Imzi+|Anm22||m+n|te-(m+n)Imzi|m+n+12|t)}.
So we get
(3.20)(1j!)2∑n=1∞[∑t=0jmax{|Ct|,|Dt|}{∑m=1∞(|αn11|+|αn21|)(|Anm11||m+n+12|te-(m+n+(1/2))Imzi+|Anm12||m+n|te-(m+n)Imzi|m+n+12|t)∑m=1∞(|αn11|+|αn21|)}+∑t=0j|Dt|{∑m=1∞|αn22|(|Anm21||m+n+12|te-(m+n+(1/2))Imzi+|Anm22||m+n|te-(m+n)Imzi|m+n+12|t)}]=(1j!)2∑n=1∞gn(z).
Using the boundness of Anmij and αnij, i,j=1,2 we obtain that
(3.21)gn(z)≤max{|Ct|,|Dt|}M∑t=0j∑m=1∞{|m+n+12|te-(m+n+(1/2))Imzi+|m+n|te-(m+n)Imzi|m+n+12|t},
where
(3.22)M=max{(|αn11|+|αn21|)|Anm11|,|αn22||Anm21|,(|αn11|+|αn21|)|Anm12|,|αn22||Anm22|}.
If we take max{|Ct|,|Dt|}M=N, we can write
(3.23)gn(z)≤N∑t=0je-nImzi∑m=1∞{(m+n+12)te-mImzi+(m+n)te-mImzi}=Ne-nImzi{∑m=1∞2e-mImzi+∑m=1∞e-mImzi((m+n+12)+(m+n))+⋯+∑m=1∞e-mImzi((m+n+12)j+(m+n)j)}≤Ne-nImzi∑m=1∞∑t=0je-mImzi((m+n+12)t+(m+n)t)≤Be-nImzi,
where
(3.24)B=A∑m=1∞∑t=0je-mImzi((m+n+12)t+(m+n)t).
Therefore we have
(3.25)(1j!)2∑n=1∞gn(z)≤(1j!)2∑n=1∞Be-nImzi<∞.
From (3.17) and (3.25) dj/dλjVn(λi)∈ℓ2(ℕ,ℂ2), j=0,1,…,mi-1, i=1,2,…,k.
On the other hand, sinceImzi=0 for j=0,1,…,mi-1; i=k+1,k+2,…,ν  using (3.12) we find that
(3.26)∑n=1∞|αn11it(n+12)teizi(n+(1/2))|2=∞,
but the other terms in (3.12) belongs ℓ2(ℕ,ℂ2), so dj/(dλj)En(1)(λ)∉ℓ2(ℕ,ℂ2). Similarly from (3.13) we get dj/(dλj)En(2)(λ)∉ℓ2(ℕ,ℂ2), then we obtain that dj/(dλj)Vn(λi)∉ℓ2(ℕ,ℂ2), j=0,1,…,mi-1; i=k+1,k+2,…,ν.Let us introduce Hilbert spacej=0,1,2,…(3.27)H-j(ℕ)={y=(yn(1)yn(2):∑n∈ℕ(1+|n|)-2j(|yn(1)|2+|yn(2)|2))<∞},
with
(3.28)∥y∥-j2=∑n∈ℕ(1+|n|)-2j(|yn(1)|2+|yn(2)|2).Theorem 3.3.
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Using (3.4), (3.14) we have
(3.29)∑n∈ℕ(1+|n|)-2(j+1)(|1j!{djdλjEn(1)(λ)}λ=λi|2+|1j!{djdλjEn(2)(λ)}λ=λi|2)=∑n∈ℕ(1+|n|)-2(j+1)(j!)2{|∑t=0jCt{dtdλtfn(1)(zi)}|2+|∑t=0jDt{dtdλtfn(2)(zi)}|2}≤1(j!)2∑n=1∞(1+|n|)-2(j+1){(∑t=0j|Ct{dtdλtfn(1)(zi)}|)2+(∑t=0j|Dt{dtdλtfn(2)(zi)}|)2},
for j=0,1,2,…,mi-1,  i=k+1,k+2,ν. Since Imzi=0, using (3.29) we obtain
(3.30)∑n∈ℕ(1+|n|)-2(j+1)1(j!)2(∑t=0j|Ct{dtdλtfn(1)(zi)}|)2=1(j!)2∑n=1∞{∑t=0j(1+|n|)-(j+1)(n+12)t|αn11||Ct|+∑t=0j|Ct||αn11|(1+|n|)-(j+1)∑m=1∞|Anm11|(m+n+12)t+|Anm12|(m+n)t}2=1(j!)2∑n=1∞{(∑t=0j(1+|n|)-(j+1)(n+12)t|αn11||Ct|)2+2(1+|n|)-2(j+1)|αn11|2[∑t=0j(n+12)t|Ct|]×[∑t=0j|Ct|∑m=1∞|Anm11|(m+n+12)t+|Anm12|(m+n)t]+(∑t=0j|Ct|(1+|n|)-(j+1)|αn11|∑m=1∞|Anm11|(m+n+12)t+|Anm12|(m+n)t)2}.
Using (3.30), (2.1), and (2.7) we first obtain that
(3.31)(∑t=0j|Ct||αn11|(1+|n|)-(j+1)∑m=1∞(|Anm11|(m+n+12)t+|Anm12|(m+n)t))2≤4(∑t=0j|Ct||αn11|m=1∞(1+|n|)-(j+1)(m+n+12)tC×∑j=n+⌈m/2⌉∞(|1-aj|+|1+bj|+|pj|+|qj|)e-εjδeεjδ)2≤4{∑t=0j|αn11|∑m=1∞(1+|n|)-(j+1)(m+n+12)tCexp(-ε(n+m4)δ)×∑j=n+⌈m/2⌉∞eεjδ(|1-aj|+|1+bj|+|pj|+|qj|)}2≤C1(∑t=0j(1+|n|)-(j+1)∑m=1∞(m+n+12)texp(-ε(n+m4)δ))2≤C1(∑t=0j(1+|n|)-(j+1)∑m=1∞(m+n+12)texp(-ε(n+m4)1/2))2≤C1(∑t=0j(1+|n|)-(j+1)∑m=1∞(m+n+12)texp(-ε24(n12+m12)))2=C1(1+|n|)-2(j+1)exp(-ε24n1/2)(∑t=0j∑m=1∞(m+n+12)texp(-ε24m1/2))2=Gexp(-ε24n12)(1+|n|)-2(j+1),
where
(3.32)C1=(2C|αn11|∑j=n+⌈m/2⌉∞eεjδ(|1-aj|+|1+bj|+|pj|+|qj|))2,G=C1(∑t=0j∑m=1∞(m+n+12)texp(-ε24m1/2))2.
So we get from (3.31)
(3.33)∑n=1∞(∑t=0j|Ct|(1+|n|)-(j+1)|αn11|∑m=1∞|Anm11|(m+n+12)t+|Anm12|(m+n)t)2≤G∑n=1∞exp(-ε24n1/2)(1+|n|)-2(j+1)<∞.
Secondly, using (3.30) and (3.31) we obtain that
(3.34)∑n=1∞2{[∑t=0j|αn11||Ct|(1+|n|)-(j+1)(n+12)t]×[∑t=0j|Ct||αn11|∑m=1∞(1+|n|)-(j+1)((m+n+12)t|Anm11|+(m+n)t|Anm12|)]}≤∑n=1∞[∑t=0j(1+|n|)-2(j+1)(n+12)texp(-ε24n1/2)G1/2]<∞,
and also the first part of the (3.31) obviously convergent so, we get from (3.33) and (3.34)
(3.35)∑n∈ℕ(1+|n|)-2(j+1)1(j!)2(∑t=0j|Ct{dtdλtfn(1)(zi)}|)2<∞,
and similarly
(3.36)∑n∈ℕ(1+|n|)-2(j+1)1(j!)2(∑t=0j|Dt{dtdλtfn(2)(zi)}|)2<∞.
Finally dj/(dλj)Vn(λi)∈H-(j+1)(ℕ), j=0,1,2,…,mi-1,  i=k+1,k+2,…,ν.



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*Source: 924628-2012-10-21.xml*