# Principal Functions of Non-Selfadjoint Sturm-Liouville Problems with Eigenvalue-Dependent Boundary Conditions

Authors: Nihal Yokuş
Journal: Abstract and Applied Analysis
 (2011)

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## Abstract

We consider the operatorL generated in L2(ℝ+) by the differential expression l(y)=−y″+q(x)y, x∈ℝ+:=[0,∞) and the boundary condition y′(0)/y(0)=α0+α1λ+α2λ2, where q is a complex-valued function and αi∈ℂ, i=0,1,2 with α2≠0. In this paper we obtain the properties of the principal functions corresponding to the spectral singularities of L.


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## Body

## 1. Introduction

LetT 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.
Ifλ=λ0 is a pole of the resolvent of T and λ0∈σc(T), but λ0∉σd(T), then λ0 is called a spectral singularity of T.Let us consider the nonselfadjoint operatorL0 generated in L2(ℝ+) by the differential expression(1.1)l0(y)=-y′′+q(x)y,x∈R+,
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 ofL0 [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 operatorL1 generated in L2(ℝ+) by (1.1) and the boundary condition(1.2)∫0∞K(x)y(x)dx+αy′(0)-βy(0)=0,
in which K∈L2(ℝ+) is a complex valued function and α,β∈ℂ, was investigated in detail by Krall [4–8] 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 [11–20]. 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.LetL denote the operator generated in L2(ℝ+) by the differential expression(1.3)l(y)=-y′′+q(x)y,x∈R+
and the boundary condition(1.4)y′(0)y(0)=α0+α1λ+α2λ2,
where q is a complex-valued function and αi∈ℂ, i=0,1,2 with α2≠0. 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 equation(2.1)-y′′+q(x)y=λ2y,x∈R+
related to the operator L.Now we will assume that the complex valued functionq is almost everywhere continuous in ℝ+ and satisfies the following: (2.2)∫0∞x|q(x)|dx<∞.
Let φ(x,λ) and e(x,λ) denote the solutions of (2.1) satisfying the conditions(2.3)φ(0,λ)=1,φ′(0,λ)=α0+α1λ+α2λ2,lim⁡x→∞e(x,λ)e-iλx=1,λ∈C¯+,
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 ℂ+:={λ:λ∈ℂ,Imλ>0} and continuous in ℂ¯+={λ:λ∈ℂ,Imλ≥0}.In addition, Jost solution has a representation ([22])(2.4)e(x,λ)=eiλx+∫x∞K(x,t)eiλtdt,λ∈C¯+,
where the kernel K(x,t) satisfies(2.5)K(x,t)=12∫(x+t)/2∞q(s)ds+12∫x(x+t)/2∫t+x-st+s-xq(s)K(s,u)duds+12∫(x+t)/2∞∫st+s-xq(s)K(s,u)duds
and K(x,t) is continuously differentiable with respect to its arguments. We also have(2.6)|K(x,t)|≤cw(x+t2),(2.7)|Kx(x,t)|,|Kt(x,t)|≤14|q(x+t2)|+cw(x+t2),
where w(x)=∫x∞|q(s)|ds and c>0 is a constant.Letê±(x,λ)  denote  the solutions of (2.1) subject to the conditions(2.8)lim⁡x→∞e±iλxê±(x,λ)=1,lim⁡x→∞e±iλxêx±(x,λ)=±iλ,λ∈C¯±.
Then(2.9)W[e(x,λ),ê±(x,λ)]=∓2iλ,λ∈C±,W[e(x,λ),e(x,-λ)]=-2iλ,λ∈R=(-∞,∞),
where W[f1,f2] is the Wronskian of f1 and f2, ([23]).We will denote the Wronskian of the solutionsφ(x,λ) with e(x,λ) and e(x,-λ) by E+(λ) and E-(λ), respectively, where(2.10)E+(λ):=e′(0,λ)-(α0+α1λ+α2λ2)e(0,λ),λ∈C¯+,E-(λ):=e′(0,-λ)-(α0+α1λ+α2λ2)e(0,-λ),λ∈C¯-,
and ℂ¯-={λ:λ∈ℂ,Imλ≤0}. Therefore E+ and E- are analytic in ℂ+ and ℂ-={λ:λ∈ℂ,Imλ<0}, respectively, and continuous up to real axis.The functionsE+ and E- are called Jost functions of L.


## 3. Eigenvalues and Spectral Singularities ofL

Let(3.1)G(x,t;λ)={G+(x,t;λ),λ∈C+,G-(x,t;λ),λ∈C-
be the Green function of L (obtained by the standard techniques), where(3.2)G+(x,t;λ)={-φ(t,λ)e(x,λ)E+(λ),0≤t≤x-φ(x,λ)e(t,λ)E+(λ),x≤t<∞G-(x,t;λ)={-φ(t,λ)e(x,-λ)E-(λ),0≤t≤x-φ(x,λ)e(t,-λ)E-(λ),x≤t<∞.
We will denote the set of eigenvalues and spectral singularities of L by σd(L) and σss(L), respectively. From (3.1)–(3.2)(3.3)σd(L)={λ:λ∈C+,E+(λ)=0}∪{λ:λ∈C-,E-(λ)=0},σss(L)={λ:λ∈R*,E+(λ)=0}∪{λ:λ∈R*,E-(λ)=0},
where ℝ*=ℝ∖{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 ℂ¯+  and ℂ¯-, respectively.Definition 3.1.
The multiplicity of zero of the functionE+(or E-) in ℂ¯+(or ℂ¯-) is called the multiplicity of the corresponding eigenvalue and spectral singularity of L.
We see from (2.9) that the functions(3.4)ψ+(x,λ)=Ê+(λ)2iλe(x,λ)-E+(λ)2iλê+(x,λ),λ∈C+,ψ-(x,λ)=Ê-(λ)2iλe(x,-λ)-E-(λ)2iλê-(x,λ),λ∈C-,ψ(x,λ)=E+(λ)2iλe(x,-λ)-E-(λ)2iλe(x,λ),λ∈R*
are the solutions of the boundary value problem
(3.5)-y′′+q(x)y=λ2y,x∈R+,y′(0)y(0)=α0+α1λ+α2λ2,
where
(3.6)Ê±(λ)=ê±′(0,λ)-(α0+α1λ+α2λ2)ê±(0,λ).
Now let us assume that
(3.7)q∈AC(R+),lim⁡x→∞q(x)=0,sup⁡x∈R+[eεx|q′(x)|]<∞,ε>0.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 ofL

In this section we assume that (3.7) holds. Let λ1,…,λj and λj+1,…,λk denote the zeros of E+ in ℂ+ and E- in ℂ- (which are the eigenvalues of L) with multiplicities m1,…,mj and mj+1,…,mk, respectively. It is obvious that from definition of the Wronskian(4.1){dndλnW[ψ+(x,λ),e(x,λ)]}λ=λp={dndλnE+(λ)}λ=λp=0
for n=0,1,…,mp-1,p=1,2,…,j, and(4.2){dndλnW[ψ-(x,λ),e(x,-λ)]}λ=λp={dndλnE-(λ)}λ=λp=0
for n=0,1,...,mp-1,p=j+1,...,k.Theorem 4.1.
The fallowing formulae:(4.3){∂n∂λnψ+(x,λ)}λ=λp=∑m=0nAm(λp){∂m∂λme(x,λ)}λ=λp,n=0,1,…,mp-1,p=1,2,…,j, where
(4.4)Am(λp)=(nm){∂n-m∂λn-mÊ+(λ)}λ=λp,(4.5){∂n∂λnψ-(x,λ)}λ=λp=∑m=0nBm(λp){∂m∂λme(x,-λ)}λ=λp,n=0,1,…,mp-1,p=j+1,…,k, where
(4.6)Bm(λp)=(nm){∂n-m∂λn-mÊ-(λ)}λ=λp
holds.Proof.
We will proceed by mathematical induction, we prove first (4.3). Let n=0. From (4.1) we get
(4.7)ψ+(x,λp)=a0(λp)⋅e(x,λp),
where a0(λp)≠0. Let us assume that for 1≤n0≤mp-2, (4.3) holds; that is,
(4.8){∂n0∂λn0ψ+(x,λ)}λ=λp=∑m=0n0Am(λp){∂m∂λme(x,λ)}λ=λp.
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
(4.9)[-d2dx2+q(x)-λ2]∂n∂λny(x,λ)=2λn∂n-1∂λn-1y(x,λ)+n(n-1)∂n-2∂λn-2y(x,λ).
Writing (4.9) for ψ+(x,λ) and e(x,λ), and using (4.8), we find
(4.10)[-d2dx2+q(x)-λ2]fn0+1(x,λp)=0,
where
(4.11)fn0+1(x,λp)={∂n0+1∂λn0+1ψ+(x,λ)}λ=λp-∑m=0n0+1Am(λp){∂m∂λme(x,λ)}λ=λp.
From (4.1) we have
(4.12)W[fn0+1(x,λp),e(x,λp)]={dn0+1dλn0+1W[ψ+(x,λ),e(x,λ)]}λ=λp=0.
Hence there exists a constant an0+1(λp) such that
(4.13)fn0+1(x,λp)=an0+1(λp)e(x,λp).
This shows that (4.3) holds for n=n0+1.
Similarly we can prove that (4.5) holds.Definition 4.2.
Letλ=λ0 be an eigenvalue of L. If the functions
(4.14)y0(x,λ0),y1(x,λ0),…,ys(x,λ0)
satisfy the equations
(4.15)l(y0)-λ0y0=0,l(yj)-λ0yj-yj-1=0,j=1,2,…,s,
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 ofL are defined similarly.Now using (4.3) and (4.5) define the functions(4.16)Un,p(x)={∂n∂λnψ+(x,λ)}λ=λp=∑m=0nAm(λp){∂m∂λme(x,λ)}λ=λp,n=0,1,…,mp-1,…p=1,2,…,jand(4.17)Un,p(x)={∂n∂λnψ-(x,λ)}λ=λp=∑m=0nBm(λp){∂m∂λme(x,-λ)}λ=λp,n=0,1,…,mp-1,p=j+1,…,k.Then forλ=λp,p=1,2,…,j,j+1,…,k,(4.18)l(U0,p)=0,l(U1,p)+11!∂∂λl(U0,p)=0,l(Un,p)+11!∂∂λl(Un-1,p)+12!∂2∂λ2l(Un-2,p)=0,n=2,3,…,mp-1,hold, wherel(u)=-u′′+q(x)u-λ2u 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,…,Ump-1,p are the associated functions of U0,p ([25, 26]).U0,p,U1,p,…,Ump-1,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(4.19)Un,p∈L2(R+),n=0,1,…,mp-1,p=1,2,…,j,j+1,…,k.Proof.
Let0≤n≤mp-1 and 1≤p≤j. Using (2.6) and (3.7) we obtain that
(4.20)|K(x,t)|≤ce-ε(x+t)/2.
From (2.4) we get
(4.21)|{∂n∂λne(x,λ)}λ=λp|≤xne-xIm⁡λp+c∫x∞tne-ε(x+t)/2e-tIm⁡λpdt,
where c>0 is a constant. Since Imλp>0 for the eigenvalues λp,p=1,…,j, of L, (4.21) implies that
(4.22){∂n∂λne(x,λ)}λ=λp∈L2(R+),n=0,1,…,mp-1,p=1,2,…,j.
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 0≤n≤mp-1 and j+1≤p≤k.Letμ1,…,μv, and μv+1,…,μl be the zeros of E+ and E- in ℝ*=ℝ∖{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:(4.23){∂n∂λnψ(x,λ)}λ=μp=∑m=0nCm(μp){∂m∂λme(x,λ)}λ=μp,n=0,1,…,np-1,p=1,2,…,v,where(4.24)Cm(μp)=-(nm){∂n-m∂λn-mE-(λ)}λ=μp,{∂n∂λnψ(x,λ)}λ=μp=∑m=0nDm(μp){∂m∂λme(x,-λ)}λ=μp,n=0,1,…,np-1,p=v+1,…,l,where(4.25)Dm(μp)=(nm){∂n-m∂λn-mE+(λ)}λ=μp.
Now define the generalized eigenfunctions and generalized associated functions corresponding to the spectral singularities of L by the following:(4.26)υn,p(x)={∂n∂λnψ(x,λ)}λ=μp=∑m=0nCm(μp){∂m∂λme(x,λ)}λ=μp,n=0,1,…,np-1,p=1,2,…,v,
(4.27)υn,p(x)={∂n∂λnψ(x,λ)}λ=μp=∑m=0nDm(μp){∂m∂λme(x,-λ)}λ=μp,n=0,1,…,np-1,p=v+1,…,l.Thenυn,p,n=0,1,…,np-1,p=1,2,…,v,v+1,…,l, also satisfy the equations analogous to (4.18).υ0,p,υ1,p,…,υnp-1,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(4.28)υn,p∉L2(R+),n=0,1,…,np-1,p=1,2,…,v,v+1,…,l.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 spaces(4.29)Hn={f:∫0∞(1+x)2n|f(x)|2dx<∞},n=1,2,…,H-n={g:∫0∞(1+x)-2n|g(x)|2dx<∞},n=1,2,…,
with(4.30)‖f‖n2=∫0∞(1+x)2n|f(x)|2dx;‖g‖-n2=∫0∞(1+x)-2n|g(x)|2dx,
respectively. Then(4.31)Hn+1⫋Hn⫋L2(R+)⫋H-n⫋H-(n+1),n=1,2,…,
and H-n is isomorphic to the dual of Hn.Theorem 4.5.
One has(4.32)υn,p∈H-(n+1),n=0,1,…,np-1,p=1,2,…,v,v+1,…,l.Proof.
From (2.4) we have
(4.33)∫0∞(1+x)-2(n+1)|(ix)neiμpx|2dx<∞,∫0∞(1+x)-2(n+1)|∫x∞(it)nK(x,t)eiμptdt|2dx<∞.
Using (4.26), (4.33) we obtain
(4.34)υn,p∈H-(n+1),n=0,1,…,np-1,p=1,2,…,v.
In a similar way, we find
(4.35)υn,p∈H-(n+1),n=0,1,…,np-1,p=v+1,…,l.Let us choosen0 so that(4.36)n0=max⁡{n1,n2…,nv,nv+1,…,nl}.By Theorem4.5 and (4.31) we get following theoremTheorem 4.6.
One has(4.37)υn,p∈H-n0,n=0,1,…,np-1,p=1,2,…,v,v+1,…,l.



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