Non-self-adjoint difference operators and their spectrum

Robert Howard Wilson

Abstract

Initially, this paper is a discrete analogue of the work of Brown et al. (1999 Proc. R. Soc. A 455, 1235–1257) on second-order differential equations with complex coefficients. That is, we investigate the general non-self-adjoint second-order difference expressionEmbedded Imagewhere the coefficients pn and qn are complex and Δ is the forward difference operator, i.e. Δxn=xn+1xn. Properties of the so-called Hellinger–Nevanlinna m-function for the recurrence relation Mxn=λwnxn, where the wn are real and positive, are examined, and relationships between the properties of the m-function and the spectrum of the associated operator are explored. However, an essential difference between the continuous and the discrete case arises in the way in which we define the operator natural to the problem. Nevertheless, analogous results regarding the spectrum of this operator are obtained.

Keywords:

1. Introduction

Brown et al. (1999) extend the work of Sims (1957) on the limit point, limit circle classification for the differential equationEmbedded Image(1.1)under general separated boundary conditions, where p and q are both complex valued and w is a positive weight function. The general case illustrates how the classification of equation (1.1) involves a weighted Sobolev space (which is not apparent in the work of Sims) as well as L2(a, b; wdx). Furthermore, by defining operators natural to the problem, Brown et al. (1999) relate the spectral properties of these operators to the analytic properties of the associated m-function.

The first objective of this paper is to establish a limit circle, limit point theory of the Hellinger–Nevanlinna circles for the difference equationEmbedded Image(1.2)whereEmbedded Image(1.3)Embedded Image(1.4)and Δ is the forward difference operator, i.e. Embedded Image. The coefficients Embedded Image Embedded Image are complex, as is the spectral parameter λ, and Embedded Image is real. Thus a non-self-adjoint difference equation analogous to equation (1.1) is produced. Note that the forward difference operator Δ is linear and, unless otherwise stated,Embedded ImageIt is also worth noting here that the difference expression in the form (1.4) is referred to as a three-term recurrence relation. This is the form commonly used by Atkinson (1964) and Akhiezer (1965) in their respective investigations into the corresponding self-adjoint case. However, for the most part, we will use the form (1.3) so that comparisons can be made with the work of Brown et al. (1999). A notable exception to this is in the work of §7, where the form (1.4) is used.

In addition to the above conditions on the coefficients, we also assume thatEmbedded Image(1.5)Then, using a similar framework to the continuous case in setting up the problem, we shall see how, in the discrete case, the limit circle, limit point classification of equation (1.2) also has three distinct possibilities.

The second problem the paper examines is the relationship between the properties of the associated m-function and the spectrum of the underlying operator. This is a different but essentially related problem to the work of the general theory, and it is also where this paper differs significantly from the work of Brown et al. (1999). In order to define an operator acting inEmbedded Image(1.6)we set the initial value of the coefficient p, i.e. p−1, equal to zero. This is consistent with the approach of Akhiezer (1965) in defining the corresponding self-adjoint operator. Nevertheless, the nested circle analysis holds and analogous results to the continuous case regarding the spectrum of the operator are obtained. This work leads onto the main result of the paper, where investigation of the numerical range of our operator exposes further spectral properties which are not apparent in the continuous case.

Finally, we examine the specific example where the coefficients of our recurrence relation remain complex but constant. This essentially transforms the problem into a self-adjoint one. Thus, by implementing results from the self-adjoint case and relating them to the operator theory outlined above, we obtain information concerning the spectrum of the operator.

2. Description of the problem

Let Embedded Image and Embedded Image. Then for Embedded Image, where ab, we have the following summation by parts formula:Embedded Image(2.1)In addition, if equation (1.5) and the preceding conditions on the coefficients pn, qn and wn are satisfied, thenEmbedded Image(2.2)Moreover,Embedded Image(2.3)andEmbedded Image(2.4)whereEmbedded Image(2.5)The results (2.2)–(2.4) can be readily proved by induction and are analogous to those obtained in the theory of differential equations. In particular, equation (2.3) and (2.4) are analogues of the Green's formula. We continue with a similar construction to the continuous case and define the following set. LetEmbedded Image(2.6)where Embedded Image denotes the closed convex hull (i.e. the smallest closed convex set containing the exhibited set). We shall assume that the set Embedded Image. It should also be noted that the complement in Embedded Image of the closed convex set Q can have one or two connected components.

Taking Embedded Image, let Embedded Image be its unique nearest point in Q and let L=L0) denote any line that touches, but does not cross, the boundary of Q at Embedded Image. (In the continuous case L would be the tangent to Q at Embedded Image, for almost all boundary points Embedded Image of Q). We then perform a transformation of the complex plane Embedded Image and a rotation through an angle η=η(λ0)∈(−π, π], so that the image of L now coincides with the imaginary axis. Furthermore, the images of λ0 and the set Q now lie in the negative and non-negative half planes respectively. Therefore, Embedded Image (and 0<r<∞),Embedded Image(2.7)andEmbedded Image(2.8)With respect to some Embedded Image, and for such admissible values of Embedded Image and η, we define the half planeEmbedded Image(2.9)For different values of λ0 we get different values of Embedded Image and η. Also, for all Embedded Image,Embedded Image(2.10)where Embedded Image is the distance from λ to the boundary Embedded Image. It should also be noted that Embedded Image is the union of the half planes Embedded Image over the set S of admissible values of η and Embedded Image

The general theory will be subject to the conditionEmbedded Image(2.11)for some fixed Embedded Image. This is required to ensure that the radii of the Hellinger–Nevanlinna circles (established in §3) are real and non-negative. We also define the setEmbedded Image(2.12)Note that α is fixed andEmbedded Image(2.13)is also closed and convex. Furthermore, Q(α)⊇Q with equality satisfied if α satisfies equation (2.11) for all η such that Embedded Image, for some Embedded Image. It shall be shown that this is precisely the case when we define our operator in §5. Nevertheless, we continue initially by considering the general case.

3. Characteristics of the circles

Let Embedded Image and Embedded Image be solutions of equation (1.2) which satisfy the following initial conditions:Embedded Image(3.1)where Embedded Image. The left-hand side of equation (2.3) is now zero, and it follows from equation (3.1) that, with a=0 and b=n−1,Embedded Image(3.2)That is, θ and ϕ are linearly independent solutions of equation (1.2). By considering those solutions of equation (1.2), which satisfy the boundary conditionEmbedded Image(3.3)at n=k, where Embedded Image, it follows that on setting z=cot β,Embedded Image(3.4)

For η satisfying equation (2.11) and Embedded Image, the function zf(λ, z, k) maps the half plane Embedded Image onto a closed disc Dk(λ) in Embedded Image, which has radius rk(λ) equal toEmbedded Image(3.5)and centreEmbedded Image(3.6)

First we set Embedded Image, so that equation (3.4) now becomesEmbedded Image(3.7)The critical point of equation (3.7) is Embedded Image, and we require this point to be such that Embedded Image is negative. Upon calculation, we find thatEmbedded Imagewhere by equations (2.2) and (3.1) it follows thatEmbedded Image(3.8)Hence, by equations (2.7) and (2.11), Embedded Image as required. Therefore, when equation (2.11) is satisfied zf(λ, z, k) maps Embedded Image onto a closed disc Dk(λ). By the properties of this transformation, the centre ak(λ) of the disc Dk(λ) corresponds to the reflection of the critical point in the imaginary axis. Therefore,Embedded ImageFinally, the point Embedded Image is mapped onto a point on the circle Ck(λ) (bounding Dk(λ)). That is, Embedded Image. Therefore, using equation (3.2), we find that the radius rk(λ) of Ck(λ) is given byEmbedded Image ▪

If m<k, then Dk(λ)⊂Dm(λ). That is, the discs Dk(λ) (with boundary circle Ck(λ)) are nested as k→∞.

The method used to determine this result is analogous to that used in the work of Brown et al. (1999) in the continuous case. Accordingly, just a brief outline of the proof is given here.

Let ψk(l)=θk+k. Then the inverse function of equation (3.4) can be written asEmbedded ImageIt follows from theorem 3.1 that l=l(λ)∈Dk(λ) if and only if Embedded Image. Using equation (2.2) and the initial conditions (3.1) this can be written in the same form as equation (3.5), so that equivalently lDk(λ) if and only ifEmbedded Image(3.9)By equations (2.7) and (2.10), the sum in the above inequality is positive. Therefore, if m<k:Embedded ImageHence, Dk(λ)⊂Dm(λ). That is, the discs Dk(λ) are nested as k→∞. ▪

For Embedded Image, as k→∞, the discs Dk(λ) contract either to a disc D(λ) (with boundary circle C(λ)) or to a point m(λ). These represent the limit circle and limit point cases respectively.

Now we defineEmbedded Image(3.10)where m(λ) is either a point in D(λ) or the limit point otherwise. In the limit circle case the radius r(λ) has to be strictly positive, which means that (by equation (3.5))Embedded ImageMoreover, for Embedded Image, an arbitrary solution y={yn} of our difference equation (1.2) satisfiesEmbedded Image(3.11)if and only ifEmbedded ImageHowever, since Re[(λEmbedded Image)eiη]=−δ, it follows thatEmbedded ImageTherefore, for Embedded Image and Embedded Image, a solution y={yn} of the difference equation (1.2) satisfies equation (3.11) if and only ifEmbedded Image(3.12)in which caseEmbedded Image(3.13)This result enables us to give the following full characterization of our difference equation (1.2). Each of the three following cases are distinct, and the uniqueness referred to is up to constant multiples.

For Embedded Image and Embedded Image, the following distinct limit circle, limit point cases are possible, where cases I and II are sub-cases of the limit point case.

  1. There exists a unique solution of equation (1.2) satisfying equation (3.12), and this is the only solution satisfying equation (3.13).

  2. There exists a unique solution of equation (1.2) satisfying equation (3.12), but all solutions of equation (1.2) satisfy equation (3.13).

  3. All solutions of equation (1.2) satisfy equation (3.12) and hence satisfy equation (3.13).

Explicit examples of each of the three cases can be found for non symmetric Sturm–Liouville equations, as well as for difference equations, in the work of Bennewitz & Brown (2002).

The next theorem in this section shows that the classification of our difference equation (1.2) into case I, II, or III is independent of the choice of λ.

  1. If Embedded Image and Embedded Image hold for some Embedded Image, then the same is true for all Embedded Image.

  2. If θ and ϕ (and hence all solutions of equation (1.2)) satisfy equation (3.12) for some Embedded Image (case III), then all solutions of equation (1.2) satisfy equation (3.12) (and hence also equation (3.13)) for all Embedded Image.

  1. The proof applies the method of variation of parameters and a general stability theorem as in the self-adjoint case. For the full argument see Atkinson (1964; §5.6).

  2. The result follows a similar argument to part (i), except that now we begin by assuming that all solutions of equation (1.2) satisfy equation (3.12) for some Embedded Image ▪

4. Analytic properties and representations of the m-function

Here we take a closer look at some of the analytic properties of the m-function(s) introduced in the previous section, and we apply these results to obtain various formulae for the m-function. These often help us to determine a number of the function's properties. Note, however, how its representation often depends on which of the three cases we are in. We shall denote by Embedded Image the function m(λ) defined on Embedded Image

If the conditions of theorem 3.1 are satisfied, then in each of the three cases defined in §3, there exists an analytic m-function Embedded Image for Embedded Image.

In cases I and II, Embedded Image is regular throughout Embedded Image. Furthermore, in case I, m(λ) is well defined on each of the possible two connected components of Embedded Image

In case III, given any point m0C(λ0), Embedded Image, there exists a function Embedded Image, which is analytic in Embedded Image and Embedded Image. Moreover, for any Embedded Image, there exists a function Embedded Image which is analytic in Embedded Image such that Embedded Image.

The proof follows a similar argument to that in the work of Sims (1957). Consider any complex number, say z0, such that Embedded Image. Now for any k, where k<l<∞, consider the sequenceEmbedded ImageSince Embedded Image, it follows that f(λ, z0, l) is in Dl(λ) and, hence, is analytic for Embedded Image. Furthermore, we have that the centre, ak(λ) (see equation (3.6)), and radius, rk(λ) (see equation (3.5)), are continuous functions of λ (for Embedded Image) and, hence, are uniformly bounded in every compact subset of Embedded Image. Therefore, sinceEmbedded Imageit follows that f(λ, z0, l) is also uniformly bounded with respect to l in every compact subset of Embedded Image. Hence, for any compact subset G of Embedded Image, there exists a sequence such that, as n→∞, f(λ, z0, n) converges to a limit m(λ) that is analytic in G (by Vitali's theorem; see Titchmarsh (1939)). Moreover, since G is arbitrary it follows that m(λ) is analytic in Embedded Image.

For the second part of the theorem we need to show that in case I Embedded Image if Embedded Image. Note that in this case, for Embedded Image is the unique solution of equation (1.2) in Embedded Image. Therefore, for some constant, say A(λ), we have Embedded Image, and using the initial conditions (3.1), we find that Embedded Image.

In case I, we define m(λ) on each connected component of Embedded Image by Embedded Image, for Embedded Image. If Embedded Image has two connected components, say Embedded Image, and m(1), m(2) are the m-functions defined on them, we setEmbedded Image

In case III, m(λ) is not uniquely defined. Indeed, there are as many analytic m-functions as there are points on (or within) a limit circle. These m-functions are obtained by a similar limiting process as used above, corresponding to different choices of zi(λ0, m0), i.e.Embedded ImageFor example, consider a point, say m0, inside or on the limit circle C(λ0). Choosing zi(λ0,m0) will then give us an analytic m-function such that m(λ0)=m0. ▪

The following is a discrete analogue of a lemma in Titchmarsh (1962); note also that the coefficients of the corresponding difference equation are now considered complex.

Let Embedded Image and define Embedded Image, where Embedded Image is either the limit point or a point in D(λ). Then,Embedded Image(4.1)Note that in case I, equation (4.1) holds for all Embedded Image

Let Embedded Image. If Embedded Image then f(λ, z, k) defined in §3 lies on the disc Dk(λ) and Embedded Image satisfiesEmbedded Image(4.2)and similarly for λ′. Combining the results of equation (4.2) above for λ and λ′ and with Embedded Image we haveEmbedded Image(4.3)where, by equations (2.3) and (3.1),Embedded Image(4.4)Embedded Image(4.5)Embedded Image(4.6)In cases II and III, equations (4.4)–(4.6) are bounded and hence equation (4.1) follows on selecting the f(λ, z, k) to be such that f(λ,k)→m(λ), f(λ′, k)→m(λ′) as k→∞. In case I, by equations (2.7), (2.10), (2.11) and (3.5),Embedded Imageand similarly for λ′. Since Embedded Image, Embedded Image, equation (4.1) now follows from equations (4.3)–(4.6). ▪

  1. For all Embedded ImageEmbedded Image(4.7)This continues to hold for all Embedded Image in case I.

  2. For a fixed Embedded Image, in cases II and III, we haveEmbedded Image(4.8)This defines a meromorphic function in Embedded Image and has a pole at λ if and only ifEmbedded Image

  1. Let xn=ψn(λ) and yn=ψn(λ′) be solutions of our difference equation (1.2) for Embedded Image. Then by equation (2.3) it follows thatEmbedded Image(4.9)Passing to the limit as k→∞ in equation (4.9) gives us the required result, since Embedded Image by lemma 4.2, and from equations (3.1) and (3.10):Embedded Image

  2. For the second part, note that in cases II and III both Embedded Image and Embedded Image. Therefore, equation (4.8) follows upon substituting Embedded Image into equation (4.7) above.

 ▪

Let equation (1.2) be in case I and define the setsEmbedded Image(4.10)Embedded Image(4.11)Note QN is the set Q where the sequences run from N torather than 0 to ∞; hence the set QN(α) is defined in similar way to Q(α) in equation (2.13). Now m(λ) is defined throughout Embedded Image, and has a meromorphic extension to Embedded Image with poles only in Q(α)\Q(α).

For nN, we defineEmbedded Image(4.12)where mN(λ) denotes the limit point and N replaces 0 in the initial conditions (3.1). Note that, by theorem 4.1, mN(.) is analytic throughout each of the two possible connected components of Embedded Image. Furthermore, equation (4.12) can be uniquely extended to Embedded Image with Embedded Image and Embedded Image analytic in Embedded Image for fixed n. We know that in case I there exists a unique solution of equation (3.12) in Embedded Image (up to constant multiples) say, ψn(λ). Therefore, there exists a constant A(λ) such thatEmbedded Image(4.13)Embedded Image(4.14)Substituting A(λ) from equations (4.13) into equation (4.14) and applying the initial conditions (3.1) yieldsEmbedded Image(4.15)giving us m(λ) as a meromorphic function in Embedded Image with isolated poles at the zeros of the denominator. ▪

5. The operator Embedded Image

The aim of this section is to define operators generated by the non-self-adjoint difference equation (1.2). Some preliminary results are first required. We begin with the following definition.

For Embedded Image, letEmbedded Image(5.1)where Embedded Image and Embedded Image are solutions of equation (1.2), which satisfy equations (3.1) and (3.10), respectively. Then, for Embedded Image we define:Embedded Image(5.2)

Note that since ϕn and ψn are solutions of equation (1.2), substitution of the function Embedded Image into our three-term recurrence relation equation (1.4) givesEmbedded Image(5.3)

Note also that for Embedded Image, if fn is supported away from infinity, then it follows from lemma 4.2 thatEmbedded Image(5.4)

Lemma 5.3 allows us to state that in case I equation (5.4) holds for all Embedded Image, and hence in all cases, since in cases II and III the above sum is bounded as n→∞ if Embedded Image and also Embedded Image is zero by lemma 4.2. However, we first require the following essential theorem.

Let Embedded Image and Embedded Image. Then, in every case with Embedded Image and for any ϵ>0,Embedded Image(5.5)In particular Ψ(λ) is bounded andEmbedded Image(5.6)where ‖.‖ denotes the Embedded Image norm.

Let fn,N=0 for nN, fn,N=fn for n<N and Ψn,N=Embedded Imageλfn,N. Then, by equations (2.2) and (5.3), it follows that (with Embedded Image and yn=Ψn,N)Embedded Imageby equation (3.1) and again using equation (2.2). Therefore (as a result of equation (3.9) of the nesting property and equation (2.11))Embedded ImageThus, for any K<N,Embedded ImageAs N→∞, Ψn,NΨn. Hence, equation (5.5) follows by first letting N→∞ and then K→∞. The second part of the theorem follows from equations (2.7) and (2.10), and by setting Embedded Image in equation (5.5), since then we obtainEmbedded Image

For Embedded Image and Embedded Image,Embedded Image(5.7)

Let fn,k=0 for nk and fn,k=fn for n<k, so that as k→∞Embedded Image(5.8)where Embedded Image It should also be noted thatEmbedded Image(5.9)andEmbedded Image(5.10)by equation (5.4). Therefore, again using equation (2.3) (with xn=Embedded Imageλfn and yn=ψn(λ′)), it follows thatEmbedded Imageas N→∞ (by equation (5.10)). This in turn tends to zero as k→∞ by equations (5.8) and (5.9). ▪

We can now proceed to define the operator associated with our difference equation (1.2). It should be noted that in order to define operators acting in Embedded Image we requireEmbedded Image(5.11)to act from Embedded Image into itself. That is, we require the sequences on the two sides of equation (5.11) to have the same index set, namely Embedded Image. However,Embedded Imageand in order for condition (5.11) above to be satisfied, either of the following methods can be adopted.

We could identify the sequence Embedded Image and {xn : n=−1, 0, 1, …; x−1=0} and set p0=p−1+p0. This is similar to the method adopted by Atkinson (1964; §6.4) when investigating the corresponding self-adjoint operator.

The alternative method, and the one that we will adopt, is to set p−1=0. It then follows thatEmbedded Image(5.12)This approach can be seen in the work of Akhiezer (1965) again in his investigations into the self-adjoint case.

We now choose Embedded Image to be the solution of Mx=λwx, which satisfies ϕ0=1, so that in particularEmbedded Image(5.13)The solution ϕ is therefore uniquely defined. Let Embedded Image now be determined by n=λwnθn (n≥1) withEmbedded Image(5.14)It should be noted that the first term in equation (3.5) no longer appears and as a result the condition (2.11) in §2 is redundant. Hence, the set Q(α) is replaced by the set Q in the theory. In addition, [θ, ϕ]n=[θ, ϕ]1=−1, so that θ, ϕ are linearly independent. Also, the nested sequence analysis of §3 continues to hold with the initial conditions (5.13) and (5.14) instead of equation (3.1).

For solutions of Mxn=λxnwn, it is sometimes helpful to think of Mx0=λx0w0 as a λ dependent boundary condition which, in view of equations (5.13) and (5.14), is satisfied by ϕ but not by θ.

Let Embedded Image, (η,Embedded Image)∈S, be fixed and setEmbedded Image(5.15)

The following details should be noted here.

  1. From the preceding argument, no left-hand boundary condition is required in the definition of Embedded Image

  2. Since a finite sequence {un}, i.e. one whose components un are eventually zero, clearly lies in Embedded Image, and the set of finite sequences is dense in Embedded Image it follows that Embedded Image is dense in Embedded Image.

  3. From equation (5.3), Embedded Imageλ is a right inverse of Embedded Image. It is also a left inverse by the following argument. Again, by equation (5.3),Embedded ImageBut if λ is an eigenvalue of Embedded Image with eigenvectors φ, say, then from equation (5.3), φ=0. Hence, λ is not an eigenvalue of Embedded Image and Embedded Image. For Embedded Image and Embedded Image, we haveEmbedded Image

In case I,Embedded Image(5.16)In cases II and III, D1 is the direct sum,Embedded Image(5.17)where 〈.〉 is the linear span.

From its definition, we find that Embedded Image. Now for Embedded Image, let uD1 and set Embedded Image. Then Embedded Image,Embedded ImageWith Embedded Image and using equations (5.13) and (5.14),Embedded ImageAlso, since (Mλw)ϕ(λ′)=0,Embedded ImageHence, [vu, ϕ(λ′)]1, so that (vu) and ϕ(λ′) each satisfy (Mλw)y=0 and the initial condition [y, ϕ(λ′)]1=0. It follows from the existence and uniqueness theorem for difference equations that for some constant Embedded Image,Embedded ImageTherefore, in case I, since Embedded Image and Embedded Image the constant Embedded Image. In cases II and III, equation (5.17) follows if uD1. Then, there exists Embedded Image such thatEmbedded Imagewhere Embedded Image and Embedded Image. ▪

6. Spectral properties of Embedded Image

Following the definition of our operator Embedded Image, we now seek to establish some spectral properties.

The resolvent set of the operator T is defined to beEmbedded Imageand Embedded Image is called the spectrum of T. If we denote the spectrum of T by σ(T), then we can define the essential spectrum, σe(T), to be the complement in Embedded Image of the setEmbedded Imagewhere the map T is Fredholm if its range, R(T), is closed and its nullity, nul T, and deficiency, def T, are finite.

Firstly, we note that, for any Embedded Image, (η,Embedded Image)∈S, the operator defined in equation (5.15) or (5.16) in case I is J-self-adjoint and quasi-m-accretive (see Edmunds & Evans 1987, ch. III). Moreover, Embedded Image and for any Embedded Image, Embedded Image which follows from theorem 5.2 and the definition of the resolvent set.

  1. In case I,Embedded Image(6.1)Embedded Image(6.2)and in Embedded Image consists of eigenvalues of finite geometric multiplicity.

  2. In cases II and III, Embedded Imageλ is compact for any Embedded Image and Embedded Image consists only of isolated eigenvalues of finite algebraic multiplicity (in Embedded Image).

(i) We have seen earlier that in case I theorem 5.2 continues to hold for all Embedded Image. Hence, Embedded Image lies in Embedded Image and equation (6.1) follows. For equation (6.2) we argue as follows. LetEmbedded ImageThen, (following from definition 5.1) we setEmbedded Imagewhere, for nN,Embedded ImageThus, Embedded Image satisfiesEmbedded ImagewhereEmbedded ImageTherefore, rankEmbedded Image since rangeEmbedded Image lies in the linear span of {e0, …, eN−1}, where (en)j=δnj (the Kronecker delta). Furthermore,Embedded ImageHence, Embedded Image is a bounded operator of finite rank and therefore compact. Now we have to show thatEmbedded Imageis also compact. Let Embedded Image in Embedded Image. That is, for all Embedded Image,Embedded ImageWe now haveEmbedded Imageand given ε>0, ∃K such that for kKEmbedded ImageHence,Embedded Imagewhich implies Embedded Image in Embedded Image as k→∞. That is, Embedded Image is compact. Hence, Embedded Image is compact for all N and so applying Weyl's theorem, see Edmunds & Evans (1987; p. 418), it follows that Embedded Imageλ and Embedded Image have the same essential spectrum. This means that if we define QN to be the closed convex set Q, when the indices run from N to ∞ (rather than 0 to ∞; see theorem 4.4), then Embedded Image has the essential spectrumEmbedded ImageSince this is true for all N, equation (6.2) is proved. Furthermore, if we assume that Embedded Image, then from definition 6.1 of the essential spectrum, it follows that Embedded Image This impliesEmbedded ImageIf a=0, then Embedded Image sinceEmbedded ImageBut the range is closed since Embedded Image is Fredholm. Thus, Embedded Image exists with domain Embedded Image and Embedded Image is closed. Now it follows from the Closed Graph Theorem that Embedded Image is bounded on Embedded Image. Thus, Embedded Image Therefore, if Embedded Image, we must haveEmbedded ImageThat is, λQ\Q is an eigenvalue of finite geometric multiplicity.

In cases II and III, the compactness of Embedded Imageλ, for Embedded Image follows, since Embedded Imageλ is Hilbert–Schmidt. Hence, in cases II and III, Embedded Image consists of isolated eigenvalues (in Embedded Image) having finite algebraic multiplicity. ▪

Let equation (1.2) be in case I. In Q\Q, the spectrum, Embedded Image consists only of isolated eigenvalues. These points are poles of the meromorphic extension of Embedded Image.

As in the proof of theorem 4.4 of §4, for nN, let us define Embedded Image where Embedded ImageN(λ) denotes the limit point and N replaces 0 in the initial conditions for the operator problems (5.13) and (5.14). Using the same method as in the proof of theorem 4.4, but with the initial conditions (5.13) and (5.14), we obtain the following meromorphic extension of Embedded Image:Embedded ImageNow, assume that λQ\Q is such that the above meromorphic extension of Embedded Image is regular at λ. Since we are in case I, we have Embedded Image for Embedded Image and some constant K(λ), where Embedded Image. Also, by theorem 5.2 (applied to {N, …,∞}),Embedded Imageis bounded on Embedded Image for N close to ∞ (so that λ∉QN). It should also be noted that for this λQ\Q it follows that equation (5.3) is satisfied by (Embedded Imageλf)n. Therefore, for any Embedded Image and some constant K1(λ), we have thatEmbedded Imagei.e. Embedded Image is bounded. Hence, Embedded Image. We shall subsequently show that Embedded Image is in fact analytic on Embedded Image so that any pole of Embedded Image in Q\Q lies in Embedded Image. ▪

Before proceeding to prove that Embedded Image is in fact analytic on Embedded Image, we require the following result.

For all Embedded Image (η,Embedded Image)∈S we haveEmbedded Image(6.3)Embedded Image(6.4)Embedded Image(6.5)where (.,.) denotes the inner product in Embedded Image.

It can be readily verified that corollary 4.3 still holds for the new initial conditions (5.14) and (6.5). Hence, by substituting equation (6.5) into equation (4.7) the identity (6.3) follows, since Embedded Image.

From equation (3.10), for Embedded Image, we have Embedded Image. Therefore, equation (6.4) follows upon substitution of the initial conditions (5.13) and (5.14).

For equation (6.5) we let Embedded Image. Then, from the definition of ψn(λ), it follows that Embedded Image. Moreover, Embedded Image, which gives usEmbedded Imageas required. ▪

For Embedded Image (η,Embedded Image)∈S and Embedded Image we define Embedded Image(.) on Embedded Image byEmbedded Image(6.6)whereEmbedded Image(6.7)

Let Embedded Image (η,Embedded Image)∈S and define Embedded Image by equation (6.6). ThenEmbedded Image(6.8)Moreover, equations (4.7) and (6.3) hold for all Embedded Image It follows that Embedded Image is analytic on Embedded Image and in cases II and III, equations (4.8) and (6.6) define the same meromorphic extension of Embedded Image. In case I, equation (6.6) defines the same meromorphic extension to Embedded Image as that in equation (4.15).

Since we have Embedded Image, it follows that, for constants A and B,Embedded Image(6.9)Using equations (5.13) and (5.14) and definition 6.5 it can be shown that A=1 and B=Embedded Image(λ), as required.

For the second part of the lemma we argue as follows. Multiplying both sides of equation (6.7) by Embedded Image, and summing over n from one to N (<∞), with respect to a weight function wn, gives usEmbedded ImageHowever, by equations (5.4) and (6.7),Embedded ImageHence, if we denote (.,.) to be the Embedded Image inner product, and since Embedded Image it follows from equations (4.7), (5.2) and (6.6) thatEmbedded Imageas required. ▪

These results allow us to say more about our operator within the set Q. If we define, for Embedded Image and Embedded ImageEmbedded ImageandEmbedded Imagethen for Embedded Image (Embedded Image, in cases II and III) we have Embedded Image. Moreover, it follows from equation (6.6) and the previous lemma that in cases II and III, λ is a pole of Embedded Image if and only if λ is an eigenvalue of Embedded Image.

It should be noted here that, up to this point, we have only been able to state that the spectrum of our operator lies within the unbounded set Q. However, we can now go on to show that under certain conditions the spectrum can be restricted to a bounded set. Let the numerical range Π(T) of the operator T be the set of complex numbersEmbedded Image(6.10)In general, Π(T) is neither open nor closed, even when T is a closed or bounded operator. However, it is convex. Therefore, the closure Γ(T) of Π(T) in Embedded Image is a closed convex set. Its complement in Embedded Image has either one or two connected components. Furthermore, if λ∉Γ(T) it follows that nul(Tλ)=0 and def(Tλ)=const. in each connected component. If one of these constants is zero, then the associated connected component lies in the resolvent set of the operator T (see Edmunds & Evans (1987); theorem III 2.3).

Suppose there exists only one connected component of Embedded Image. Since we have that any λ∉Q lies in the resolvent set Embedded Image of Embedded Image then defEmbedded Image for all Embedded Image. Hence, Embedded Image lies in the resolvent set Embedded Image i.e. Embedded Image.

Further investigation of the numerical range reveals the following. For Embedded Image and p−1=0, suppose thatEmbedded Image(6.11)This is the so-called strong limit point condition when the coefficients are real (see Brown et al. (1993)). Then,Embedded Image(6.12)where ∑′ indicates that un≠0. Thus, from equation (6.10), the numerical range of Embedded Image is now the convex setEmbedded ImageFrom the theory of §2, for all Embedded Image, 0<r<∞ and Embedded Image,Embedded Image(6.13)by equation (2.8). Therefore, for all Embedded Image and Embedded Image,Embedded ImageIt follows (along with remark 6.7) that Embedded Image Furthermore, considerEmbedded Image(6.14)If Embedded Image, we haveEmbedded ImageDenoting the operator of multiplication by qn as Embedded Image, thenEmbedded Imagewhere Embedded Image If Embedded Image, then Embedded Image and Embedded Image, the disc centre of the origin and radius K. Hence, if Embedded Image (which is shown to imply equation (6.11) in remark 6.8 below), it follows from equation (6.14) that Embedded Image. In particular, this means that Embedded Image is bounded. Indeed, it lies in B(0, 4k+K).

  1. If |pn|≤kmin(wn, wn+1), thenEmbedded ImageSince, Embedded Image, it follows that equation (6.11) is satisfied.

  2. If |pn|≤kmin(wn, wn+1) and Embedded Image, then M is in case I. For suppose there exist linearly independent solutions u, v of (Mλ)y=0 for some Embedded Image which are in Embedded Image and that 1=[u, v]=pn(un+1vnunvn+1), Embedded Image. Then, as in the previous remark,Embedded Imagefrom which it follows that Embedded Image, which is a contradiction if Embedded Image.

Therefore, in summary we have shown the following result.

Suppose that (the strong limit point condition) equation (6.11) holds. Then for all Embedded Image, Embedded Image. Furthermore, if |pn|≤k min(wn, wn+1) and |qn|≤K then Embedded Image lies in B(0, 4k+K). Hence, the spectrum Embedded Image of our operator Embedded Image lies within a bounded set.

7. Constant coefficients case

Finally, we shall examine the spectral theory of our operator by considering the specific case where the coefficients of our difference equation are constant. Again, let us define (with p−1=0)Embedded Image(7.1)where bn=pn+pn−1+qn. Furthermore, let Embedded Image, Embedded Image, Embedded Image, Embedded Image satisfyEmbedded Image(7.2)andEmbedded Image(7.3)Embedded Image(7.4)Note that equations (7.3) are exactly the conditions considered in the operator theory of §5. Clearly, conditions (7.4) satisfy the work §3 since they are obtained on taking α=0 in equation (3.1) and performing a shift of index (e.g. ϕ0ϕ1). Thus, we also have, for Θ and Φ:Embedded Image(7.5)For Embedded Image, the map zF(λ, k) has similar properties to the function zf(λ, k) of §3. Therefore, we define Embedded Image and m to be the Hellinger–Nevanlinna functions in terms of equations (7.3) and (7.4), respectively, and we let Ψn(λ)=Θn(λ)+M(λ)Φn(λ).

Now taking the coefficients of equation (7.1) to be constants, i.e. pn=p, qn=q, wn=w, (n≥0), with p≠0, we haveEmbedded Image(7.6)It follows that for n≥1, if Mxn=λwxn,Embedded Image(7.7)where Embedded Image Therefore, making the coefficients constant has effectively transformed the problem into the self-adjoint case, since now all the complex elements have been absorbed into the parameter μ. We find that the recurrence relation (7.7) has two linearly independent solutions Embedded Image, where Embedded Image. The argument in Brown et al. (1993, §5) shows that Embedded Image is the unique (up to constant multiples) solution of Embedded Image, which lies in l2. Therefore, M is in the limit point case if the coefficients are real, and in case I if they are complex.

Since we are in the limit point case (i.e. case I), for some Embedded Image, we have Embedded Image and from equations (7.3) and (7.4) it then follows thatEmbedded Image(7.8)Furthermore, by lemma 6.6, we know that the resolvent set of the operator Embedded Image defined by M (see definition 5.4) coincides with the set in Embedded Image in which Embedded Image(λ) is analytic. This now means that given properties of m(.) we can also determine properties of Embedded Image. Using equation (7.4) we find thatEmbedded Image(7.9)Therefore, m(μ) is analytic in Embedded Image; hence m(.) is analytic outsideEmbedded ImageIf we let λ=λ1+iλ2, b=b1+ib2 and p=p1+ip2, where λ1, λ2, b1, b2, p1 and p2 are all real constants, then it follows that (assuming p1≠0) m(.) is analytic, except whenEmbedded Image(7.10)andEmbedded Image(7.11)It should be noted that the spectrum of the operator Embedded Image is determined by the function Embedded Image(λ), since this is the function determined by the initial conditions natural to the operator theory of §5. However, having considered the situation when the function Embedded Image(λ) is analytic, it is now possible to investigate the analytic nature of the function Embedded Image(λ). From equation (7.8), we haveEmbedded ImageHowever, from equation (7.9) we also know thatEmbedded Imagethus, Embedded Image(λ) is analytic for Embedded Image, except for poles, whenEmbedded Image(7.12)Equation (7.12) can now be solved for μ and hence a solution can be given in terms of λ (since Embedded Image). We will consider the following two cases.

  1. Special case: Embedded Image. In this example the relationship (7.12) becomesEmbedded Imagewhich gives us Embedded Image. Thus, Embedded Image(λ) has a simple pole at Embedded Image.

  2. General case: Embedded Image. Here equation (7.12) holds if and only ifEmbedded ImageEmbedded Imagewhere b0=p+q. Thus, in the general case, Embedded Image(λ) has a simple pole whenEmbedded Image

Acknowledgements

I would sincerely like to thank the referees for their detailed comments regarding the paper. A great deal of appreciation also goes to Professor W. D. Evans for his time and continual support.

Footnotes

    • Received May 20, 2003.
    • Accepted October 4, 2004.

References

View Abstract