## 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 expressionwhere the coefficients *p*_{n} and *q*_{n} are complex and Δ is the forward difference operator, i.e. Δ*x*_{n}=*x*_{n+1}−*x*_{n}. Properties of the so-called Hellinger–Nevanlinna *m*-function for the recurrence relation *Mx*_{n}=*λ**w*_{n}*x*_{n}, where the *w*_{n} 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.

## 1. Introduction

Brown *et al*. (1999) extend the work of Sims (1957) on the limit point, limit circle classification for the differential equation(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 *L*^{2}(*a*, *b*; *w*d*x*). 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 equation(1.2)where(1.3)(1.4)and Δ is the forward difference operator, i.e. . The coefficients are complex, as is the spectral parameter *λ*, and 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,It 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 that(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 in(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 and . Then for , where *a*≤*b*, we have the following summation by parts formula:(2.1)In addition, if equation (1.5) and the preceding conditions on the coefficients *p*_{n}, *q*_{n} and *w*_{n} are satisfied, then(2.2)Moreover,(2.3)and(2.4)where(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. Let(2.6)where denotes the closed convex hull (i.e. the smallest closed convex set containing the exhibited set). We shall assume that the set . It should also be noted that the complement in of the closed convex set *Q* can have one or two connected components.

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

The general theory will be subject to the condition(2.11)for some fixed . 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 set(2.12)Note that *α* is fixed and(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 , for some . 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 and be solutions of equation (1.2) which satisfy the following initial conditions:(3.1)where . 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,(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 condition(3.3)at *n*=*k*, where , it follows that on setting *z*=cot *β*,(3.4)

*For η satisfying equation* *(2.11)* *and* , *the function z*↦*f*(*λ*, *z*, *k*) *maps the half plane* *onto a closed disc D*_{k}(*λ*) *in* , *which has radius r*_{k}(*λ*) *equal to*(3.5)*and centre*(3.6)

First we set , so that equation (3.4) now becomes(3.7)The critical point of equation (3.7) is , and we require this point to be such that is negative. Upon calculation, we find thatwhere by equations (2.2) and (3.1) it follows that(3.8)Hence, by equations (2.7) and (2.11), as required. Therefore, when equation (2.11) is satisfied *z*↦*f*(*λ*, *z*, *k*) maps onto a closed disc *D*_{k}(*λ*). By the properties of this transformation, the centre *a*_{k}(*λ*) of the disc *D*_{k}(*λ*) corresponds to the reflection of the critical point in the imaginary axis. Therefore,Finally, the point is mapped onto a point on the circle *C*_{k}(*λ*) (bounding *D*_{k}(*λ*)). That is, . Therefore, using equation (3.2), we find that the radius *r*_{k}(*λ*) of *C*_{k}(*λ*) is given by ▪

*If m*<*k*, *then D _{k}*(

*λ*)⊂

*D*

_{m}(

*λ*).

*That is*,

*the discs D*

_{k}(

*λ*) (

*with boundary circle C*

_{k}(

*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 *ψ*_{k}(*l*)=*θ*_{k}+*lϕ*_{k}. Then the inverse function of equation (3.4) can be written asIt follows from theorem 3.1 that *l*=*l*(*λ*)∈*D*_{k}(*λ*) if and only if . 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 *l*∈*D*_{k}(*λ*) if and only if(3.9)By equations (2.7) and (2.10), the sum in the above inequality is positive. Therefore, if *m*<*k*:Hence, *D*_{k}(*λ*)⊂*D*_{m}(*λ*). That is, the discs *D _{k}*(

*λ*) are nested as

*k*→∞. ▪

*For* , *as k*→∞, *the discs D _{k}*(

*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 define(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))Moreover, for , an arbitrary solution *y*={*y*_{n}} of our difference equation (1.2) satisfies(3.11)if and only ifHowever, since Re[(*λ*−)e^{iη}]=−*δ*, it follows thatTherefore, for and , a solution *y*={*y*_{n}} of the difference equation (1.2) satisfies equation (3.11) if and only if(3.12)in which case(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* *and* , *the following distinct limit circle*, *limit point cases are possible*, *where cases I and II are sub-cases of the limit point case*.

*There exists a unique solution of equation*(1.2)*satisfying equation*(3.12),*and this is the only solution satisfying equation*(3.13).*There exists a unique solution of equation*(1.2)*satisfying equation*(3.12)*, but all solutions of equation*(1.2)*satisfy equation*(3.13).*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*.

*If**and**hold for some*,*then the same is true for all*.*If θ and ϕ (and hence all solutions of equation*(1.2))*satisfy equation*(3.12)*for some**(case III), then all solutions of equation*(1.2)*satisfy equation*(3.12)*(and hence also equation*(3.13)*) for all*.

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).

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 ▪

## 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 the function *m*(*λ*) defined on

*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* *for* .

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

*In case III*, *given any point m*_{0}∈*C*_{∞}(*λ*_{0}), , *there exists a function* , *which is analytic in* *and* . *Moreover*, *for any* , *there exists a function* *which is analytic in* *such that* .

The proof follows a similar argument to that in the work of Sims (1957). Consider any complex number, say *z*_{0}, such that . Now for any *k*, where *k*<*l*<∞, consider the sequenceSince , it follows that *f*(*λ*, *z*_{0}, *l*) is in *D*_{l}(*λ*) and, hence, is analytic for . Furthermore, we have that the centre, *a*_{k}(*λ*) (see equation (3.6)), and radius, *r*_{k}(*λ*) (see equation (3.5)), are continuous functions of *λ* (for ) and, hence, are uniformly bounded in every compact subset of . Therefore, sinceit follows that *f*(*λ*, *z*_{0}, *l*) is also uniformly bounded with respect to *l* in every compact subset of . Hence, for any compact subset *G* of , there exists a sequence such that, as *n*→∞, *f*(*λ*, *z*_{0}, *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 .

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

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

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 *z*_{i}(*λ*_{0}, *m*_{0}), i.e.For example, consider a point, say *m*_{0}, inside or on the limit circle *C*_{∞}(*λ*_{0}). Choosing *z*_{i}(*λ*_{0},*m*_{0}) will then give us an analytic *m*-function such that *m*(*λ*_{0})=*m*_{0}. ▪

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* *and define* , *where* *is either the limit point or a point in D*_{∞}(*λ*). *Then*,(4.1)*Note that in case I,* *equation* (4.1) *holds for all*

Let . If then *f*(*λ*, *z*, *k*) defined in §3 lies on the disc *D*_{k}(*λ*) and satisfies(4.2)and similarly for *λ*′. Combining the results of equation (4.2) above for *λ* and *λ*′ and with we have(4.3)where, by equations (2.3) and (3.1),(4.4)(4.5)(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),and similarly for *λ*′. Since , , equation (4.1) now follows from equations (4.3)–(4.6). ▪

*For all*(4.7)*This continues to hold for all**in case I*.*For a fixed*,*in cases II and III*,*we have*(4.8)*This defines a meromorphic function in**and has a pole at λ if and only if*

Let

*x*_{n}=*ψ*_{n}(*λ*) and*y*_{n}=*ψ*_{n}(*λ*′) be solutions of our difference equation (1.2) for . Then by equation (2.3) it follows that(4.9)Passing to the limit as*k*→∞ in equation (4.9) gives us the required result, since by lemma 4.2, and from equations (3.1) and (3.10):For the second part, note that in cases II and III both and . Therefore, equation (4.8) follows upon substituting into equation (4.7) above.

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

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

## 5. The operator

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

Note that since *ϕ*_{n} and *ψ*_{n} are solutions of equation (1.2), substitution of the function into our three-term recurrence relation equation (1.4) gives(5.3)

Note also that for , if *f*_{n} is supported away from infinity, then it follows from lemma 4.2 that(5.4)

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

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

Let *f*_{n,N}=0 for *n*≥*N*, *f*_{n,N}=*f*_{n} for *n*<*N* and *Ψ*_{n,N}=_{λ}*f*_{n,N}. Then, by equations (2.2) and (5.3), it follows that (with and *y*_{n}=*Ψ*_{n,N})by 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))Thus, for any *K*<*N*,As *N*→∞, *Ψ*_{n,N}→*Ψ*_{n}. 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 in equation (5.5), since then we obtain

*For* *and* ,(5.7)

Let *f*_{n,k}=0 for *n*≥*k* and *f*_{n,k}=*f*_{n} for *n*<*k*, so that as *k*→∞(5.8)where It should also be noted that(5.9)and(5.10)by equation (5.4). Therefore, again using equation (2.3) (with *x*_{n}=_{λ}f_{n} and *y*_{n}=*ψ*_{n}(*λ*′)), it follows thatas *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 we require(5.11)to act from into itself. That is, we require the sequences on the two sides of equation (5.11) to have the same index set, namely . However,and in order for condition (5.11) above to be satisfied, either of the following methods can be adopted.

We could identify the sequence and {*x*_{n} : *n*=−1, 0, 1, …; *x*_{−1}=0} and set *p*_{0}=*p*_{−1}+*p*_{0}. 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 that(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 to be the solution of *M*_{x}=*λwx*, which satisfies *ϕ*_{0}=1, so that in particular(5.13)The solution *ϕ* is therefore uniquely defined. Let now be determined by *Mθ*_{n}=*λw*_{n}*θ*_{n} (*n*≥1) with(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 *Mx*_{n}=*λx*_{n}*w*_{n}, it is sometimes helpful to think of *Mx*_{0}=*λx*_{0}*w*_{0} 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* , (*η*,)∈*S*, *be fixed and set*(5.15)

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

Since a finite sequence {

*u*_{n}}, i.e. one whose components*u*_{n}are eventually zero, clearly lies in , and the set of finite sequences is dense in it follows that is dense in .From equation (5.3),

_{λ}is a right inverse of . It is also a left inverse by the following argument. Again, by equation (5.3),But if*λ*is an eigenvalue of with eigenvectors*φ*, say, then from equation (5.3),*φ*=0. Hence,*λ*is not an eigenvalue of and . For and , we have

*In case I,*(5.16)*In cases II and III, D*_{1} *is the direct sum,*(5.17)*where* 〈.〉 *is the linear span*.

From its definition, we find that . Now for , let *u*∈*D*_{1} and set . Then ,With and using equations (5.13) and (5.14),Also, since (*M*−*λ*′*w*)*ϕ*(*λ*′)=0,Hence, [*v*−*u*, *ϕ*(*λ*′)]_{1}, so that (*v*−*u*) and *ϕ*(*λ*′) each satisfy (*M*−*λ*′*w*)*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 ,Therefore, in case I, since and the constant . In cases II and III, equation (5.17) follows if *u*∈*D*_{1}. Then, there exists such thatwhere and . ▪

## 6. Spectral properties of

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

*The resolvent set of the operator T is defined to be**and* *is called the spectrum of T*. *If we denote the spectrum of T by σ*(*T*), *then we can define the essential spectrum*, *σ*_{e}(*T*), *to be the complement in* *of the set**where 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 , (*η*,)∈*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, and for any , which follows from theorem 5.2 and the definition of the resolvent set.

*In case I,*(6.1)(6.2)*and in**consists of eigenvalues of finite geometric multiplicity*.*In cases II and III,*_{λ}is compact for any*and**consists only of isolated eigenvalues of finite algebraic multiplicity*(*in*).

(i) We have seen earlier that in case I theorem 5.2 continues to hold for all . Hence, lies in and equation (6.1) follows. For equation (6.2) we argue as follows. LetThen, (following from definition 5.1) we setwhere, for *n*≥*N*,Thus, satisfieswhereTherefore, rank since range lies in the linear span of {*e*_{0}, …, *e*_{N−1}}, where (*e*_{n})_{j}=*δ*_{nj} (the Kronecker delta). Furthermore,Hence, is a bounded operator of finite rank and therefore compact. Now we have to show thatis also compact. Let in . That is, for all ,We now haveand given *ε*>0, ∃*K* such that for *k*≥*K*Hence,which implies in as *k*→∞. That is, is compact. Hence, is compact *for all N* and so applying *Weyl's theorem*, see Edmunds & Evans (1987; p. 418), it follows that * _{λ}* and have the same essential spectrum. This means that if we define

*Q*

_{N}to be the closed convex set

*Q*, when the indices run from

*N*to ∞ (rather than 0 to ∞; see theorem 4.4), then has the essential spectrumSince this is true for all

*N*, equation (6.2) is proved. Furthermore, if we assume that , then from definition 6.1 of the essential spectrum, it follows that This impliesIf

*a*=0, then sinceBut the range is closed since is Fredholm. Thus, exists with domain and is closed. Now it follows from the Closed Graph Theorem that is bounded on . Thus, Therefore, if , we must haveThat is,

*λ*∈

*Q*\

*Q*

_{∞}is an eigenvalue of finite geometric multiplicity.

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

*Let equation* (1.2) *be in case I*. *In Q*\*Q*_{∞}, *the spectrum*, *consists only of isolated eigenvalues*. *These points are poles of the meromorphic extension of* .

As in the proof of theorem 4.4 of §4, for *n*≥*N*, let us define where _{N}(*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 :Now, assume that *λ*∈*Q*\*Q*_{∞} is such that the above meromorphic extension of is regular at *λ*. Since we are in case I, we have for and some constant *K*(*λ*), where . Also, by theorem 5.2 (applied to {*N*, …,∞}),is bounded on for *N* close to ∞ (so that *λ*∉*Q*_{N}). It should also be noted that for this *λ*∈*Q*\*Q*_{∞} it follows that equation (5.3) is satisfied by (_{λ}*f*)_{n}. Therefore, for any and some constant *K*_{1}(*λ*), we have thati.e. is bounded. Hence, . We shall subsequently show that is in fact analytic on so that any pole of in *Q*\*Q*_{∞} lies in . ▪

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

*For all* (*η*,)∈*S we have*(6.3)(6.4)(6.5)*where* (.,.) *denotes the inner product in* .

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 .

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

For equation (6.5) we let . Then, from the definition of *ψ*_{n}(*λ*), it follows that . Moreover, , which gives usas required. ▪

*For* (*η*,)∈*S and* *we define* (.) *on* *by*(6.6)*where*(6.7)

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

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

For the second part of the lemma we argue as follows. Multiplying both sides of equation (6.7) by , and summing over *n* from one to *N* (<∞), with respect to a weight function *w*_{n}, gives usHowever, by equations (5.4) and (6.7),Hence, if we denote (.,.) to be the inner product, and since it follows from equations (4.7), (5.2) and (6.6) thatas required. ▪

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

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 numbers(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 is a closed convex set. Its complement in has either one or two connected components. Furthermore, if *λ*∉*Γ*(*T*) it follows that nul(*T*−*λ*)=0 and def(*T*−*λ*)=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* . *Since we have that any λ*∉*Q lies in the resolvent set* of *then def* *for all* . *Hence*, *lies in the resolvent set* *i.e.* .

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

*If*|*p*_{n}|≤*k**min*(*w*_{n},*w*_{n+1}),*then**Since*, ,*it follows that equation**(6.11)**is satisfied*.*If*|*p*_{n}|≤*k**min*(*w*_{n},*w*_{n+1})*and**, then M is in case I. For suppose there exist linearly independent solutions u*,*v of*(*M*−*λ*)*y*=*0 for some**which are in**and that*1=[*u*,*v*]=*p*_{n}(*u*_{n+1}*v*_{n}−*u*_{n}*v*_{n+1})*,*.*Then, as in the previous remark,**from which it follows that**, which is a contradiction if*.

Therefore, in summary we have shown the following result.

*Suppose that* (*the strong limit point condition*) *equation* (6.11) *holds*. *Then for all* , . *Furthermore, if* |*p*_{n}|≤*k* min(*w*_{n}, *w*_{n+1}) *and* |*q*_{n}|≤*K then* *lies in B*(0, 4*k*+*K*). *Hence, the spectrum* *of our operator* *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)(7.1)where *b*_{n}=*p*_{n}+*p*_{n−1}+*q*_{n}. Furthermore, let , , , satisfy(7.2)and(7.3)(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. *ϕ*_{0}↦*ϕ*_{1}). Thus, we also have, for *Θ* and *Φ*:(7.5)For , the map *z*↦*F*(*λ*, *k*) has similar properties to the function *z*↦*f*(*λ*, *k*) of §3. Therefore, we define and *m* to be the Hellinger–Nevanlinna functions in terms of equations (7.3) and (7.4), respectively, and we let *Ψ*_{n}(*λ*)=*Θ*_{n}(*λ*)+*M*(*λ*)*Φ*_{n}(*λ*).

Now taking the coefficients of equation (7.1) to be constants, i.e. *p*_{n}=*p*, *q*_{n}=*q*, *w*_{n}=*w*, (*n*≥0), with *p*≠0, we have(7.6)It follows that for *n*≥1, if *Mx*_{n}=*λwx*_{n},(7.7)where 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 , where . The argument in Brown *et al*. (1993, §5) shows that is the unique (up to constant multiples) solution of , which lies in *l*^{2}. 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 , we have and from equations (7.3) and (7.4) it then follows that(7.8)Furthermore, by lemma 6.6, we know that the resolvent set of the operator defined by *M* (see definition 5.4) coincides with the set in in which (*λ*) is analytic. This now means that given properties of *m*(.) we can also determine properties of . Using equation (7.4) we find that(7.9)Therefore, *m*(*μ*) is analytic in ; hence *m*(.) is analytic outsideIf we let *λ*=*λ*_{1}+i*λ*_{2}, *b*=*b*_{1}+i*b*_{2} and *p*=*p*_{1}+i*p*_{2}, where *λ*_{1}, *λ*_{2}, *b*_{1}, *b*_{2}, *p*_{1} and *p*_{2} are all real constants, then it follows that (assuming *p*_{1}≠0) *m*(.) is analytic, except when(7.10)and(7.11)It should be noted that the spectrum of the operator is determined by the function (*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 (*λ*) is analytic, it is now possible to investigate the analytic nature of the function (*λ*). From equation (7.8), we haveHowever, from equation (7.9) we also know thatthus, (*λ*) is analytic for , except for poles, when(7.12)Equation (7.12) can now be solved for *μ* and hence a solution can be given in terms of *λ* (since ). We will consider the following two cases.

## 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.

- © 2005 The Royal Society