## Abstract

We give a characterization of the spectral properties of linear differential operators with constant coefficients, acting on functions defined on a bounded interval, and determined by general linear boundary conditions. The boundary conditions may be such that the resulting operator is not selfadjoint. We associate the spectral properties of such an operator *S* with the properties of the solution of a corresponding boundary value problem for the partial differential equation ∂_{t}*q*±i*Sq*=0. Namely, we are able to establish an explicit correspondence between the properties of the family of eigenfunctions of the operator, and in particular, whether this family is a basis, and the existence and properties of the unique solution of the associated boundary value problem. When such a unique solution exists, we consider its representation as a complex contour integral that is obtained using a transform method recently proposed by Fokas and one of the authors. The analyticity properties of the integrand in this representation are crucial for studying the spectral theory of the associated operator.

## 1. Introduction

In this paper, we study the following two objects:

(1) A linear constant-coefficient differential operator

*S*defined on a domain of the form sufficiently smooth and satisfying*n*prescribed boundary conditions}.(2) An initial-boundary value problem (IBVP) for the linear evolution partial differential equation (PDE)

*q*_{t}(*x*,*t*)±i*Sq*(*x*,*t*)=0,*x*∈(0,1)*t*∈(0,*T*), with*S*as in (1), an initial condition*q*(*x*,0)=*q*_{0}(*x*) and*n*given boundary conditions.

The boundary conditions, assumed to be linear, can be prescribed at either end of the interval [0,1] or can couple the two ends.

It is to be expected that the objects (1) and (2) are closely related. For each of these objects, it is natural to formulate a basic question, whose answer depends on the specific boundary conditions. Namely, given a set of *n* boundary conditions,

(Q1) Does the resulting operator

*S*admit a basis of eigenfunctions, in any appropriate sense?(Q2) Does the resulting IBVP admit a unique solution representable by a discrete series expansion in the eigenfunctions of

*S*?

Although it should be clear that these are the same question posed in different contexts, very little is explicitly known beyond the classical cases when the spatial operator has a known basis of eigenfunctions. This basis can be used after the separation of variables to express the solution of the boundary value problem.

In this paper, we give an explicit connection between the two problems in general; we give a link between the solutions of (1) and (2), and we show precisely how the answer to (Q1) and (Q2) is related. In particular, the rigorous answer to one question can be given through answering the other. Our results are true for general *n*, however they are new and interesting in particular for *n* odd.

Since in general *S* will not be self-adjoint, we expect that any spectral decomposition involves not only *S* but also the adjoint *S**. In terms of the PDE problem, we will see that this is reflected in the need to consider both the initial time and the final time problems (the evolution with reversed time direction).

### (a) The operator problem

We consider the linear ordinary differential operator *S*, given by
1.1defined on the domain given by
1.2where
1.3By , we denote the *L*^{2} closure of . Here, the *order* *n*≥2 is an integer and the *boundary coefficient matrix* , encoding the given boundary conditions, is of rank *n* and given, in reduced row-echelon form, by
1.4The numbers *α*_{jr}, *β*_{jr} are called the *boundary coefficients*.

This operator has been studied at least since Birkhoff [1]. Depending on the particular entries of the matrix *A*, the operator may or may not be selfadjoint. The theory of the selfadjoint case was fully understood by the time Dunford & Schwartz [2] presented it.

Locker [3,4] used the theory of Fredholm operators to study the non-selfadjoint case. He defined the *characteristic determinant*
1.5where and the entries of the matrix *M*(*ρ*) are given by

It is known that, provided Δ≠0, if Δ(*σ*)=0 then *σ*^{n} is an eigenvalue of *S*. Furthermore, the algebraic multiplicity of *σ*^{n} as an eigenvalue of *S* is equal to the order of *σ* as a zero of Δ. Locker showed that, for Birkhoff-regular operators, the generalized eigenfunctions form a complete system. However, he gives no general statement about the cases that do not satisfy these regularity conditions.

### (b) The partial differential equation problem

In a separate development, a novel transform method for analysing IBVPs was developed by Fokas (see [5] for an overview). The method was applied to IBVPs posed for evolution equations on the half-line by AS Fokas & LY Sung (1999, unpublished data) and on the finite interval by Fokas & Pelloni [6] with simple, uncoupled boundary conditions. In the study of Smith [7], Fokas’ method was applied to IBVPs whose spatial part is given by the operator *S*, namely those of the form
1.6with prescribed boundary conditions and an initial condition *q*_{0}(*x*)=*q*(*x*,0), assumed smooth to avoid technical complications. Usually, the initial condition is assumed to be in . However, the same results hold assuming that *q*_{0}∈*AC*^{n}. Indeed, in this case, the uniform convergence of the integral representation (see equation (1.7)), the polynomial decay rate of the integrand and the explicit exponential *x* dependence imply that the solution *q* belongs to the same class. In what follows we assume *q*_{0}∈*AC*^{n}.

This method yields an integral representation of the solution of the IBVP in the form:
1.7where the quantities , *ζ*^{±}, Δ_{PDE}, *σ*_{k} and *Γ*^{±} are defined below in definitions 2.1 and 2.4. In many cases, including all problems with *n* even, the integrals in equation (1.7) both evaluate to zero [7]. We study these cases here.

In Pelloni [8,9] and then in greater generality in Smith [10], this method is used to characterize boundary conditions that determine well-posed problems, and problems whose solutions admit representation by series. To achieve this characterization, the central objects of interest are the *PDE characteristic matrix* (see definition 2.1) and its determinant Δ_{PDE}.

Note that in this work, by ‘well-posed’, we mean existence and uniqueness of a solution and make no claim to continuity with respect to data. By ‘ill-posed’, we mean that existence or uniqueness fails. The results of A.S. Fokas & L.Y. Sung (1999, unpublished data), Pelloni [8] and Smith [7] establish that a problem is well posed if and only if it admits a solution through the method of Fokas.

The present work details results connecting the spectral theory of *S* with the behaviour of the associated IBVPs for the PDE (1.6), as well as the one obtained from the same set of boundary conditions but for the PDE
1.8We refer to the latter in the sequel as the *final time boundary value problem*.

### (c) Summary of the main results

For an operator *S* of the type given by (1.1), and the associated IBVP and final-boundary value problems, we prove the following:

—

*If the eigenfunctions of*.*S*and*S** form a biorthogonal basis of and the IBVP is well posed, then its solution is representable as a seriesThis is the content of proposition 2.7. It follows from this result that if a series representation does not exist, then the eigenfunctions of

*S*and*S** cannot form a basis of . What is interesting is that we can use the PDE approach to obtain results on*S*in cases that are not covered by usual operator theoretic techniques. In §4, we provide an example when (Q1) cannot be answered by the usual tests involving projector norms, but may be settled through this result and a negative answer to (Q2).—

*If the IBVP and final-boundary value problems are well posed, then the eigenfunctions of*.*S*and*S** form a complete biorthogonal system inThis is the content of theorem 2.6. The conclusion does not imply that the eigenfunctions necessarily form a basis. However, the integral representation (1.7) can

*always*be deformed to derive a*series representation*for the solution of the IBVP in terms of the eigenfunctions.—

*The departure of the family of eigenfunctions of*.*S*and*S** from being a biorthogonal basis can be estimated in terms of the integrand in the representation of the solution of the associated IBVPThis is the content of theorem 2.12. This departure is quantified in the notion of ‘wildness’ [11]. Indeed, if the eigenfunctions of

*S*and*S** form a wild system in*L*^{2}[0,1], then we provide an estimate of the wildness of the system in terms of the quantities used to determine whether the IBVP and final-boundary value problems are well posed.

### (d) Outline of paper

In §2, we review the necessary definitions and notation. Following this, we precisely state and prove the results described above.

Each of §§3 and 4 is devoted to the analysis of an example that illustrates the above general results. We compare and contrast the results obtained through the new theorems with those yielded by Davies' wildness method.

## 2. Complete and basic systems of eigenfunctions

### (a) Notation, definitions and preliminary results

In this paper, we make extensive use of the notation developed in Smith [7]. We refer to that paper for details, but we list here some of the notation used throughout the rest of this work.

*The IBVP Π(n,A,a,q_{0})*: find

*q*∈

*AC*

^{n}([0,1]×[0,

*T*]) which satisfies the linear, evolution, constant-coefficient PDE 2.1subject to the initial condition 2.2and the boundary conditions 2.3where the quadruple is such that

(

*Π*1) the*order**n*≥2,(

*Π*2) the*boundary coefficient matrix**A*is in reduced row-echelon form,(

*Π*3) the*direction coefficient*has the specific value*a*=±*i*,(

*Π*4) the*initial datum**q*_{0}is compatible with the boundary conditions in the sense 2.4

Given a problem *Π*=*Π*(*n*,*A*,*a*,*q*_{0}), we define the corresponding *final time problem* *Π*′=*Π*(*n*,*A*,−*a*,*q*_{0}).

We assume that the boundary conditions are homogeneous to aid the comparison with *S*, the differential operator representing the spatial part of the PDE problem *Π*. There is no loss of generality in this assumption. Without this restriction, *Π* is no more difficult to solve; the solution simply contains an additional term represented as an integral along the real line [7].

### Definition 2.1

Let , be the boundary coefficients of the operator *S*^{⋆}, adjoint to *S*. We define
2.5
2.6
and then
2.7is called the *PDE characteristic matrix*. The determinant Δ_{PDE} of is called the *PDE characteristic determinant*.

### Remark 2.2

The PDE characteristic matrix is a realization of Birkhoff's characteristic matrix for *S*^{⋆} and also represents the Dirichlet-to-Neumann map for the problem *Π*. Indeed, it is through this matrix that the unknown (Neumann) boundary values are obtained from the (Dirichlet) boundary data of the problem. Smith [7] uses a different but equivalent definition of that generalizes the construction via determinants and Cramer's rule originally found in A.S. Fokas & L.Y. Sung (1999, unpublished data). The validity of the new definition is established in Fokas & Smith [12] and the equivalence is explicitly proved in a forthcoming article.

### Remark 2.3

In definition 2.1, we construct via the boundary conditions of *S*^{⋆}. It is possible to make an alternative but equivalent definition of via an explicit construction from the boundary conditions of *S* itself. For the examples considered in §§3 and 4, this is a simple matter. Indeed, provided the boundary conditions of *S* are non-Robin, Smith ([10], lemma 2.14) provides a simple construction. This can be done for general boundary conditions [7] and can easily be coded to be done automatically.

### Definition 2.4

Let be a sequence containing each non-zero zero of Δ_{PDE} precisely once. We define the index sets , . Let 3*ϵ* be the infimal separation of the zeros *σ*_{k}. Then the contours *Γ*^{±} are the positively oriented boundaries of
2.8

The minor *X*^{rj}(*ρ*) is the (*n*−1)×(*n*−1) submatrix of whose (1,1) entry is the (*r*+1,*j*+1) entry. This is used to construct the spectral functions
2.9and
2.10where

### Definition 2.5

We say the IBVP is *well conditioned* if it satisfies

* ζ^{±}(ρ) is entire and the ratio*
2.11

Otherwise, we say that the problem is *ill conditioned*.

Well-conditioning of an IBVP is not a classical definition and is unrelated to the concept of conditioning that appears in numerical analysis. Conditioning, in the sense of definition 2.5, is necessary for well-posedness, but is also central to the validity of a series representation. Indeed, switching the direction coefficient *a*↦−*a* in the PDE (1.6) switches which sectors are enclosed by the contours *Γ*^{±} thus, by Jordan's lemma, well-conditioning of the problem with the opposite direction coefficient is equivalent to the two integrals in (1.7) vanishing [7].

The reader will recall that a system in a Banach space is said to be *complete* if its linear span is dense in the space and such a system is a *basis* if for each *f* in the space there exists a unique sequence of scalars such that

### (b) Well-posed partial differential equation systems and bases of eigenfunctions

It is well known (see §12.5 in [13]) that if the zeros of the characteristic determinant Δ of *S* are all simple then the eigenfunctions of *S* form a complete system in . This theorem is proved using an analysis of Green's functions of both the operator *S* and its adjoint *S*^{⋆}. We prove the following result without directly analysing the adjoint operator.

### Theorem 2.6

*Let S be such that the zeros of Δ*_{PDE} *are all simple. Let Π= Π(n,a,A,q*_{0}*,0) be an IBVP associated with A and Π′ be the corresponding problem with the opposite direction coefficient, Π(n,−a,A,q*_{0}*,0). If Π is well posed and Π′ is well conditioned in the sense of definition 2.5 then the eigenfunctions of S form a complete system in* *.*

Rather than analysing both the original operator *S* and the adjoint operator *S*^{⋆}, one needs information on both the IBVP and final-boundary value problems associated with the operator *S*.

A stronger, but essentially straightforward, result in the reverse direction is

### Proposition 2.7

If the eigenfunctions of *S* form a basis in and, for some *a*, the associated IBVP *Π* is well posed, then *Π*′ is well conditioned.

Further, if are the eigenfunctions of *S*, with corresponding eigenvalues then there exists a sequence biorthogonal to such that the Fourier expansion
2.12converges to the solution of *Π*.

Indeed, in the notation of proposition 2.7, each *ψ*_{k} is an eigenfunction of the adjoint operator *S*^{⋆} with corresponding eigenvalue [14].

The above results are essentially the translation into operator theory language of results proved in the study of Smith [10]. Here, we extend the parallelism between PDE and operator theory in important ways. Namely, under some further assumptions, we construct explicitly the eigenfunctions of the differential operator directly from the PDE characteristic matrix. The construction does not require knowledge of the integral representation even implicitly, as neither *Π* nor *Π*′ need be well posed.

In the sequel, we assume that the boundary conditions are non-Robin and that a technical symmetry condition always holds, see conditions A.1 and A.2 in the appendix. We also define 2.13so that 2.14

In the next proposition, we characterize the eigenfunctions of *S* in terms of the PDE characteristic matrix and the spectral functions.

### Proposition 2.8

*For each* *and for each* *j*∈{1,2,…,*n*}, *the function*
2.15*is an eigenfunction of S with eigenvalue* . *Further*,
2.16
2.17
*and*
2.18*where* *is the corresponding eigenfunction from the adjoint operator S*^{⋆} *and* *C*_{j} *is a non-zero real scalar quantity depending only upon j*.

### Remark 2.9

The proposition above requires that the boundary conditions be non-Robin and obey the symmetry condition. These requirements may not be sharp but we have been unable to find an example failing either condition for which the result holds.

By proposition 2.8, the spectral functions of the original and adjoint problems, which we denote by *ζ*_{j}, , obey the identity
2.19

The function *q*_{0}(*x*) denotes the initial datum of the IBVP. Hence it can be chosen arbitrarily in . The particular choice is admissible since is by definition. With this choice, equation (2.19) yields
2.20where *Q*_{k} is the projection operator
2.21considered by Davies [11]. Note that the latter equality follows from equation (2.18).

By a simple change of variables, we find
2.22We therefore deduce the following important result, which gives a way to control the norms of the projection operators *Q*_{k} explicitly in terms of the spectral functions associated with the corresponding initial and boundary value problem.

### Proposition 2.10

*Let S be the operator associated with Π*. *Then the eigenfunctions* *and* *of S and of its adjoint satisfy*
2.23

### Remark 2.11

This result implies that we can estimate ∥*Q*_{k}∥ using only the spectral functions of the IBVP and final-BVPs, whose construction is algorithmic.

Conversely, this proposition has an important consequence, namely an estimate on the unboundedness of the spectral functions in terms of the ‘wildness’ of the family of biorthogonal eigenfunctions of *S*. (Following Davies [15], we say that a biorthogonal system is *wild*, if the corresponding projection operators are not uniformly bounded in norm.) We illustrate the result of this theorem in the two examples we consider in §§3 and 4.

### Theorem 2.12

*Let q*_{0} *be any admissible initial condition for the boundary value problem, and let* be any sequence such that

—

*as*.—

*|ρ*_{k}*|<|ρ*_{k+1}*|*—

*(ρ*_{k}*) is bounded away from the set of zeros of Δ*_{PDE}*, uniformly in k:*

*Then*

### (c) Sketch of proofs

### Proof of theorem 2.6.

As *Π* is well posed and *Π*′ is well conditioned, by Smith [7,16] the solution *q* of the problem *Π* can be expressed using a series as
As each *σ*_{k} is a simple zero of Δ_{PDE}, the series is separable into *x*-dependent and *t*-dependent parts
2.24and
2.25so that
2.26Further, Smith ([7], lemma 6.1) guarantees the existence of a non-zero complex constant *C* such that *σ*_{k}=*Ck*+*O*(1) as , which, by Sedletskii ([17], theorems 3.3.3 and 4.1.1), guarantees that is a minimal system in *L*^{2}[0,*T*].

As *q* is the solution of *Π*, *q* satisfies
The minimality of the *t*-dependent system means that this implies each *ξ*_{k} satisfies the boundary conditions of *S*, so .

As *q* satisfies the PDE,
so, by minimality of , each *ξ*_{k} is an eigenfunction of *S* with eigenvalue .

Evaluating equation (2.26) at *t*=0 yields an expansion of *q*_{0} in the system . □

### Remark 2.13

We have to require the zeros of Δ_{PDE} are all simple. It would be desirable to be able to say that the zeros of Δ and Δ_{PDE} are all the same and of the same order. It has been shown that this holds under certain symmetry restrictions on the boundary conditions [10] and has been established in particular for all possible third-order boundary conditions.

### Proof of proposition 2.7.

As is a basis, the Fourier expansion
converges. By Smith [16], well-posedness of *Π* guarantees that the *σ*_{k} are arranged in such a way that the exponential functions are bounded uniformly in *k*, hence that the series (2.12) converges for all *t*∈[0,*T*]. The eigenfunctions all satisfy the boundary conditions of the operator so the Fourier series satisfies the boundary conditions of the initial-value problem. The Fourier series also satisfies the PDE. So we have a series representation of the solution and *Π*′ must be well conditioned. □

### Proof of proposition 2.8.

Let *B*_{l} be the *l*th boundary condition of *S*. As the boundary conditions are non-Robin, they each have an order *m*_{l}. Hence
2.27The bracketed expression is an entry from row *r* of the characteristic matrix of *S*. Provided the boundary conditions also satisfy the symmetry condition, an algebraic manipulation yields that each column of the characteristic matrix of *S* is a scalar multiple of a column of (see the proof of theorem 4.15, [[10]]). So either is the determinant of a matrix with a repeated column or . In either case, , so . Finally, .

Let the map be given by the permutation (1,*n*,*n*−1,…,3,2), whose sign is (−1)^{⌈n/2⌉−1}. Because the boundary conditions obey conditions A.1–A.2,
2.28where the real constant
and *β*_{l} is the coupling constant appearing in the *l*th column of (1 if there is no coupling constant in that column).

Indeed, as *S* is a closed operator, densely defined on *L*^{2}[0,1], the eigenvalues of *S*^{⋆} are the points , and are the zeros of the adjoint PDE characteristic matrix ([[10]], theorem 4.15). Note also that the construction of the adjoint boundary conditions from the boundary conditions of the original problem ([[13]], theorem 3.2.4) ensures that the adjoint boundary conditions also satisfy conditions A.1–A.2.

As the boundary conditions are non-Robin, the only columns that may appear in are
2.29where *l* may vary over {0,1,…,*n*−1}. To each of these corresponds a unique column in , with the same values of *l* as each column in , respectively,
2.30Hence, to construct from we apply the following operations:

— For all

*r*, multiply the*r*th row by .— For all

*m*, multiply the*m*th column by*β*_{m}.— Apply the permutation to the row index.

— Take the complex conjugate of each entry.

This justifies equation (2.28).

By equation (2.15), the eigenfunctions of the adjoint operator are
2.31By the definition of , . Hence
2.32Hence, by equation (2.28),
Hence, by the definition of *ζ*_{j}, it follows that . □

## 3. Third-order coupled and uncoupled examples

In this section, we outline the analysis of a particular class of boundary value problems, depending on a real parameter *β*, for the third-order PDE *q*_{t}=*q*_{xxx}. Namely, we consider the following problem:
3.1where is a known function.

In the limit as the constant *β*→0, the second boundary condition at *x*=0 is *q*_{x}(0,*t*)=0. The spectral properties of this limiting case are very different from the case *β*≠0, when the coupling between the first-order derivatives is lost. Hence we refer to the boundary conditions corresponding to the value *β*=0 as *uncoupled*.

In this section, we analyse the behaviour of the associated differential operator in the two cases. To avoid technicalities, and to concentrate on the *β*=0 limit, we assume in what follows that *β*∈(−1,1).

### (a) The associated differential operator

Let *S*^{β} be the differential operator corresponding to the boundary value problem (3.1), hence specified by *n*=3 and by the boundary coefficient matrix
3.2

Setting we find that the characteristic determinant (1.5) is given by 3.3in particular, 3.4

In all these cases, the PDE discrete spectrum is equal to the discrete spectrum of the operator [10].

A calculation of the associated polynomials shows that the differential operator *S*^{β} is Birkhoff regular if *β*≠0. On the other hand, the differential operator *S*^{0} obtained when *β*=0 is degenerate irregular by Locker's [4] classification.

Although the only difference between the coupled and uncoupled operators is the first boundary condition, it is expected from the classification result that the operators have very different behaviour. This difference is reflected in the spectral behaviour of the two differential operators, as is shown in §3*b*. The IBVPs also have very different properties. These are discussed in §3*d*.

### (b) The spectral theory

In this section, we use operator theoretic results to investigate whether the eigenfunctions of *S*^{β} form a basis.

*The case β≠0.* It is shown in Smith [10] that this differential operator is regular, hence by the theory of Locker [3] we conclude that the eigenfunctions form a complete system in .

*The case β=0.* Since this differential operator is degenerate irregular, Locker's theory does not apply. Indeed, the proof of the following result can be found in Smith [10] and also in Papanicolaou [18].

### Theorem 3.1

*Let S*^{0} *be the differential operator corresponding to β=0. Then the eigenfunctions of S*^{0} *do not form a basis in* *.*

The proof is based on the following steps:

— The eigenvalues of

*S*^{0}are the cubes of the non-zero zeros of the exponential polynomial 3.5The non-zero zeros of expression (3.5) may be expressed as complex numbers*σ*_{k},*ωσ*_{k},*ω*^{2}*σ*_{k}for each , where Re(*σ*_{k})=0 and*Im*(*σ*_{k})>0. Then*σ*_{k}is given asymptotically by 3.6— For each ,

*ϕ*_{k}is an eigenfunction of*S*^{0}with eigenvalue , where 3.7— The adjoint operator (

*S*^{0})^{⋆}has eigenvalues , , and eigenfunctions 3.8— Let

*Ψ*_{k}(*x*)=*ψ*_{k}(*x*)/〈*ψ*_{k},*ϕ*_{k}〉. Then there exists a minimal such that is a biorthogonal sequence in*AC*^{n}[0,1]. Moreover, 3.9— The eigenfunctions have the same norm and it grows

*at a greater rate*than their inner product. 3.10— Assume

*Y*=1 (if*Y*>1 the biorthogonal sequence is not complete). Then the projections*Q*_{k}=∥*ϕ*_{k}∥∥*Ψ*_{k}∥ are well defined, and 3.11Hence the biorthogonal sequence is wild. Now the results of Davies ([11], ch. 3) show that is not a basis in*AC*^{n}[0,1].

### (c) The case *β*=0 as a limit

We now consider the uncoupled case as the limit *β*→0 of such calculations for the coupled operator. The zeros of Δ^{β}_{PDE} are given by
3.12and the eigenfunctions of *S* and *S*^{⋆} are given by equations (3.7) and (3.8), respectively, using the new *σ*_{k}. After a suitable scaling, the eigenfunctions of the operator and its adjoint form a biorthogonal sequence.

A direct computation shows that for *β*≠0, the fastest-growing terms in cancel out so that, for large *k*,
(This cancellation does not occur in the case *β*=0.) This causes the projection operators *Q*_{n} to be uniformly bounded in terms of the parameter *β* for *β*∈[−1+*ε*,−*ε*] for every 0<*ε*<1. However, the bound is not uniform as *ε*→0. This lack of uniformity is reflected in the transition from a regular to a degenerate irregular problem.

It is useful to compare the positioning of the eigenvalues, . Asymptotic estimates as well as numerical evidence suggest that for any particular *β*∈(−1,0) the zeros of Δ_{PDE} are distributed approximately at the crosses in figure 1; the solid rays and line segments represent the asymptotic locations of the zeros; the dashed lines are ∂*D*, the contours of integration in the associated IBVP. As *β*→0^{−}, hence , the solid rays move further from the origin, leaving the complex plane entirely in the limit, so that the solid line segments emanating from the origin extend to infinity.

### (d) The PDE theory

We now show, using the Fokas method, that while the IBVPs is well posed for any value of *β*, the solution admits a series representation only if *β*≠0, in agreement with the operator theory result of §3*c*.

*The case β≠0*

It is already well known [10,19] that in this case we have the following result.

### Theorem 3.2

*The IBVP associated with (S*^{β}*,i) is well posed and its solution admits a series representation.*

*The case β=0*

### Theorem 3.3

*The IBVP associated with (S*^{0}*,i) is well posed but the problem (S*^{0}*,−i) is ill conditioned.*

### Proof.

The proof of the well-posedness claim in this statement can be found in Smith [10]. However, for this example, we now show that the statement ‘*ζ*^{±}(*ρ*)/Δ_{PDE}(*ρ*)→0 as from within the sets enclosed by *Γ*^{±}’ does not hold, implying that (*S*^{0},−*i*) is ill conditioned.

The reduced global relation matrix in this case is given by
hence its determinant Δ_{PDE}(*ρ*)=Δ^{0}(*ρ*) given by (3.4), and the functions
As *a*=*i*, the regions of interest are
and

We consider the particular ratio
3.13For , Re(i*ω*^{r}*ρ*)<0 if and only if *r*=2 so we approximate ratio (3.13) by its dominant terms as from within ,
We expand the integrals from in the numerator and multiply the numerator and denominator by *e*^{−iωρ} to obtain
3.14

Let be a strictly increasing sequence of positive real numbers such that , *R*_{j} is bounded (uniformly in *j* and *k*) away from and as . Then from within . We evaluate ratio (3.14) at *ρ*=*ρ*_{j},
3.15The denominator of ratio (3.15) is bounded away from 0 by the definition of *R*_{j} and the numerator tends to for any non-zero initial datum. □

### Remark 3.4

In the proof of theorem 3.3, we use the example of the ratio *ζ*_{3}(*ρ*)/Δ_{PDE}(*ρ*) being unbounded as from within . It may be shown using the same argument that *ζ*_{2}(*ρ*)/Δ_{PDE}(*ρ*) is unbounded in the same region and that both these ratios are unbounded for using *ρ*_{j}=*R*_{j} e^{i(11π/6)} for appropriate choice of . However, the ratio
is bounded in hence it is possible to deform the contours of integration in the upper half-plane. This permits a partial series representation of the solution to the IBVP.

### Remark 3.5

For all *β*∈(−1,1) the final time boundary value problem is ill posed. The asymptotic location of the zeros of Δ_{PDE}, along rays wholly contained within means that for non-zero initial data the solution exhibits instantaneous blow-up. Nevertheless, for all *β*∈(−1,0)∪(0,1) the final time problem is well conditioned. In the case *β*=0, the final-time problem becomes ill conditioned and *S* becomes degenerate irregular under Locker's classification.

When *β*=±1, *S* is self-adjoint and both the IBVP and final-boundary value problems are well posed. For |*β*|>1, the final-boundary value problem remains well posed but the IBVP becomes ill posed. Thus, the self-adjoint cases represent the transitions between well-posedness of the IBVP and final-boundary value problems. Analogous to the *β*=0 case, in the limit , the IBVP becomes ill conditioned, the solution to the final-boundary value problem may not be represented as a series and *S* becomes degenerate irregular.

### (e) Comparison

The explicit computation of the operator norms in §3*d* requires the evaluation of the biorthogonal family of eigenfunctions and the precise asymptotics for the corresponding eigenvalues.

On the other hand, the integral representation of the solution of the boundary value problem can be constructed algorithmically from the given data, without the need for any precise asymptotic information about the eigenvalues, except their asymptotic location (always along a ray for odd-order problems; (see theorem 6.3 in [7]). This is sufficient for a direct analysis of the terms that blow up and prevent deformation of the contour of integration and a residue computation around the eigenvalues, thereby precluding a series representation of the solution.

In the example mentioned earlier, the particular term in the integral representation exhibiting this blow-up is the term 3.16where the right-hand side is obtained by an integration by parts. Note that in particular, we can choose .

Comparing this with expression (3.10),
it is evident that the lack of boundedness of the norms of the operators, responsible for the lack of the properties of a basis for the eigenfunctions biorthogonal family, is exactly the same lack of boundedness in the integrand of the integral representation for the solution of the PDE, yielding a barrier to the contour deformation. Indeed, using the notation of theorem 2.12, we have shown that, for this example,
This is a tighter bound on the blowup of ∥*Q*_{k}∥ than that obtained in §2. No examples have been found that violate the tighter bound but an example is presented in §4 for which ∥*Q*_{k}∥=*O*(1) while the spectral ratio grows exponentially with *k*.

## 4. Third-order pseudo-periodic examples

In this section, we outline the analysis of another class of boundary value problems, depending on a real parameter *β*, for the linearized Korteweg–de Vries equation. Namely,
4.1where is a known function.

For all *β*≠0, these are pseudo-periodic boundary conditions. In the limit as the constant *β*→2, the boundary conditions fall into the special class of pseudo-periodic conditions for which the solution cannot be represented as a discrete series ([7], §5). As in §3, the spectral properties of this limiting case are very different from the case *β*≠2.

In this section, we analyse the behaviour of the associated differential operator in the two cases. To avoid technicalities, and to concentrate on the *β*=2 limit, we assume in what follows that *β*∈(2−*ϵ*,2].

### (a) The associated differential operator

For the real parameter *β*∈(2−*ϵ*,2], we investigate the differential operator *S*^{β} with pseudo-periodic boundary coefficient matrix
and the associated IBVP and final-boundary value problems *Π*^{β} and *Π*^{′β}.

### Remark 4.1

The restriction from to *β*∈(2−*ϵ*,2] is not of any consequence other than notational convenience but the cases *β*=−1, *β*=0 and require special treatment.

Indeed, is equivalent to the final-boundary value problem *Π*′ being well posed but with solution lacking a series representation (as *Π* is ill conditioned) and, as *S*^{1/β} is the adjoint of *S*^{β} for any *β*≠0, the below analysis carries over to this case with a relabelling between *S* and *S*^{⋆}.

If *β*=0 then the boundary conditions are no longer pseudo-periodic. A description of well-posedness for this case is given in Smith [16].

If *β*=−1 then the operator is periodic hence, from the classical theory, its eigenfunctions form a basis and both the problems *Π* and *Π*′ are well posed.

### (b) The spectral theory

In this section, we attempt to use operator theoretic results to investigate whether the eigenfunctions of *S*^{β} form a basis.

*The case β<2*. It is shown by Smith [10] that this differential operator is regular, hence by the theory of Locker [3] we conclude that the eigenfunctions form a complete system in .

*The case β=2*. Since this differential operator is degenerate irregular, Locker's theory does not apply. However, in this example, we are

*unable*to apply Davies' method to discern whether the eigenfunctions form a basis. The eigenfunctions form a

*tame*(in the sense of [15]) system, which is a necessary but not sufficient condition for a basis.

Indeed, following the same outline method as in §4*b*, we obtain

— The eigenvalues of

*S*^{2}are the cubes of the non-zero zeros of the exponential polynomial 4.2The non-zero zeros of expression (4.2) may be expressed as complex numbers*σ*_{k},*ωσ*_{k},*ω*^{2}*σ*_{k}for each , where Re(*σ*_{k})=0 and*Im*(*σ*_{k})<0. Then*σ*_{k}is given asymptotically by 4.3— Let 4.4Then, for each ,

*ϕ*_{k}is an eigenfunction of*S*^{2}with eigenvalue .— The adjoint operator (

*S*^{2})^{⋆}has eigenvalues , corresponding to eigenfunctions 4.5and there are at most finitely many eigenfunctions of (*S*^{(2)})^{⋆}that are not in the set .— Let 4.6Then there exists a minimal such that is a biorthogonal sequence in

*AC*^{n}[0,1]. Moreover 4.7— The eigenfunctions have the same norm and it grows

*at the same rate*as their inner product. 4.8— Then the projection

*Q*_{k}has norm ∥*ϕ*_{k}∥∥*Ψ*_{k}∥, which is bounded uniformly in*k*. From this result, it is impossible to determine whether the eigenfunctions form a basis or not.

### (c) The PDE theory

As shown in Smith ([7], example 5.2), *Π* is ill posed if and only if *β*=2. Via proposition 2.7, this yields the result that the analysis of §4*b* could not—the eigenfunctions do not form a basis.

### Proposition 4.2

Let and let *ρ*_{k}=*R*_{k} e^{iπ/6}. Then, using the notation of Smith [7], the ratio

### Proof.

A quick calculation yields
4.9hence
4.10The spectral function
is independent of *β*.

By the definition of *ρ*_{k}, the functions
grow exponentially with *k*, while
decay. Hence
Also
Note that *q*_{T}(1)=*q*(1,*T*)=−*q*(0,*T*)=−*q*_{T}(0), by the first boundary condition. Hence, provided we can be sure that *q*_{T}(0)≠0, 2*q*_{T}(1)−*q*_{T}(0)≠0. □

As , and *R*_{k} was chosen to ensure that Δ_{PDE}(*ρ*_{k}) is bounded away from 0, . Hence, by Smith ([7], theorem 1.1), *Π* is ill posed.

The rate of blowup exhibited in proposition 4.2 is maximal in the sense that for any sequence such that |*σ*_{k−1}|<|*ρ*_{k}|<|*σ*_{k}| and for any *j*∈{1,2,3},

The problem *Π*′ is well conditioned for all *β*∈(2−*ϵ*,2]. Indeed, for any sequence with and , we find the asymptotic behaviour:

### (d) Comparison

In order to find the asymptotic behaviour of ∥*Q*_{k}∥, the complex calculation outlined in §4*b* is necessary. However, the result we obtain is that the projection operators are uniformly bounded in norm, from which we cannot discern whether the eigenfunctions form a basis.

The calculation required to prove proposition 4.2 is relatively simple and from that result, via proposition 2.7, it follows that the eigenfunctions are not a basis.

## 5. Conclusions

In this paper, we have gathered and summarized old and new results on a newly analysed correspondence between the spectral theory of linear differential operators with constant coefficients and the analysis and solution of IBVPs for linear constant coefficient evolution PDEs. We also presented two specific examples to illustrate the power of this connection for inferring results on the spectral structure of the operator.

In §2, we developed a new method for showing that the eigenfunctions of certain differential operators do not form a basis. This method relies crucially upon finding a well-posed IBVP whose solution cannot be represented as a series.

In §§3 and 4, we compare the new method to the established method of Davies by applying each method to examples. The calculations we present suggest that the PDE approach is more straightforward in deriving estimates for the boundedness of projector operators, hence results on the existence of eigenfunction bases. Indeed, it is sufficient to estimate the boundedness of functions constructed algorithmically in certain well-defined complex directions.

The second example represents a case where the new method yields a result but the operator-theoretic methods we considered do not. Indeed, we show that the solution of the only well-posed IBVP cannot be represented as a discrete series hence, by proposition 2.7, the eigenfunctions cannot form a basis. But the eigenfunctions are not wild, indeed the associated projection operators are uniformly bounded in norm, so we cannot reach the same conclusion using, e.g. the operator-theoretic framework of Davies.

The remainder of §2 investigates the relation between the two methods. Indeed, for the class of operators we discuss, determining the wildness of the eigenfunctions is equivalent to the calculation of precisely the same quantities used to determine well-posedness of the associated IBVPs.

It is expected that the well-posedness of both the IBVP and final-boundary value problems is sufficient to guarantee that the projection operators are uniformly bounded in norm.

The applicability of the new method has only been shown for eigenfunctions of the class of differential operators considered herein, whereas Davies' method could be applied to any complete biorthogonal system, whether it is constructed from the eigenfunctions of differential operator or not. However, it should be possible to extend the new method, along with the results of Smith [7] to a wider class of differential operators, providing a powerful tool to investigate the spectral properties of linear differential operators. For example, throughout this work, we have assumed that *S*=(−*i*∂_{x})^{n}. A general constant-coefficient linear differential operator may have more terms, but its principal part could always be represented by such an operator *S*. As the spectral behaviour of the operator is governed by its principal part, we expect the above results to carry over to such operators.

## Acknowledgements

We are very grateful to E. B. Davies, M. Marletta and the reviewers for their useful suggestions. The research leading to these results has received funding from EPSRC and the European Union's Seventh Framework Programme (FP7-REGPOT-2009-1) under grant agreement no 245749.

## Appendix A

The statements of results 2.8–2.12 all require the following additional conditions:

### Condition A.1

The boundary coefficient matrix *A* is *non-Robin*:

None of the boundary conditions represent couplings between different orders of boundary function.

That is, for each *k*∈{1,2,…,*n*}, if *α*_{kj}≠0 or *β*_{kj}≠0 then *α*_{kr}=0=*β*_{kr} for all *r*≠*j*.

Note the following contrast with Robin's original definition. Our Robin/non-Robin classification is independent of coupling between the two ends of the interval; the boundary condition *q*_{x}(0,*t*)=*q*(1,*t*) is of Robin type and couples the ends of the interval.

### Condition A.2

Recall that *A* is reduced row-echelon form. The boundary conditions are such that if the boundary function of order *r* at one end corresponds to a pivoting entry in the boundary coefficient matrix *A* then the boundary function of order *n*−1−*r* at the other end must correspond to a non-pivoting entry in *A*. Further, the coupling constants for coupled boundary conditions of order *r* and *n*−1−*r* are equal.

For simple boundary conditions, this means that if the boundary function of order *r* at one end is specified then the boundary function of order *n*−1−*r* at the other end must not be specified.

- Received January 11, 2013.
- Accepted March 21, 2013.

- © 2013 The Author(s) Published by the Royal Society. All rights reserved.