FINITE EIGENFUNCTION APPROXIMATIONS FOR CONTINUOUS SPECTRUM OPERATORS

In this paper, we introduce a new formulation of the theory of continuous spectrum eigenfunction expansions for self-adjoint operators and analyze the question of when operators may be approximated in an operator norm by finite sums of multiples of eigenprojections of multiplicity one. The theory is designed for application to ordinary and partial differential equations; relationships between the abstract theory and differential equations are worked out in the paper. One motivation for the study is the question of whether these expansions are susceptible to computation on a computer, as is known to be the case for many examples in the discrete spectrum case. The point of the paper is that continuous and discrete spectrum eigenfunction expansions are treated by the same formalism; both are limits in an operator norm


INTRODUCTION
Eigenfunction expansions may be considered as an abstraction of the idea of approximating complicated waves by finitely many standing waves.For discrete spectrum eigenfunction expansions associated with a self-adjoint operator H in a Hilbert space L2(X,p) the rate of convergence of the expansion has been the subject of a great deal of research.In the continuous spectrum case, sums are replaced by integrals, and the question of whether the integral can be approximated by a finite sum has not been studied.This paper begins such a study; first, however, we indicate why the question is important, and what sort of answers we look for.It is helpful to take a naive look at the method of separation of variables, or eigenfunction expansions.
One purpose of an eigenfunction expansion is to convert continuous data, such as functions, into elements of :n, vectors formed from the coefficients of the func.tion in the expansion.A problem, such as a partial differential equation, in a function space is then transferred to :n, solved there (partial differential equations often transform into ordinary differential equations with constant coefficients), and then the solution to the original problem is obtained by transforming back, that is, by summing the eigenfunctions with the recalculated coefficients.This involves a certain error.The error is measured using two Hilbert spaces and Z.
An eigenprojection in B(,Z), the bounded operators from into Z, of a self-adjoint operator H in a Hilbert space L. is an operator in B(Y,Z) of the form P()= F()F, where F E Z c ', and where H takes a dense locally convex vector space W contained in continuously into itself, and H'F ,F.Ideally, the operator which solves the original problem is the limit in B(Y,Z) of finite sums of multiples of eigenprojections.We call this property the discrete approximation property, for the remainder of this introduction.
If the discrete approximation property holds, the eigenfunctions used to perform the expansion do not have to be recalculated for each new function being expanded, and only finitely many coefficients must be calculated.In other words, up to a certain error, functions become elements of Cn, and semigroups generated by the original self-adjoint operator become semigroups of diagonal matrices.Continuous spectrum expansions have at present no such theory; these are modelled on the inverse Fourier transform, so instead of finite sums one must work with integrals which in general only converge in the mean, and convergence in operator norm is not discussed.
One consequence of this is that the theory of continuous spectrum eigenfunction expansions appears to have no computational significance; it seemingly cannot be put onto a computer.This situation, which if true would lead to problems of whether the theory is well-posed in any reasonable sense, contradicts the intuition gained from the Fourier transform; the Fourier transform is well-known to be computationally significant.The purpose of this paper is to begin a continuous spectrum theory modelled on the discrete spectrum case, where finite sums appear instead of integrals.In order to do this, it has been useful to reformulate the existing theory of these expansions.More about this reformulation will be given later in this introduction.
The discrete approximation property is shown in section 3 to follow if the operator in question is a convergent operator-valued integral, in B(Y,Z), of eigenprojections, which in the continuous spectrum case must go from a space Y smaller than L. to a space Z large enough to contain the eigenfunctions.Hence we must study when the expansion is such an integral.This is shown in section 3 to be true when the measure of A is finite, with respect to an invariant measure depending for a cyclic subspace only upon Z.This measure is the one which normalizes the eigenfunctions in Z.
For Sturm-Liouville theory for the Dirichlet problem on a fini'te interval, with H and Z L(R), the sort of convergence we study is well known to occur, as it does in many other discrete spectrum problems.Even in discrete spectrum problems, however, the calculation of appropriate spaces and Z is often nontrivial.In this paper, in the discrete or continuous spectrum case, they are calculated using a priori estimates on the domain of H. Eigenfunctions satisfy the equation H'F ,F in a certain dual space and are members of Z.
To study the discrete approximation property for an arbitrary bounded continuous function the separation of variables scheme discussed earner works.
It is not quite accurate to say that existing theories completely ignore these questions.A little bit of thought will convince one that for a bounded set A, the inverse Fourier transform representation of the spectral projection P(A) for the self-adjoint operator associated with id/dx has discrete approximation properties from, say, Y {0: 0/w E L2} to Z {F: wF E L2} where w is a well-behaved L 2 function.From the equivalence, shown in section 5, of the approximation property for P(A) with the compactness of P(A) in B(Y,Z), together with known results about SchrCdinger operators, it is not difficult to establish the same properties for the Schrdinger operator with a well-behaved potential, where A is a bounded set of energies.However, in situations like this, the compactness of the spectral projection is known beforehand, and may be used as in Section 5 to produce the approximation property.A more interesting problem is that of of unbounded sets A. With the Y and Z above, for the inverse Fourier transform, it turns out that some unbounded sets A have this property and others do not.The compactness of P(A) is only known as a consequence of the theory.The results of this paper are oriented toward the study of which sets A have this property.As an example, the general Sturm-Liouville case on a half-line is studied in section 4. Much sharper results for short-range potentials are given by D.
B. Hinton and the author in [4], using the results of this paper.Examples from partial differential equations also fit easily into the formalization of this paper and explicit examples are given.Here the results are less sharp unless one restricts to a single cyclic subspace.The heart of the paper is the operator valued integrals of Section 3, together with their relation to the approximation problem.In section 5, we show the equivalence of the two problems, compactness and approximation.Using the results of sections 3, 4 and 5 of the paper, we see that certain spectral projections P(A) ae compact as operators between spaces where they are not already known to be compact by a priori estimates.A simple example is given in section 4, which is about second order ordinary differential operators.
The theory of continuous spectrum eigenfunction expansions is a very old one, going back to Gelfand's work in the 1950's.The book of Berezanskii [1] is a fundamental reference, but it is difficult to extract specific information from such a general theory.The work of Simon  [8], which is functional-analytic though it is specifically slanted toward SchrSdinger operators, is a clear and rigorous approach to the theory with a lot of specific information.
The paper of Poerschke-Stolz-Weidmann [5] is more general, and also has more elenentary proofs.This paper has been followed up by Poershke and Stolz [6], who give applications of their results to scattering theory.
With such a large and excellent literature, why give yet another approach to the whole theory?We do so, partly to obtain the crucial assertion iii) of Lemma 1.6, which we need for the basic problems discussed earlier, but also to be able to analyze the expansion in a format based simply upon a priori estimates on the domain of powers of the self-adjoint operator H which is being decomposed, so as to make the results as concrete as possible for applications to differential equations.The question of what is needed about an a priori estimate in order to do this is answered in the paper.Our proofs are self-contained, since once the formalism is set up and Lemma 1.6 is proved, the inverse Fourier transform (Lemma 3.4) and the Fourier transform (Theorem 1.8) follow quite directly from the spectral theorem; to attempt to invoke other results would introduce technical difficulties.The estimates on the eigenfunctions in this paper contained in Assertion ii) of Theorem 1.8 do not follow (at least directly) from other results; and as was remarked earlier the kind of strong convergence of the integrals in the inverse transform contained in Lemma 3.4 is not studied at all in existing literature.On the other hand, our hypotheses are different from those of other approaches such as [5] and [8]; for example, our theory also demands more smoothness on the coefficients when applied to differential equations, as we discuss below.
The relationship between this work and that of [5], [6] and [8] is an interesting question for future research.
The formalism of our theory of continuous spectrum expansions depends on the introduction of a locally convex space W with certain properties, such that H takes W continuously into itself.It is needed in order to have a core where all operations make sense, and from which estimates may be extended by the closed graph theorem.It also allows us to say what an eigenfunction ; it is just an element of W' such that H'F AF.This, together with regularity theorems for the domain of H, if H is a differential operator, is what turns an abstractly defined eigenfunction into a concrete object such as a smooth function.For example, if W C(fI), and f is an open subset of a C (R) manifold, and H is generated by a hypoelliptic differential expression, then W' is the space of distributions, so that if F W' and H'F AF, then F is a C (R) function.If the operator H is, for example, associated with a Dirichlet problem, smootheness of F is needed to show that F vanishes on the boundary of fl.It should be noted that the smoothness of F does not follow only from the fact that F W', which is implied by virtually any theory, but from this fact ogether with the fact that H'F AF.Of course, there are many approaches to these expansions which imply, for example for SchrSdinger operators, that the generalized eigenfunctions satisfy the differential equation in a distributional sense and hence classically, but these assertions are shown as consequences of specific properties of the examples being studied.Since such assertions are necessary for applications, we build them into the theory, producing a more powerful structure.
Motivated by the above discussion, in examples studied in this paper we often take W to be C(fl); this causes us to assume smoothness of the coefficients when H is a differential operator.
However, other choices of W would perhaps allow more general coefficients.This is another subject for further work.
The author has been fortunate enough to have many discussions of this theory with many different mathematicians over a period of some years.He would like especially to thank Christer Bennewitz, Rainer Hempel, and Don Hinton, although discussions with a number of others have also been very helpful.He would also like to thank W. D. Evans, University College, Cardiff, and the British SERC for support during the author's very pleasant four-onth stay in Cardiff in the spring of 1987, when this paper was begun, and Peter Hislop and the University of Kentucky for their hospitality in the Spring semester of 1991, when the research for the paper was finished.

BASIC FORMALISM AND L 2 ESTIMATES
In this section we develop the basic formalism and eigenvector estimates for our theory.We shall need to introduce some basic spaces W and W'.W is contained in the domain of the self-adjoint operator H, and W' is its dual space under a certain topology.One may think of W as like C(IR n) and W' as the space of distributions on in; also one may sometimes wish to think of W as the rapidly decreasing functions and W' as the tempered distributions.In order to handle the case where W C(In), we need to assume that W is an inductive limit of Frechet spaces, rather than a Frechet space itself, since C(i n) under the usual topology is not metrisable. (See Proposition 5, p. 125, Robertson and Robertson [7].)The purpose of these topological vector spaces is to get a precise definition of what the eigenfunctions are: they are just elements F of W' such that H'F AF for some A, where H' is the transpose of H.The structure of W is needed for an application of the closed graph theorem to obtain a priori estimates from assertions about the domain of H. Notation 1.1.Let b be a Hilbert space, and let H be a self-adjoint operator with domain a dense subspace of b and range contained in .If e E D, let S e denote the closed linear span of {P(A)e A is a Borel subset of}, where for any Borel set A, P(A) is the spectral projection associated with A by the spectral theorem.Let e(A) [P(A)e,e], where [, denotes the inner product of D. (In this paper, will always be L2(X,p), where p is a positive measure on X).Note that e is a positive Borel measure on , such that e() Ilell 2. Note also that the restriction of H to S e is a self-adjoint operator which is unitarily equivalent to the operator in L2(e) which maps f(A) to Af(A).Assumption 1.2.Throughout the paper, we shall assume the following hypotheses: i) H is a self-adjoint operator with domain a dense subset of I and range contained in f L2(X,p), where X is a locally compact Hansdorff space, and is a positive regular Borel measure on X such that the measure of every compact set is finite; ii) W is the inductive limit of a sequence {Vn} of separable Frechet spaces such that for each n, V n is algebraically and topologically contained in or equal to Vn+l; (hence W is complete, by Prop.3, p.128, [5]); (note that a subbase for the topology of W is the set of all absolutely convex subsets U of W such that Uf]V n is open in V n for every n; recall that W is metrisable if and only if for some M, V n V M for n > M, by Prop.5, p. 129, [7]); iii) W' is the dual of W, and W' is given the topology oW',W) of pointwise convergence on W; (recall that a neighborhood subbase about 0 for this topology is the set of neighborhoods UCx,) {F: IFCx)l < }); iv) w c domain H and H is a continuous linear transformation from W into W; v) W is contained and dense in LI(X,)IL2(X, and the identity mapping from W into L l(X,p) and L2(X,fl is continuous; vi) for any open set in X, if C c(X denotes the continuous complex-valued functions of compact support in X, and if is any element in Cc(X which is 'supported in F, there is a sequence {n } of elements of W, each supported in r, such that Cn converges in L2(X,p to .
Remarks: We shall sometimes assume the following estimate; we shall explicitly state this assumption each time.
Estimate 1.3.(an a priori estimate) There exists a 1-1 continuous linear transformation B from W onto W and a positive function f E L2(X,p), such that multiplication by f maps W continuously into W and such that the linear transformation B' has the property that there exists a positive integer N such that {B'}/f L(R)(X,p) for all in the domain of HN. (Note that by v) of Assumption 1.2, L2(X,p is naturally embedded in W').
Reraar.We now make the initial definition of the eigenfunctions, as linear functionals on a dense subspace of W over the rationals.The work consists of showing that they belong to a natural space.Note that W has a countable dense set by hypothesis ii) above.Let S' be a countable dense subspace of W over the rationals.
Notation,: Let e E h.Let Ce be the unitary mapping from S e onto L2(ae) such that (e(e)= 1, and such that Ce(H)(,)= ,e()()), for all in D(H)0Se, and such that (e(P(A)) x(A)(e(), where x(A) denotes the characteristic function of the Borel set A. Let U e CeoP(Se ), where P(Se) denotes the orthogonal projection onto S e.Note that the existence and uniqueness of e follows from the spectral theorem.Let S S' + HS'.For each i E S, select an everywhere defined representative of Ue(Bi ).Denote the above representative by gi" Definition 1.4.For each , e , define Z,,e on S by Z,,e(i) gi(,), so that if U e is the mapping discussed above which arises from the spectral theorem, then Z,,e(i)= Ue(Bi)( for almost every , with respect to a e.Note that, if A 0 is the complement of the set of , such that ,,e is a linear functional on S over the rationals, then ae(0) 0. Lemma 1.5.If there exists a positive constant M such that II(n'0)/fll(R) <_ U(ll011 + HNolI) 1/2 for aU 0 the domain of HN, where f is as in Estimate 1.3, then for almost every x with respect to p the following is true: for any orthogonal set {ei} in the d'omain of HN, such that He i/s also an orthogonal set, and such that Ileill 2 + IlaNeill 2 x for aU i, then(ii=l IB'eil2(x)) 1/2 <_ Mr(x).
Proof: It can be proved by a technique like that of Weidmann [9], p. 140, that if T is a bounded operator from a separable Hilbert space h into L (X,#) and IITII is the associated operator norm, then for almost every x with respect to #, it is true that for any orthonormal set {ei} in h, (ZilTei(x)12) 1/2 <_ IITII.The idea of the proof is to construct a mapping Q from X into h such that Tg(x) [Q(x),g].Then one can see from the hypothesis that for almost every x, Q(x) has norm less than or equal to IITII.Then for each fixed x, the Schwartz inequality gives the desired result.To make this proof rigorous demands careful attention to sets of measure 0 with respect to .Now if we let Tg be (B'g)/f, and h be the domain of H N with norm Ilgll-Ilgll / IIHNglI, the result follows.
Another way to construct the mapping Q which does not pay such careful attention to sets of measure 0 is due to C. Bennewitz and the author: Note that by the Gdfand representation theorem for commutative Banach algebras L (X,p) is isomorphic and isometric as a Banach algebra to the algebra C(Y) where Y is the maximal ideal space of the Banach algebra L(R)(X,p), a compact Hausdorff space.Let E be the isometry from L(R)(X,p) onto C(Y).Define to be the operator ET.Then at every y e Y, define (y(g) to be ETg(y).It is dear from the hypotheses that for all y e Y, the linear functional (y is in the dual space of h, and has norm less than or equa to M IITII.Thus Qy(g) [g,a] for some a e h, with Ilall <_ M. It follows that for each orthonormal set {ei} of elements of h, and for each finite N, (ziN=l IETei(Y)l 2) 1/2 e C(Y) as a function of y, with supremum norm less than or equal N 2)112 to M. Hence for each N, (i=1 Tei(x)l e L(R)(X,p) as a function of x with L(R) norm less than or equal to M, since the map E is an isometric isomorphism and takes absolute values to absolute values.It follows immediately that for almost every x in X with respect to p, (i=l [Tei (x)12)1/2 -< M, as we desired to show.
Lemma 1.6.Let n be a positive integer.Assume the hypotheses of Estimate 1.3.Then the following hold: i) B is a homeomorphism from W onto W, and B' is a homeomorphism from W' onto W'.
ii) There exists a constant K such that .forany e domain HN, II(B')/t]](R) _< K(11112 + ]]H N]]2 ).iii) Let {(i)}ik=l be any pairwise disjoint collection of Borel subsets of R; let K be as in ii) above.Suppose that e is in the domain of HN, and that {0i} is any collection of elements of W such that 0 i]12 <-1.Then i=1 ] (i) UeB0i dae (A) -< where [[el[h 2 Ile][ + [IHNe[[, r is the union of the supports of the functions 0 i, and l(r) is the characteristic function of r.
Proof."To prove the first conclusion, we note that the mapping B is a 1-1 mapping from W onto itself.This implies that B is a homeomorphism, by a result of Dieudonne and Schwartz (see the discussion on page 124 of [7]).However, it is now easy to prove this result from later work on webs; for completeness we give the proof.It is clear that the graph of B -1 is closed.By Theorem 2, page 158, [7], we see that if E is a Frechet space, and F is a separated convex space with a completing web, then mappings from E onto F with (sequentially) closed graph are continuous.It is dear from the definition of a compatible web that a Freehet space or the strict inductive limit of Freehet spaces has a compatible web, which is completing by Lemma 1, p. 156, [7].Hence, W has a completing web.To show the continuity of B-1, we need only show that the restriction of B -1 to each space V n is continuous, since for an open set U, B-I(u) is the union of its intersection with each Vn, and since a set is open in W if and only if its intersection with each V n is open.But each V n is a Frechet space, and it is obvious from the continuity of B that the graph of the restriction of B -1 to each V n is sequentially closed.Ther.eforeB -1 is continuous, as we desired to show.That B' is a homeomorphism is immediate.
The second conclusion is a consequence of the dosed graph theorem, because the mapping T taking the domain of HN, with graph norm, into L(R)(X,p) has closed graph, where T {B'}/f.To see this, suppose that Cn is a Cauchy sequence of elements of the domain of HN, in graph norm, and that Ten is Cauchy in L(R)(X,p).Then Cn converges in L2(X,# to an element of the domain of HN.We must show that Ten converges to T. Since Cn converges to in L2(X,p), then Cn converges to in W', where we have embedded L2(X,p into W' using v) of Assumption 1.2 by mapping g to the linear functional Fg such that Fg() [,g].Since B' is continuous from W' into W', then B'n converges in W' to B'.But by hypothesis, (B'n)/f is Cauchy in L(R)(X,p) and therefore converges to an element 0 of L(R)(X,p).The preceding argument, using v) of Assumption 1.2 for Ll(X,p), shows that convergence in L(R)(X,p) implies convergence in W'.However, multiplication by f takes W' continuously into itself, where by definition fF() F(f), because the transpose of a continuous map from W into W is a continuous map from W' into W'.But we have embedded L2(X,p into W' using an embedding map E such that E(f) fE().Thus B'n converges to f0in W'.Thus B' f0in W'.Hence B'(x) fx) for almost every x in X with respect to p. Therefore (B')/f , as we desired to show.The second assertion is proved.
The corollary is proved.
Theorem 1.8.Assume the hypotheses of Estimate 1.3.Suppose that e is in the domain of HN, and that IIh is as in Assertion iii) of Lemma 1.6.Then there exists a subset of such that ae($ \ () 0 and such that for every E (, there ezists a unique element Z,,e in W' such that Z,,e is not the zero functional and ZA,e agrees with ZA,e on S. Furthermore, if FA,e is defined on by the relation F,,e (B')-IzA then: i) H'F,e F, where H' is the conjugate of the restriction of H to W; furthermore, for any 0 e W, F/3,e (0) Ue(0)(/3) .foralmost every 13 with respect to ae; ii) S'F,,e { L2(X,p); iii) if a()= B'F then a is a measurable nction with respect to a e from into L2(X,p) in the sense that e > 0 there exists a compact set F such that ae( \F) < e and such that the restriction of a to r is a continuous function from F into L2(X,p); iv) if3 is a Borel subset of X and A is a Borel subset of , then ]A [[:(/3)a(')ll2dae < q'K[IP(A)ellhll2:(/3)f[12' where y,(3) is the characteristic nction of3.
Proof" B' is a homeomorphism from W' onto W'.By the preceding lemma, for almost every the functional Z,,e has a unique extension Z,,e to W. There exists a unique element of L2(X,p which agrees with Z,e on W; denote this also by Z,e.For any element of S', Z,e() B'F2,e() F2,e(B) for almost every , with respect to a e.But also Z2,e() Ue(B)()0 for almost every ).
If 6 is the set of such that Z,e is the zero functional, then for every in W, Ue() vanishes on 6.By continuity of U e from L2(X,p into L2(ae) and since W is dense in L2, it follows that U e(e vanishes on 6.But this function is identically equal to 1 almost everywhere with respect to a e. Hence ae(6) 0. Thus Z,k,e is non-zero for almost every .
By the spectral theorem for self-adjoint operators in a Hilbert space, if B 0, we see that F,e(H0 Z,e(B-1H0 Ue(H0)(, ,Ue(0)(, except for a fixed set of measure 0 with respect to a e, which is the union of exceptional sets for each in the countable set S'. Since F,,e E W', and since the range of the restriction of B to S' is dense in the range of B and therefore dense in W, the first conclusion follows immediately.The second assertion was proved at the outset. We prove the third assertion.Note that if Be e S, ,e(B) is measurable by definition.
Note that [B'F,,e, F,,e(B) Ue(B) almost everywhere.By passing to the limit in Lemma 1.6 we obtain that for any open set F in X, and any partition {f(i)} of A, and any set of elements i e L2(X,p), each supported in F, such that I1ill 1, i=l'tA I[n'fA,e'i] dae -< Now use assertion iii) to select for every e > 0 a compact set R of A such that ae(A\e) < and such that the restriction of B'FA,e to R is a continuous function from R into L2(X,p ).It follows that, if ]]2,r denotes the norm of L2(r,p), .tell B'F A,e]12,rdae -< qKII 2:(r)fl1211P(e)ell h.
From the monotone convergence theorem, it follows that A llB'FA,ell2,rdre <-Kllx(r)fll211P(A)ellh Selecting a countable decreasing chain of open sets r with intersection essentially equal to /, and using the monotone convergence theorem again, we easily complete the proof of iv), and hence the proof of Theorem 1.8 is completed.
2. SOME REPRESENTATIVE EXAMPLES In this section, we take a look at some representative examples to motivate the theory.
Ezsmile 2.1.Let X [R n and p be Lebesgue measure.Let W denote C([Rn).It is well-known (see p. 75, [7]) that W satisfies the hypotheses above.Suppose that H is a self-adjoint operator in L2({ n) such that Hfor all in the domain of H, where -is a partial differential expression with C (R) coefficients.Suppose further that for some positive integer N, all elements of the domain of H N lie in L(R)(iRn).Let a be any bounded, L2, positive element of C(R)(n).Let B be multiplication by ; then B' is also multiplication by .Let f in Estimate 1.3 be w.Theorem 1.8 then yields that, for any e in the domain of HN, the eigenfunction FA,e has the property that FA,e is an L 2 function, which has the properties of B'FA,e in this theorem.Here rFA,e AFA,e in the sense of distributions.
Ezsmple 2.2.Suppose that H is a self-adjoint operator in L2(iRn such that Hf r/f for all f in the domain of H, where r/is a partial differential expression with C (R) coefficients such that each derivative of any coefficient of r/has at most polynomial growth at infinity.Suppose that H has the additional property that for any positive integer j, there exists a positive integer N(j) such that the domain of H N(j) is contained in the Sobolev space HJ(n).Letdenote the differential expression defined by -= ii=l 2/bx-1.Let B be the operator on W defined by B (Ixl 2 + l)-(n+)/4, where r is large enough such that for any g such that g 6 H2r(iRn), it follows that g e L(R)(n).Let W be the space of rapidly decreasing functions on n.By using the Fourier transform, we see that B satisfies the hypotheses of Estimate 1.3.W satisfies Assumption 1.2 since it is a Frechet space.Direct calculation shows that B'(8)/(Ix12+l)"-(n+)/4) E L(R)(iR n) for all 8 in the domain of HN(j), where is large enough that for 8 i HJ([Rn), DaB e L(R)( n) for lal _< 2r.By Theorem 1.8, we see that rr{(Ixl 2 + 1)-(n+)/4FA,e } e L2(n).Since FA,e is in W', the space of tempered distributions, we see from the ordinary Fourier transform that (II + l)-(n+0/4F, e e L(R)(n).
Ezample 2.3.Let H be any self-adjoint operator in L2(iRn such that H8 r/# for all 8 in the domain of H, where r/is a partial differential expression with C (R) coefficients, such that each derivative of each coefficient of r/has at most polynomial growth at infinity.Let W be the space of rapidly decreasing functions.Suppose that W is contained in the domain of H. Let e L2(iRn ).

APPROXIMATION
In ts section we study the mn problem of the paper.spon 3.1.Let Y be a subspace of L2(X, wch is so a Banach space with norm [[y, and which h the properties that a) W is a dense subspace of Y, b) there ests a constt Let A denote the continuous extension of B -1 as operator om Y into L2(X, ).
Rr: We now ve the deflation of agonzation, d introduce a sctr meure wch we denote by e" Since the properties of e e very importer for o theory, it is usef to note that by an ementy ccation it follows that the deflation of e does not depend upon e, but only upon S e.In other words, if S e Sp then e f" It should so be remarked that the following deflation has been made qte detled cause it sms usef for later appcation to state the appromation properties we obtn completely.Dtion 3.2.Suppose the hotheses of Assumption 3,1, Assumption 1.2 d Estimate 1.3 hold, d that e domn (HN).Let e be the positive meure on defined by the relation [[B'F,e[[de, where F,e is in Threm 1.8.Let Q r(H), where r is a unded d e continuous fction om the sctrum of the restriction of H to S e into K.Let be a Bor subset of .We say that P(A)P(Se) Q is doiMe in B(Y,Z) th resct to H d e if B'P(A)P(Se)Qg L2(X, for g Y, d a) for every > 0 there ests a positive integer k and a fite sint faly {Ai}=lof subsets of i such that e (Ai) is fiMte for every d such that there ests a set of reM humors {Ai}= 1 th A Ai0A d th the property that, inting P(A)P(Se)Q0 cocy into W', II{P()P(Se)Q-=1 e(AinA)7(Ai)RAi,e}(O)l[Z [[0l[y for O e Y, where b) A is in the complement of the ceptionM set of Theorem 1.8, so that in ptic FAi,e is in Threm 1.8 d B'FAi,e L2(X,p th ]]B'FA,e]]2 # 0, and where c) RAi,e($ B GAi,e(A)GAi,e for any Y, where GAi,e FAi,e/]]B'FAi,e]2, d where B 'GA,e denotes the complex conjugate of the functionM B'GA,e.The complex conjugate appes agMn, cause we e worMng in W'.Note that wle the points A depend on A, the number k and the sets A do not; these depend oy upon .Note Mso that RAi,e agrs with G Ai,e($)GAi,e on W, d that by by,thesis b) of Assumption 3.1, together th the fact that B'GAi,e L2(X,), it follows that RAi,e B(Y,Z).
Let M be a family of bounded continuous functions from the spectrum of the restriction of H to S e into t:.Let QM (r(H)lr E M}.We say that P(A)P(Se)QM is simultaneously diagonalizable in B(Y,Z) with respect to H and e if for every > 0 there exist A and h as in a), b) and c) above such that 1=1 (0)llz _< ,ll011y for all 0 E Y and all r c M. Theorem 3.3.Suppose that the hypotheses of Assumptions 3.1 and I. hold, and that Estimate 1.3 holds, and e domain HN.Let G,,e F,,e/IIB'F,,ell2.Then Assertion i) below implies Assertion ii), which in turn implies Assertion iii).i) A is a Borel subset of such that {B'G,,e , c A} is precompact in L2(X,p ).
ii) ] A IIB'F,,elldae(') < (R)" (an elementary computation using the spectral theorem shows that /'A IIB'F,ell2d% !ZX IIB'F,gll2dag_ if g is another cyclic vector for the subspace Se.) iii) Let Q r(H), where r is a bounded continuous function from the spectrum of the restriction of H to S e into f.Then P(A)P(Se) Q is diagonalizable with respect to H and e in B,(Y,Z).IfM is a set of bounded continuous functions from the spectrum of the restriction of H to S e into f which is uniformly bounded and equicontinuous on A, then P(A)P(Se)Q M is simultaneously diagonalizable with respect to H and e in B(Y,Z).
Proof: We note that Theorem 1.8 guarantees that B'F into L2(X,p with respect to ae, in the sense that for every e > 0 there exists a compact set K such that ae(\K < and such that the restriction of into L2(X,p ).
We show that Assertion i) implies Assertion ii).In fact, if Assertion i) holds, there exists a finite set {i}ik=l of points of A such that for all A, there exists a 'i such that IIB'G,,e-B'G,i,ell2 < /2.Select a set {i}ik=x of elements of W such that [B'G,i,e,i] > 3/4, and such that I1i112 -Let A t'kl=l Ai, where for , E A i, IIB'G,,e-B'G,,elI2.<_ /2.We may assume without loss of generality that the sets A are disjoint.It follows that for all , C Ai, I[B'GA,e,i]I > 1/4.Hence I[B'FA,e'i]I > IIB'F,,elI2/4" But [i,B'FA,e] Ue(Bi)(A ).Hence I A IIB'FA,elldae(A < 16ii=l/A IUe(Bi)]2dae(A).Since Si W, the integral on the fight is finite by the spectral theorem.
To complete the proof of the theorem, we need a lemma, which has some independent interest.
Proof" Part iii) follows om the finiteness of the measure e' together with the boundedness of B'G,e in L2(X,p ).Since B is 1-1 d onto, part i) follows well.Since e is fite on A, L(A,e) c Ll(e).We show that @() W'.By Threm 1.8, we know that is a measurable function from a into L2(X,p ).Furthermore, @()() [1][2, so we s that () agrees on W with a unique element of L2(X,p) defined a the esz representation threm for Hilbert spaces by the relationsp @()(0)= la ()[0,B'G,e]d%() for 0e L2(X,p).
Part iv) is proved.Since (B') -1 is a continuous fine mapping om L2(X,p into W', (canse the injection of L2(X,p into W' is continuous, d (B') -1 is a continuous line trsformation in W'), where we once agn identify L2(X,p th its canocM emdng into W', we may use Theorem 8.14.5, p. 562, [4] to obtn part ii) and pt v).We prove pt vi).
If fi W, it fonows from part iv) that [,@()] IA ()['n'G,e]d%()" Also, ]A ('X)['B'GA,e]d#e ('x) ]A (Ue(B)/IIB'F,x,elI2){(U e 0)/llB'G,x,ell2}d/e (A)" But [[B'F,x,e[[2dae so the integral on the right becomes -fA (Ue(B))(Ue 0)dae('X), which d# e equals [B, by the spectral theorem; this theorem also guarantees that the integrand is in Ll(ae) and also that / ( L2(e ).Since e is finite on A, it follows that / ( LI(A,e).But if a= P(A)P(Se)0 [Be, a] a(B)= B'a(), again embedding L2(X,p canonically into W'.We have therefore seen that B'a (/), as we desired to show.The lemma is proved.We now show that Assertion ii) implies Assertion iii) of the theorem.For any positive real number , select a compact subset K of A such that h is a continuous function from K into L2(X,p), and such that #e(A\K) < /.Note that {h(A)] A e K} is a compact subset of L2(X,p ).
Let {v(i)} be a finite open cover of this set where each v(i) has diameter less than .Let subsequence, which we also denote by n' we see that Ue(n)(A converges to 0 for almost every with respect to a e and Ug(n)(A converges to 1 for almost every with respect to ag.Hence, since g is absolutely continuous with respect to e' we see that for almost every A with respect to g, F,e(n converges to 0 and FA,g(n) converges to 1.But FA,e must be a multiple of F%,g for almost every % with respect to g, since both vanish at 0. This is a contradiction; the theorem is proved.Remark: We now prove a negative result, showing that some hypotheses are necessary in order to diagonalize on a Borel set A. The first operator one might wish to diagonalize is the identity operator; this gives a discrete approximation to an inverse Fourier transform.Theorem 4.6.Let H be as in Definition g.1.Suppose that P(A) is diagonalizable in B(Y w ,Zw) with respect to H. Then p(A) is a compact operator from Y w into Z w.In particular, the identity operator P() is not diagonalizable in B(Y w ,Zw) with respect to H.
Remark: Since the embedding of Y w into L 2 is continuous, as is the embedding of L 2 into Zw, it follows that P(A) is in B(Y w ,Zw).Since the previous theorem showed that there exists an dement e of the domain of H such that S e L2, we do not need to consider P(Se).
Proof" Operators with finite dimensional range are compact, as are limits in operator norm of such operators.The first conclusion is therefore immediate.The embedding of Yw into Z w is dearly not compact, since on the interval [0,1] the norm of Yw is equivalent to that of Z w and to the norm of L2([0,1]).Theorem 4.7.Let H be as in Definition 4.1; let e be any element o/the domain of H such that S e L 2. Then a) for any bounded continuous function r: spectrum (H) P(A)r(H) is diagonalizable in B(Y w ,Zw) with respect to H; b) if the essential spectrum of H is not a bounded set, there exist subsets A of the spectrum d# e dae) and such that for all N, of H such that %(5) is finite (where ae(A\[-N,N]) > 0. In particular, for such A and any bounded continuous function r(H) of H, P(A)r(H) is diagonalizable with respect o H in B(Y w ,Zw), although A is not an essentially bounded set with respect to a e.
Remark: The first conclusion of the theorem does not state that #e is finite on bounded sets.This question is a difficult one, which we do not address here.
Proof: The first assertion will be proved in more generality in Theorem 5.5 of the next section.If the second assertion of the theorem is false, then the finiteness of #e (A) implies that for some N, ae(A\[-S,S])= 0. This implies that if a e([S+i,N+i+l)) {ai: a > 0} is bounded away from 0 in (0,(R)).Since the operator H is unbounded, if F {i: a > 0}, then F is glb {#e(J)" #e(J) > 0}, then {bj: e F} is also infinite.It is also clear that if b J[N+i,N+i+l) bounded away from 0. Suppose A > N. Either A is in the exceptional set of Theorem 4.3, or there exists a decreasing tower A n of Borel sets such that A N (R) A n and such that #e (An) is finite n=l and positive for each n.Since by hypothesis, #e(An) does not approach 0, this implies that A is a point mass.It follows that there exists a countable set {An} of points of the spectrum of H such that #e([N,(R))\{An} 0. Hence, the same assertion is true for e" These points A n are eigenvalues of H. Since the essential spectrum of H is unbounded, and the multiplicity of each A n is one, it follows that there exists an unbounded set {j} of cluster points of {An}. (It should be remarked that the A n are not necessarily arranged in increasing order.) Let n be the normalized eigenfunction corresponding to the eigenvalue A n.If c n [e,n]  then e(An)= ICn 12 But FAn,e anon for some complex number n" Hence Ue(e)(An) [e,FAn, j 1 -nCn; thus n 1/Cn" Thus 2 IIFAn,ell 2 IInll2/ICnl; then #e(An IIFAn,ellwe(A n) IIWnll2.
Let j be a cluster point of {An}.Let {Ar} converge to j. [Ar,As 0 for r # s.But on any compact interval [0,M], if I([0,M]) is the characteristic function of [1,M], then Ascoli's theorem guarantees that {R([0,M])Ar } has a cluster point gM in L 2. But if 7 r [gM,I([0,M])Ar], then the sequence 7r is square summable and 7r-IIl([0,M]) A 2 r converges to 0. But since Ar is normalized in L2, and rAr ArAr we see since the sequence A r is bounded that IIA I1(R) is also bounded.It follows that for each j, there exists at least one r (actually infinitely many) Ar(j such that IIWAr(j)112 <_ 1/j9", and such that Ar(j)-jl < 1.It follows that the sequence Ar(j) approaches infinity, and that if A {At(j)}, #e(A) is finite but e(A\[-N,N]) > 0 for all N.The theorem is proved.

DIAGONALIZATION IN L2(X,p
In this section we discuss diagonalization of p(A), not just P(A)P(Se) and show the equivalence of compactness of P(A) in B(Y, L2) and diagonalization in B(Y,Z) discussed in the introduction.For situations where the embedding of Y into L 2 is not compact, these properties generally hold for some but not all A. In applications to partial differential equations, it is often true that P(A) is known by other means to be compact for all bounded A; an example is the situation of Theorem 5.5.When A is unbounded but the embedding from Y into Z is not compact, compactness and diagonalization become delicate properties of A as we see for an example by the results of section 4.
Remark."The difference between the following definition and Definition 3.2 is only that we diagonalize P(A)r(H), not just P(A)P(Se)r(H ).For completeness, however, the definition is given in its entirety.
Definition 5.1.Assume the hypotheses of Assumptions 1.2 and 3.1 and Estimate 1.3.Let r be a bounded continuous function from the spectrum of H into the complexes.Let A be any Borel subset of .We say that Q--P(A)r(H) is diagonalizable in B(Y,Z) with respect to H if B'Qg E L2(X,p)for all g E Y, and the following is true: a) for every e > 0 there exists a positive integer k and a finite disjoint family {Ai}=lof m subsets of A such that there exists a finite set {ei} =1 of orthonormM elements of the domain of k such that Ai, A and such that, injecting Q0 H N and a finite set of real numbers {Ai,j}i=1 canonically into W', II{Q ii =1'=1 #ej(Ai r (i,j)Ri,j,ej}(0)ll Z < ellOlly for all Y, where b) Ai, is in the complement of the exceptional set for ej of Theorem 1.8, so that in particular F%i,j,e j is as in Theorem 1.8 and B'FA..,e.L2(X'P)' and where ,J c) RAi,j,ej(_ ) B' GAi,j,ej(A)GAi,j,e for any E , where GAi,j,ej FA',j"e'/IIB'FA"elI2'j and where B'GA,e denotes the complex conjugate of the functional B'GA,e.
Let M be a family of bounded continuous functions from the spectrum of the restriction of Hto S einto C. Let A bea Borel subset of.Let QM,A={P(A)r(H):rEM}" We say that QM,A is simultaneousl.]diagonalizable in B(Y,Z) with respect to H if for every > 0 there exists Ai, Ai,j and ej as above, which are independent of r, such that II{Q-i=lj =I #ej(Ai)(Ai,j)RAi,j,ej}(O)llZ <-ellOlly   for MI e Y and all Q e QM,6" Remark: The implication that diagonalizability implies compactness in B(Y,Z) follows from the fact that any operator with finite dimensional range is compact, and that the compact operators are a dosed subset of B(Y,Z).Theorem 5.2.Suppose that e hjpotheses of Asumption .1 and 1. hold, and that Estimate 1.3 holds.Suppose that Z contains L2(X,p), and that the injection from L2(X,p into Z is continuous.Let Ey denote tAe injection from Y into L2(X,p), and E Z denote the injection from Lf(X,p) into Z.Then i) PEy/s compact in B(Y,L2) if and only i.f P is dia7onalizable in B(Y,Z) uith respect to H, here P p(A) or P(A)P(Se); frther, EZPEy /s compact in B(Y,Z) if and only ifPEy /s compact in B(Y,L2); ii) Suppose that PEy/s compact in B(Y,L2).Then for any bounded continuous .functionr from the spectrum of H into the complezes, and any Borel subset of , including itself, Pr(H)/s diagonalizable ith respect to H. Furthermore, if M is a set of continuous fnctions from the spectrum of H into f, and M is uniformly bounded and efuicontinuous on A, then QM,A /s simultaneously diagonalizable.

Call for Papers
Thinking about nonlinearity in engineering areas, up to the 70s, was focused on intentionally built nonlinear parts in order to improve the operational characteristics of a device or system.Keying, saturation, hysteretic phenomena, and dead zones were added to existing devices increasing their behavior diversity and precision.In this context, an intrinsic nonlinearity was treated just as a linear approximation, around equilibrium points.Inspired on the rediscovering of the richness of nonlinear and chaotic phenomena, engineers started using analytical tools from "Qualitative Theory of Differential Equations," allowing more precise analysis and synthesis, in order to produce new vital products and services.Bifurcation theory, dynamical systems and chaos started to be part of the mandatory set of tools for design engineers.
This proposed special edition of the Mathematical Problems in Engineering aims to provide a picture of the importance of the bifurcation theory, relating it with nonlinear and chaotic dynamics for natural and engineered systems.Ideas of how this dynamics can be captured through precisely tailored real and numerical experiments and understanding by the combination of specific tools that associate dynamical system theory and geometric tools in a very clever, sophisticated, and at the same time simple and unique analytical environment are the subject of this issue, allowing new methods to design high-precision devices and equipment.
Authors should follow the Mathematical Problems in Engineering manuscript format described at http://www .hindawi.com/journals/mpe/.Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at http:// mts.hindawi.com/according to the following timetable: