A 2-component Camassa-Holm equation, Euler-Bernoulli Beam Problem and Non-Commutative Continued Fractions
aa r X i v : . [ n li n . S I] N ov A 2-component Camassa-Holm equation,Euler–Bernoulli Beam Problem andNon-Commutative Continued Fractions
Richard Beals * Jacek Szmigielski † November 12, 2020
Abstract
A new approach to the Euler-Bernoulli beam based on an inhomoge-neous matrix string problem is presented. Three ramifications of the ap-proach are developed:1. motivated by an analogy with the Camassa-Holm equation a classof isospectral deformations of the beam problem is formulated;2. a reformulation of the matrix string problem in terms of a certaincompact operator is used to obtain basic spectral properties of theinhomogeneous matrix string problem with Dirichlet boundary con-ditions;3. the inverse problem is solved for the special case of a discrete Euler-Bernoulli beam. The solution involves a non-commutative gener-alization of Stieltjes’ continued fractions, leading to the inverse for-mulas expressed in terms of ratios of Hankel-like determinants.
Contents -Component CH equation 53 Transfer to an interval 7 * Department of Mathematics, Yale University, New Haven, CT 06520, USA;[email protected] † Department of Mathematics and Statistics, University of Saskatchewan, 106 Wiggins Road,Saskatoon, Saskatchewan, S7N 5E6, Canada; [email protected] The Dirichlet problem for a beam 105 Spectral theory 146 Wronskians and Green’s kernels 217 The discrete beam 258 The inverse problem for the discrete beam 309 Appendix: The consistency conditions; the smooth case 36References 40
In 1993, Camassa and Holm [5] discovered the shallow water equation m t + ( um ) x + u x m = m = u − u xx , (1.1)where subscripts denote partial derivatives. The most attractive novel propertyof (1.1) is that it supports non-smooth soliton solutions. These peaked solitons( peakons ) are obtained from the ansatz u ( x , t ) = d X j = m j ( t ) e − | x − x j ( t ) | , (1.2)involving amplitudes m j ( t ) and positions x j ( t ) depending smoothly on time. Itwas shown in [5] that u given by (1.2) is a weak solution to the Camassa–Holmequation (1.1) if and only if the positions x j ( t ) and amplitudes m j ( t ) satisfy theHamiltonian system˙ x j = ∂ H ∂ m j = u ( x j ), ˙ m j = − ∂ H ∂ x j = − m j u x ( x j ), (1.3)with Hamiltonian H ( x , . . . , x n , m , . . . , m n ) = d X i , j = m i m j e − | x i − x j | ,and the convention that u x ( x i ) = D x u ® ( x i ) where f ® ( x i ) means the arith-metic mean of the left and right hand limits of f at x i .2he general solution of (1.3) (for arbitrary d ) was constructed in [2, 3, 4]using inverse spectral methods. The main premise of these papers was the re-alization that the CH equation can be viewed as an isospectral deformation ofthe classical inhomogeneous string problem studied in the 1950s by M.G. Krein[14, 13, 12] and, subsequently, by Dym and McKean in [7]; for a review see [8].In particular, the peakon solutions (1.2) were shown in [3] to be directly linkedto an isospectral deformation of discrete strings, i.e. those strings for which themass density is a linear combination of point masses. Krein had observed longago that, in the case of discrete strings, the inverse string problem can be solvedexplicitly by using results of Stieltjes on continued fractions [21]. Stieltjes’ meth-ods were applied in [3] to derive the determinantal formulas for the amplitudesand positions of peakons and, in turn, to determine the asymptotic behaviourof peakon solutions.It has been known at least since the fundamental paper [15] by Lax on theKorteweg–de Vries equation that the isospectral deformations of boundary valueproblems may lead to interesting non-linear equations. The boundary valueproblem for the KdV equation is given by the Schrödinger equation on the wholereal axis, in which case the spectrum is a union of continuous and discrete spec-tra.In the late 1970s, in a series of interesting papers [18, 19, 20], Sabatier putforward an idea that spectral problems with just discrete spectrum can also bea source of interesting non-linear problems derived via isospectral deforma-tions. In particular, the inhomogeneous string boundary value problem wassingled out as a potential source of interesting boundary problems with the dis-crete spectrum, to be subjected to isospectral deformations. Sabatier studied avery limited class of spectral deformations, applicable only to strictly positive,smooth densities. The situation changed with the discovery of the CH equation(1.1) and subsequent realization in [2, 3, 4] that the CH equation is, in disguise,an isospectral deformation of an inhomogeneous string boundary value prob-lem. This line of research was later generalized to other equations from the CHfamily ([16, 6, 10]). In the introduction to [4], the authors speculated, guidedby the work of Barcilon [1] on the Euler-Bernoulli beam problem, that higherorder boundary value problems might provide an equally rich environment forisospectral deformations. This task required a new look at the beam bound-ary value problem that would be naturally amenable to a Lax type deformation.The present paper aims to offer a new approach to the beam boundary valueproblem by rephrasing it as a matrix inhomogeneous string problem.In the remainder of this section we outline the main results of this paperand provide a context for some of the techniques used in our arguments.In Section 2 we propose a simple derivation of a system of nonlinear equa-tions that generalizes the CH equation to a new two component equation which3s structurally of the same type as the CH equation. We are keenly aware thatthere exist other two-component generalizations of the CH equation, for exam-ple, m t = ( um ) x + u x m + ρρ x , m = u − u xx , ρ t = ( ρ u ) x ,first derived using tri-hamiltonian methods by P.J. Olver and Rosenau in [17]and studied, for example, in [11]. Other generalizations have been proposed aswell [24].The equation we propose (see (2.5)) takes the form n t = ( un ) x + u x n + v n , m t = ( um ) x + u x m − v m , u xx − u = n + m ), v x = n − m ,and comes from a matrix valued Lax pair, structurally identical to the originalCH case, but involving two measures m and n rather than one.In Section 3, using a Liouville transformation, we map the problem to a fi-nite interval, following a similar procedure used in [2] to study an acoustic scat-tering problem. We note that the transformed x -member of the Lax pair (see(3.1)) is a matrix version of an inhomogeneous string boundary value problem.In Section 4 we review the pertinent facts about the Euler–Bernoulli beamproblem, which we show can be reformulated as a string problem with a matrixdensity – the same system already encountered in Section 3. We then study thebasic spectral properties of that matrix string problem.In Section 5 we reformulate the Euler–Bernoulli beam problem as a stan-dard spectral problem for a compact (in fact trace-class) operator T and westudy the properties of that operator on an appropriate Hilbert space. In partic-ular, we establish basic properties of the resolvent of that operator.In Section 6, in Theorem 6.4, we derive a closed-form expression for the re-solvent of T and introduce a special element of the resolvent, the Weyl function,which plays a key role in the formulation of the inverse problem.In Section 7 we analyze the spectral problem for a discrete Euler-Bernoullibeam, that is, a beam in which both measures m and n are chosen to be finitesums of point masses.. This is a beam counterpart of Stietljes’ string. Similar towhat occurs for the string problem, the Weyl function admits a continued frac-tion expansion (Proposition 7.2), albeit with non-commuting coefficients. Thegeneral concept of continued fractions over non-commutative rings goes backto Wedderburn [23], but our special non-commutative case is a direct general-ization of continued fractions of Stieltjes’ type, this time associated to a discreteinverse beam problem, rather than a string problem. The continued fraction ex-pansion can be rephrased in terms of non-commutative Padé approximations4nd we formulate Padé approximation conditions needed for the inverse prob-lem.In Section 8 we explicitly solve the inverse problem for the discrete Euler-Bernoulli beam. To this end we construct a sequence of non-commutative Padéapproximations using the Weyl function, or, to be more precise, the spectralmeasure, as an input data and, in the end, we recover the discrete measures m and n . In the process of solving the inverse problem we are prompted tointroduce several variations on the Hankel moment matrix that are germane tothe inverse beam problem. The final formulas bear a remarkable resemblanceto string formulas (see [4]) with the proviso that in the beam problem the usualHankel determinants of the moment matrix are replaced by determinants ofsuitably reduced Hankel matrices of moments.The paper concludes with (Appendix) Section 9, in which we provide a de-tailed analysis of the Lax pair parametrization giving rise to (2.5). -Component CH equation The Camassa-Holm (CH) equation (1.1) is the compatibility condition for a pairof scalar equations on the line, which we take for simplicity to have the form ψ xx = (1 + λ m ) ψ , ψ t = a ψ + b ψ x , (2.1)where −∞ < x < ∞ , and the subscripts in x and t represent distribution deriva-tives in x , t respectively. Remark 2.1.
In general, the derivatives we consider are distribution derivatives.It will sometimes be convenient to use D x , D t , or simply D in the case of onevariable.The CH flow corresponds to the choice b = u + λ , a = − u x m t = ( um ) x + u x m , ( u xx − u ) x = m x . (2.2)This compatibility condition holds even if m is a measure, in which case u x is ofbounded variation, u is continuous, and the term u x m means that on the sin-gular support of m the multiplier is taken to be u x ( a ) = u x ® ( a ), where f ® ( a )is [ f ( a − ) + f ( a + )]. 5 emark 2.2. Our choice of the coefficients in (2.1) differs slightly from the orig-inal Lax pair in [5]. This change results in a different relation between u and m as one can see by comparing (1.1) and (2.2).Now, we consider a two-component version Ψ xx = ( + λ M ) Ψ , M = · nm ¸ , (2.3a) Ψ t = a Ψ + b Ψ x , (2.3b)where Ψ , a and b are 2 × x , t , λ and denotes the 2 × a , b have only terms of degree 0, − λ . As shown in the Appendix, if we add the assumption that the matrices a and b are bounded as x → ±∞ , then, up to a normalization, they have the form b = · u u ¸ + λ · ¸ , a = − · u x − v u x + v ¸ . (2.4)The compatibility conditions split into two constraints( u xx − u ) x = n + m ) x , v x = n − m ,and a system of evolution equations n t = ( un ) x + u x n + v n , m t = ( um ) x + u x m − v m . (2.5)Furthermore, if we are interested in bounded u and compactly supported m , n ,then we can replace one of the constraints with a more restrictive one, keepingthe other constraint intact, u xx − u = n + m ), v x = n − m . (2.6)We assume that M is compactly supported and take as solutions of (2.6) theparticular choices u ( x , t ) = − Z ∞−∞ e − | x − y | [ m ( y , t ) + n ( y , t )] d y , (2.7) v ( x , t ) = Z ∞−∞ sgn( x − y )[ n ( y , t ) − m ( y , t )] d y . (2.8)It follows that lim x →±∞ u ( x , t ) = = lim x →±∞ u x ( x , t )lim x →±∞ v ( x , t ) = ± R ∞−∞ [ n ( y , t ) − m ( y , t )] d y . (2.9)6 emark 2.3. The integrals appearing in this paper are all Stieltjes integrals, butwe find it more convenient to write them as R f ( x ) m ( x , t ) d x , or, later in thepaper, as R f ( x ) d m ( x , t ) . The former notation is analogous to writing the pointmass at the origin as δ ( x ) d x . Proposition 2.4.
The integral Z ∞−∞ [ m ( y , t ) + n ( y , t )] d yis independent of t .Proof. According to (2.5), (2.9), D t Z ∞−∞ [ m ( y , t ) + n ( y , t )] d y = Z ∞−∞ {[ u ( n + m )] y + u y ( n + m ) + v ( n − m )} d y = Z ∞−∞ [ u y u y y − u y u + v v y ] d y = Z ∞−∞ [ u y − u + v ] y d y = Remark 2.5.
Note, however, that separately R ∞−∞ md y and R ∞−∞ nd y are not con-served. Indeed, it follows from (2.6) that D t Z ∞−∞ [ n − m ] d y = − Z ∞−∞ uv d y . As in [3], we transfer the problem on the real axis to the interval [ −
1, 1]. In thissection we will refer to the variable on the real axis as ξ and its counterpart on[ −
1, 1] by x . The functions originally defined on the real axis will carry a tilde.Thus, for example, M from the previous section will be denoted by ˜ M . We set x = tanh ξ , D x = cosh ξ D ξ Define, for general f : R → R , f ∗ (tanh ξ ) = f ( ξ ).Note that sech ξ = − x . Then a straightforward computation shows that(1 − x ) D x ( f ∗ (1 − x ) − ) = [( D ξ − f ] ∗ .7herefore[( D ξ − − λ ˜ M ) f ] ∗ = (1 − x ) [ D x − λ (1 − x ) − ˜ M ]( f ∗ (1 − x ) )and solutions to (2.3a), after flipping x with ξ and M with ˜ M , correspond tosolutions of Φ xx = λ M Φ (3.1)under the map Φ = (1 − x ) Ψ ∗ , M = (1 − x ) − ˜ M ∗ .The flow equation (2.3b) implies D t Φ = [(cosh ξ ) − D t Ψ ] ∗ = [(cosh ξ ) − ( ˜ a Ψ + ˜ bD ξ Ψ ] ∗ = [(cosh ξ ) − ˜ a Ψ + (cosh ξ ) − ˜ b cosh ξ D ξ Ψ ] ∗ = ˜ a ∗ Φ + (1 − x ) ˜ b ∗ D x ¡ (1 − x ) − Φ ¢ = ˜ a ∗ Φ + (1 − x ) ˜ b ∗ £ D x + x − x ¤ Φ = [ ˜ a ∗ + x ˜ b ∗ ] Φ + (1 − x ) ˜ b ∗ D x Φ .Thus Φ t = a Φ + b Φ x , (3.2)where a = ˜ a ∗ + x ˜ b ∗ , b = (1 − x ) ˜ b ∗ ,or, more explicitly, a = · − u x + v − u x − v ¸ − λ · β x β x ¸ , b = · u u ¸ + λ · ββ ¸ , u = (1 − x ) ˜ u ∗ , v = ˜ v ∗ , β = − x . (3.3)It can be shown that u xxx = ( β ( m + n )) x + β x ( m + n ), v x = β ( n − m ), (3.4)and that the flow of M takes the same form as the flow of ˜ M , namely, n t = ( un ) x + u x n + v n , m t = ( um ) x + u x m − v m . (3.5)In fact (3.4) and (3.5) are consequences of the compatibility conditions for theLax pair (3.1) and (3.2). It follows from (2.9) and (3.3) that u ( ± = u x ( ± = v ( − = − v ( + roposition 3.1. The integral Z − β [ m + n ] d xis independent of t .Proof. According to (3.5), (3.4) and (3.6), D t Z − β ( x )[ m ( x , t ) + n ( x , t )] d x = Z − { β ( x )[ u ( n + m )] x + u x ( n + m ) + v ( n − m )} d x = Z − [ − β x u ( m + n ) − u ( β ( m + n )) x + v v x ] d x = Z − ( − u xxx u + v x v ) d x = Z − [ u x + v ] x d x =
12 ( u x + v ) ¯¯ − = u and v , which, as canbe easily checked, provide the unique solutions of (3.4) subject to boundaryconditions (3.6). Proposition 3.2. u ( x , t ) = − Z − G D ( x , y ) ( m ( y , t ) + n ( y , t )) d y , v ( x , t ) = Z − sgn( x − y ) G D ( y , y )( n ( y , t ) − m ( y , t )) d y .Here G D ( x , y ) = ( (1 + x )(1 − y ), x < y (1 − x )(1 + y ), y < x (3.7)is the Green’s function of the classical Dirichlet string problem − D x f = λρ f , f ( − = f (0) = The Dirichlet problem for a beam
Vibrations of a beam parametrized by the interval − ≤ x ≤ D [ r D φ ] = λ m φ , D = D x ; (4.1)(see [1] or, for a comprehensive view of vibration problems in engineering, [9]).What we refer to as a beam problem is often referred to as an Euler–Bernoullibeam problem . The two functions (or positive measures) r , m are the flexuralrigidity and mass density. The spectral parameter λ denotes the square of thefrequency.Setting D ϕ = λ n ϕ , (4.2)where n = r , (4.1) becomes D ϕ = λ n ϕ , D ϕ = λ m ϕ . (4.3)The matrix form is D ϕ = λ M ϕ , ϕ = · ϕ ϕ ¸ , M = · nm ¸ . (4.4)We require that ϕ be continuous. We want to allow m and n to be finitepositive measures on the interval − ≤ x ≤
1. We assume(a) The endpoints x = ± m or n .(b) The measures m and n have the same support. (4.5)Then M ϕ should be interpreted as M ϕ = · ϕ n ϕ m ¸ . Remark 4.1.
Throughout this section and the next it is convenient to adopt the(more correct) way of writing integrals with respect to m and n , using d m ( x ), d n ( x ) rather than the symbolic m ( x ) d x , n ( x ) d x .A partial fundamental solution Φ ( x , λ ) with the property Φ ( − λ ) = D Φ ( − λ ) = ,the identity matrix, can be constructed in the form Φ ( x , λ ) = ∞ X k = λ k Φ k ( x ) (4.6)with Φ ( x ) = (1 + x ) , Φ k + ( x ) = Z x − ·Z y − M ( z ) Φ k ( z ) d z ¸ d y .To be specific, we take the integrals to run on the intervals [ − x ) and [ − y ).10 roposition 4.2. The functions Φ k ( x ) are diagonal for even k, off-diagonal forodd k. The non-zero entries Φ k , Φ k are non-negative, positive at x = , andsatisfy the estimate ≤ Φ k j ( x ) ≤ + x ) k ( m + n ) k k k k ! . j =
1, 2. (4.7) where m and n denote the total mass of m and n.Proof.
The first assertions follow by induction from the construction (4.6) andthe assumption that m and n are positive measures. To prove the estimate, let p = m + n , and define ψ k by ψ ( x ) = + x , ψ k + ( x ) = Z x − Z y − ψ k ( z ) d p ( z ) d y .Each Φ k j ( x ) is ≤ ψ k ( x ). Changing the order of integration and integrating firstwith respect to y , the result can be written as ψ k ( x ) = Z x − ( x − y k ) ψ k − d p ( y k ) = Z x − Z y k − £ ( x − y k )( y k − y k − ) ¤ ψ k − ( y k − ) d p ( y k − ) d p ( y k ) = Z x − · · · Z y − £ ( x − y k )( y k − y k − ) · · · ( y − y ) ¤ ψ ( y ) d p ( y ) d p ( y ) · · · d p ( y k ).The product in brackets is maximized when the k factors are all the same. Theinterval [ y , x ] has length at most 1 + x , and ψ ( x ) ≤
2, so0 ≤ ψ k ( x ) ≤ · + xk ¸ k Z ≤ y <···< y k < x d p ( k ) ( y , y , . . . y k ), (4.8)where p ( k ) denotes the product measure on the k -cube {( x , . . . x k ) : | x j | ≤
1} in R k . The domain of integration is one of k ! pairwise disjoint domains in thecube that are obtained by permuting the indices. Each domain has the samemeasure, so 0 ≤ Z ≤ y <···< y k < x d p ( k ) ( y , y , . . . y k ) ≤ £R x − d p ( y ) ¤ k k ! . (4.9)Since R − d p ( x ) = m + n , the estimates (4.8) and (4.9) imply (4.7). . Corollary 4.3. (a) The function Φ ( x , λ ) is continuous in both variables, and en-tire as a function of λ , for fixed x.(b) Each entry of Φ (1, λ ) is dominated by ∞ X k = a k | λ | k (2 k ) ! ≤ a | λ | ), a = p m + n ). (4.10)11 Proof. (a) The estimates (4.7) imply that the series (4.6) converges uniformly onbounded sets in [ −
1, 1] × C .(b) Since (2 k ) ! ≤ (2 k ) k k !, the estimates (4.7) imply the bound (4.10).We are interested in the Dirichlet problem for solutions of (4.4): ϕ ( ± λ ) = = ϕ ( ± λ ). (4.11)A value λ ∈ C for which a non-zero solution of (4.4), (4.11) exists will be referredto as a Dirichlet eigenvalue. Note that zero is not an eigenvalue. Proposition 4.4.
The Dirichlet eigenvalues { λ ν } are precisely the zeros of ∆ , where ∆ ( λ ) = det Φ (1, λ ). They satisfy X ν | λ ν | < ∞ . (4.12) Proof.
Any solution ϕ of the Dirichlet problem with eigenvalue λ is a linearcombination of the two columns of Φ ( · , λ ): ϕ ( x ) = Φ ( x , λ ) v where v is a constant 2-vector. The condition at x = Φ (1, λ ) v = Φ (1, λ ) =
0. Corollary 4.3 implies that | ∆ ( λ ) | is dominated by exp(4 a | λ | ). Thereforethe zeros λ ν , if numbered with | λ ν | non-decreasing, satisfy | λ ν | ≥ c ν (4.13)for some constant c >
0. [22, §8.21]. In particular, (4.12) is true.At the end of this section we will show that the zeros of ∆ are simple.If ϕ is a solution of the Dirichlet problem (4.4), (4.11), then f = D ϕ is asolution of D f = λ M ϕ . (4.14)It follows that f is a function of bounded variation which is continuous at anypoint that is not an atom. In particular, f is continuous at the endpoints ± f and similar functions, justifies the various integration-by-partsformulas in this and later sections.The following identity is fundamental to our discussion of the Dirichlet prob-lem. 12 emma 4.5. If ϕ is a solution of the Dirichlet problem with eigenvalue λ , andf = D ϕ , then Z − [ f f + f f ] d x = − λ Z − [ | ϕ | d m + | ϕ | d n ]
0. (4.15)
Proof.
It is convenient to write the left side in the form ( D ϕ , σ D ϕ ), where theinner product ( , ) is the L inner product for vector-valued functions:( f , g ) = Z − f ( x ) · g ( x ) d x = Z − [ f ( x ) g ( x ) + f ( x ) g ( x )] d x , (4.16)and σ is the matrix σ = · ¸ . (4.17)Then integration by parts gives( D ϕ , σ D ϕ ) = − ( D ϕ , σϕ ) = − λ ( M ϕ , σϕ ), (4.18)which is (4.15).If the right side of (4.15) is zero, then ϕ = m and ϕ = n . As a consequence( D ϕ , D ϕ ) = − λ ( M ϕ , ϕ ) = − λ Z − [ ϕ ϕ d m + ϕ ϕ d n ] = ϕ is constant, hence zero, a contradiction. Theorem 4.6.
The Dirichlet eigenvalues satisfy the following conditions:(a) If λ is an eigenvalue, so is − λ .(b) Each eigenvalue λ is real and its eigenspace has dimension one.Proof. If ϕ = [ ϕ , ϕ ] t is an eigenvector with eigenvalue λ , it follows immedi-ately from (4.3) that [ ϕ , − ϕ ] t has eigenvalue − λ . The complex conjugate ϕ has eigenvalue λ . Integration by parts shows that( D ϕ , ϕ ) = ( ϕ , D ϕ ) = ( D ϕ , ϕ ),so Lemma (4.5) and (4.18) show that0 λ Z − [ | ϕ | d m + | ϕ | d n ] = λ Z − [ | ϕ | d m + | ϕ | d n ].Therefore λ = λ . Then the singular matrix Φ (1, λ ) has positive diagonal entries,so it has two eigenvalues, namely 0 and tr Φ (1, λ ) >
0. Thus the eigenspace for λ has dimension 1. 13 roposition 4.7. The zeros of ∆ = det Φ (1, λ ) are simple.Proof. Write Φ (1, λ ) = · a ( λ ) b ( λ ) c ( λ ) d ( λ ) ¸ v ( λ ) = · d ( λ ) − c ( λ ) ¸ ,and let ϕ ( x , λ ) = Φ ( x , λ ) v ( λ ). The entries of Φ (1, λ ) are non-zero, so ϕ
0. Wehave ϕ ( − λ ) = and ϕ (1, λ ) = · ϕ (1, λ ) ϕ (1, λ ) ¸ = · ∆ ( λ )0 ¸ .We want to prove that the derivative D λ ϕ (1, λ ) does not vanish if λ is an eigen-value. Differentiating with respect to λ shows that at an eigenvalue( D x − λ M ) D λ ϕ = M ϕ .Therefore, by Lemma (4.5), and because all terms here are real, Z − [ | ϕ | d m + | ϕ | d n ] = ( M ϕ , σϕ ) = ( D x D λ ϕ , σϕ ) − ( λ M D λ ϕ , σϕ ) = − ( D x D λ ϕ , σ D x ϕ ) − λ ( M ϕ , σ D λ ϕ ) = − D λ ϕ · σ D x ϕ ¯¯ − + [( D λ ϕ , σ D x ϕ ) − ( λ M ϕ , σ D λ ϕ )] = − D λ ϕ (1, λ ) D x ϕ (1, λ ).Therefore D λ ∆ ( λ ) = D λ ϕ (1, λ ) In this section we rephrase the Dirichlet problem for the Euler-Bernoulli beamas a standard eigenvalue problem for a compact operator, whose eigenfunc-tions are the derivatives D ϕ .Let H be the L space of vector-valued functions on the interval [ −
1, 1], withthe inner product 4.16 and norm || f || = ( f , f ) .We also introduce an indefinite form 〈 f , g 〉 = Z − [ f ( x ) g ( x ) + f ( x ) g ( x )] d x = ( σ f , g ) = ( f , σ g ) (5.1)where σ is the matrix (4.17). 14e know that the Dirichlet eigenvalues { λ ν } are real, and we take their eigen-vectors { ϕ ν } to be real as well. Lemma 4.5 can be rephrased as 〈 f ν , f ν 〉 = − λ ν Z − [ ϕ ν ,1 d m + ϕ ν ,2 d n ]
0. (5.2)Thus 〈 f ν , f ν 〉 and λ ν have opposite signs. Because of this we index the eigenval-ues with . . . < λ < λ < < λ − < λ − < . . . . (5.3)The calculation that led to (5.2) leads to two formulations for 〈 f ν , f µ 〉 that show 〈 f ν , f µ 〉 = µ ν . (5.4)Let H ⊂ H consist of the constant functions, and let H be the orthogonalcomplement: H = ½ f ∈ H : Z − f = ¾ .Note that each f ν belongs to H , since R − f ν = ϕ ν (1) − ϕ ν (0) = D = D x that map H to H : D − g ( x ) = ( R x − g ( y ) d y − R x g ( y ) d y , g ∈ H ,0, g ∈ H , (5.5)and D − g ( x ) = Z x − ( y + g ( y ) d y + Z x ( y − g ( y ) d y . (5.6) Lemma 5.1.
For any g ∈ H ,D − g ( − = D − g (1) = D − g ∈ H . Proof.
The first pair of identities is clear for g ∈ H and true by definition for g ∈ H . The third identity follows from an easy calculation.In view of these remarks, the Dirichlet eigenvalue problem can be reformu-lated as D f = λ M D − f , f ∈ H .or, equivalently, f ∈ H , f = λ T f , T = D − M D − . (5.7)Thus the problem (4.4), (4.11) is equivalent to a standard eigenvalue problemfor the operator T mapping H to H or H to H . Note that T = H , sincethis is true, by definition of D − . 15 emma 5.2. The operator T is a compact operator in H . If f is in the image of T ,then f has bounded variation and is continuous at the endpoints x = ± .Proof. Any inverse D − takes L functions to functions that satisfy a Hölder con-dition: if u = D − f and x < y , then Du = f so | u ( y ) − u ( x ) | ≤ Z yx | f ( t ) | d y ≤ ·Z yx | f | ¸ ·Z yx d t ¸ ≤ || f || ( y − x ) .It follows that the image under D − of a bounded sequence in H is uniformlyuniformly equicontinuous. The same argument applied to the two summandsin (5.5) shows that | D − f ( x ) | ≤ || f || , so the image is also uniformly bounded.By the theorem of Ascoli–Arzelá there is a uniformly convergent subsequence,{ g n }. Thus D − is compact from H to the space of bounded continuous func-tions. The map g → D − g M from this space to H is bounded, so T is compact.Moreover g → D − g M maps to functions of bounded variation; continuity of T f at the endpoints follows from assumption (a) of (4.5).
Lemma 5.3.
If K (0) is a closed subspace of H that is invariant under T , thenK contains an eigenfunction of T .Proof. Since T is compact, every non-zero point of the spectrum is an eigen-value. This applies also to the restriction of T to the invariant subspace K . Com-pactness implies that the operator I − λ T is Fredholm with index zero. If therestriction to K has no null space, for all λ ∈ C , then it is invertible, and the in-verse is entire and bounded at λ = ∞ , leading to a contradiction. Therefore T has an eigenvector in K . Lemma 5.4.
The operator T is symmetric with respect to the indefinite form 〈 , 〉 .Proof. We may assume that f and g are in H and are smooth. Since g = D [ D − g ] and D − g ( ± = 〈 T f , g 〉 = ( T f , σ DD − g 〉 = − ( DT f , σ D − g ) = − ( M D − f , σ D − g ) = − ( D − f , M t σ D − g ). (5.8)Since M t σ is diagonal, the last expression is symmetric in f and g .Let H T be the closure in H of the range of T . Any solution of (5.7) will belongto H T . Proposition 5.5.
The span of the eigenfunctions { f ν } is dense in H T . roof. Let N T ∈ H T be the orthogonal complement of { f ν } in H T with respectto the standard inner product. Then σ N T is orthogonal to H T with respect tothe indefinite form. Since T is symmetric, σ N T is invariant for T . By construc-tion, σ N T is orthogonal to every eigenfunction. It follows from Lemma 5.3 that σ N T = (0). Therefore N T = (0). Proposition 5.6.
Let N be the null space of T , i.e. N = { f ∈ H ; T f = . ThenN = σ N . Moreover, N coincides with each of the subspaces(a) The orthogonal complement of H T with respect to ( , ) ;(b) The orthogonal complement of H T with respect to 〈 , 〉 .Proof. Note that
T f = f ∈ H , in which case σ H = H ,or f ∈ H and M ϕ =
0, where ϕ = D − f . This, in turn, is equivalent to theconditions that ϕ vanish on the support of m and ϕ vanish on the support of n . In view of condition (b) of (4.5), this is equivalent to M σϕ =
0, so T σ f = f ∈ N .The relations between N and the spaces (a), (b) follow from the identities,valid for every f , g in H :( f , T g ) = 〈 σ f , T g 〉 = 〈 T ( σ f ); 〈 T f , g 〉 = 〈 f , T g 〉 .The first identity shows that f is in space (a) if and only if T ( σ f ) =
0, whichis equivalent to
T f =
0. The second shows that f is in space (b) if and only if T f = H T into two subspaces that are orthog-onal with respect to the indefinite form: H ± T = closure of the span of { f ν : ± ν > H T by setting( f , g ) T = 〈 f , g 〉 , f , g ∈ H + T ; −〈 f , g 〉 , f , g ∈ H − T ;0, f ∈ H + T , g ∈ H − T . (5.9)Let b H T be the completion of H T with respect to the norm || f || T = ( f , f ) T . Notethat in all cases( f , f ) T = |〈 f , f 〉| = | ( f , σ f ) | ≤ || f || || σ f || = || f || . (5.10)17he { f ν } are clearly an orthonormal basis for this space. Since T f ν = λ − ν f ν , therestriction of T to H T extends to a compact self-adjoint operator in b H T witheigenvalues (in the usual sense) { λ − ν }.The orthogonal projection of b H T onto the span of the eigenfunction f ν is E ν f = 〈 f , f ν 〉〈 f ν , f ν 〉 f ν = ( f , σ f ν ) 〈 f ν , f ν 〉 f ν . (5.11)Abusing notation, we write E ν also for the kernel of the operator (5.11): E ν f ( x ) = Z − E ν ( x , y ) f ( x ) d x , f ∈ H T E ν ( x , y ) = 〈 f ν , f ν 〉 · f ν ,1 ( x ) f ν ,2 ( y ) f ν ,1 ( x ) f ν ,1 ( y ) f ν ,2 ( x ) f ν ,2 ( y ) f ν ,2 ( x ) f ν ,1 ( y ) ¸ . (5.12)We may also view the operator E ν as a projection of H onto the span of f ν . Weview the kernels E ν as belonging to the Hilbert space H (2) = L ( I × I ; M (2, C ))of mappings from the square I × I = [ −
1, 1] × [ −
1, 1] to the space M (2, C ) of com-plex 2 × D − as κ . Since | y ± | ≤ y ∈ [ −
1, 1], we have | κ | ≤ r is a bounded positive measure on the interval [ −
1, 1], then | D − r ( x ) | = ¯¯¯¯Z − κ ( x , y ) d r ( y ) ¯¯¯¯ . Lemma 5.7.
Suppose that r is a bounded positive measure and ψ is a continuousfunction on the interval [ −
1, 1] . Then | [ D − ψ r ]( x ) | ≤ r Z − ψ ( y ) d r ( y ), r = Z − d r ( y ). (5.13) Proof.
This follows by applying the Cauchy–Schwarz inequality to the term onthe right in the inequality ¯¯ [ D − ( ψ r )]( x ) ¯¯ = ¯¯¯¯Z − κ ( x , y ) ψ ( y ) d r ( y ) ¯¯¯¯ ≤ Z − | ψ ( y ) | d r ( y ). Proposition 5.8.
Each element of the kernel (5.12) has absolute value boundedby ( m + n ) | λ ν | , where m = R − d m and n = R − d n. roof. Since f ν = λ ν T f ν , we have f ν ,1 = λ ν D − ( ϕ ν ,2 n ), ϕ ν = D − f ν .By Lemma 5.7, | f ν ,1 ( x ) | ≤ λ ν n Z − ϕ ν ,2 d n ≤ n | λ ν 〈 f ν , f ν 〉| .Similarly, | f ν ,2 ( x ) | ≤ m | λ ν 〈 f ν , f ν 〉| .Therefore each entry of E ν is bounded by one of | λ ν | m , | λ ν | n , or | λ ν | ( mn ) .For later use we introduce the operator R λ defined by R λ = I − ( I − λ T ) − = − λ T (1 − λ T ) − . (5.14)The operator R λ maps H to H T and is compact, as is the extension to b H T of itsrestriction to H T . Since T f ν = λ − ν f ν , R λ f ν = − λλ ν µ − λλ ν ¶ − f ν = λλ − λ ν f ν .Therefore, we have a formal expansion R λ = X ν λλ − λ ν E ν (5.15)with a formal kernel b K λ ( x , y ) = X ν λλ − λ ν E ν ( x , y ) = X ν λλ − λ ν 〈 f ν , f ν 〉 · f ν ,1 ( x ) f ν ,2 ( y ) f ν ,1 ( x ) f ν ,1 ( y ) f ν ,2 ( x ) f ν ,2 ( y ) f ν ,2 ( x ) f ν ,1 ( y ) ¸ . (5.16)The question is: does this series converge, in some sense, to the kernel of R λ ? Theorem 5.9.
For each λ that is not in the set of eigenvalues { λ ν } , the partialsums of the series on the right in (5.16) converge weakly in H (2) . The weak limit b K λ is the kernel for R λ .Proof. For | λ ν | > | λ |, | ( λ − λ ν ) − | is less than 2/ | λ ν | . By Proposition 5.8, thecorresponding summand is O (1) as an element of H (2) . Linear combinationsof matrix functions f ( x ) g ( y ) t , f , g ∈ H are dense in H (2) . If either f or g is in19 + span{ f ν }, integration against (5.16) yields a finite sum which is ( f , R λ g ). Thisproves the weak convergence. Let b K λ be the weak limit. As an element of H (2) ,it induces a bounded operator in H . This operator agrees with R λ on a densesubspace, so it is R λ .In the next section we derive a closed-form expression for the kernel b K λ .Theorem 5.9 can be strengthened considerably under a strengthening of theassumption (4.5) (b), that m and n have the same support: namely that each isdominated by the other. This can be put in the form(b ′ ) There is a constant C such that m + n ≤ C m and m + n ≤ C n . (5.17)
Lemma 5.10.
Under assumption (5.17) , for each eigenfunction f ν , |〈 f ν , f ν 〉| ≤ ( f ν , f ν ) ≤ C |〈 f ν , f ν 〉| . (5.18) Proof.
The first inequality is (5.10). To prove the second inequality, let ϕ ν = D − f ν . Then( f ν , f ν ) = ( D ϕ ν , D ϕ ν ) = − ( D ϕ ν , ϕ ν ) = − λ ν ( M ϕ ν , ϕ ν ) (5.19)and ( M ϕ ν , ϕ ν ) = ·Z − ϕ ν ,1 ϕ ν ,2 d ( m + n ) ¸ ≤ Z − ϕ ν ,1 d ( m + n ) Z − ϕ ν ,2 d ( m + n ) ≤ C Z − ϕ ν ,1 d m Z − ϕ ν ,2 d n ≤ C ½Z − [ ϕ ν ,1 d m + ϕ ν ,2 d n ] ¾ = C λ − ν 〈 f ν , f ν 〉 . (5.20)Together, (5.19) and (5.20) establish the second inequality in (5.18).This result leads to the following strengthening of the previous convergenceresult. Theorem 5.11.
Under assumption (5.17) , for each λ that is not an eigenvalue,the series (5.16) converges in L norm to the kernel of R λ . Remark 5.12.
The preceding arguments can easily be extended to similar series.The formal series X ν λ ν E ν ( x , y )20onverges weakly to the kernel of T , and under the assumption (5.17) it con-verges in L norm. Also, under assumption (5.17) the formal series X ν E ν ( x , y )converges weakly to the kernel of the orthogonal projection of H onto H T . Let Φ ( x , λ ) be the partial fundamental matrix solution of (4.4), normalized at x = −
1, as constructed earlier. Let Ψ ( x , λ ) be the matrix solution normalized at x = D Φ = λ M Φ , Φ ( − λ ) = D Φ ( − λ ) = ; D Ψ = λ M Ψ , Ψ (1, λ ) = D Ψ (1, λ ) = − .As for Φ , condition (4.5) (a) implies that Ψ and D x Ψ are continuous at x = ± C is a matrix, let b C = C t σ . (6.1)Differentiating shows that quasi-Wronskians like b Φ x Ψ − b ΦΨ x are constant; forexample D [ b Φ x Ψ − b ΦΨ x ] = ( λ Φ t M t σ Ψ + Φ tx σ Ψ x ) − ( Φ tx σ Φ x + λ Φ tx σ M Ψ ) = M t σ = σ M t . The value of the constant can be computed by taking x = ± λ as given, λ ∈ ( C \ R )): b Φ x Φ − b ΦΦ x =
0, so b Φ − b Φ x = Φ x Φ − ; (6.2) b Ψ x Ψ − b ΨΨ x =
0, so b Ψ − b Ψ x = Ψ x Ψ − ; (6.3) b Φ x Ψ − b ΦΨ x = − C − = σ Ψ ( − λ ) = b Φ (1, λ ); (6.4) b Ψ x Φ − b ΨΦ x = C + = − σ Φ (1, λ ) = − b Ψ ( − λ ). (6.5)Combining some of these identities we find that C − = − b Φ [ b Φ − b Φ x − Ψ x Ψ − ] Ψ = b Φ A Ψ ; (6.6) C + = b Ψ [ b Ψ − b Ψ x − Φ x Φ − ] Φ = b Ψ A Φ ; (6.7)where A ( x , λ ) = Ψ x Ψ − − Φ x Φ − . (6.8)21he identities (6.4) and (6.7) lead to two additional important identities: Ψ C − − b Φ − Φ C − + b Ψ = A − − A − = Ψ x C − − b Φ − Φ x C − + b Ψ = Ψ x Ψ − A − − Φ x Φ − A − = . (6.10)The identities (6.2), (6.3), (6.6), (6.7), (6.8), (6.9) and (6.10) call for some dis-cussion. Lemma 6.1.
Suppose λ is not real. Then the matrix functions Φ ( x , λ ) and Ψ ( x , λ ) are invertible for x ∈ ( −
1, 1] , x ∈ [ −
1, 1) , respectively.Proof.
Suppose that Φ ( x , λ ) is not invertible at some point x in the interval( −
1, 1). Then x > − λ is a Dirichlet eigenvalue for the beam problem re-stricted to the interval [ − x ]. Therefore λ is real. The same argument appliesto Ψ for an interval [ x , 1].It follows that all the expressions above are well-defined when λ is not realand | x | <
1. Moreover, (6.4) and (6.5) imply that C − and C + are invertible if λ isnot real. In turn, these imply that A is invertible if λ is not real and | x | <
1. Insummary,
Corollary 6.2.
The identities (6.2) – (6.10) are valid for each non-real λ and | x | < . Theorem 6.3.
The matrix functionG λ ( x , y ) = ( Ψ ( x , λ ) C − − b Φ ( y , λ ), y < x ; Φ ( x , λ ) C − + b Ψ ( y , λ ), y > x . (6.11) is the Green’s kernel for the equation D u − λ Mu = f , λ ∉ R .Proof. Let u ( x ) = Z − G λ ( x , y ) f ( y ) d y = Ψ ( x ) Z x − C − − b Φ ( y ) f ( y ) d y + Φ ( x ) Z x C − + b Ψ ( y ) f ( y ) d y . (6.12)Then, using (6.9), Du ( x ) = [ Ψ ( x ) C − − b Φ ( x ) − Φ ( x ) C − + b Ψ ( x )] f ( x ) + Ψ x ( x ) Z x − C − − b Φ ( y ) f ( y ) d y + Φ x ( x ) Z x C − + b Ψ ( y ) f ( y ) d y = + Ψ x ( x ) Z x − C − − b Φ ( y ) f ( y ) d y + Φ x ( x ) Z x C − + b Ψ ( y ) f ( y ) d y .22herefore, using (6.10), D u ( x ) = λ M ( x ) u ( x ) + [ Ψ x C − − b Φ − Φ x C − + b Ψ ]( x ) f ( x ) = λ M ( x ) u ( x ) + f ( x ).The kernel G λ can be used to calculate the kernel for the operator R λ de-fined in (5.14). Given f ∈ H with g = D f integrable, consider the inhomoge-neous problem ( D − λ M ) u = g (6.13)with solution u ( x ) = Z − G λ ( x , y ) g ( y ) d y .Let v = Du , so (6.13) is equivalent to D v − λ M D − v = D f or v − λ T v = f , which is the same as v = f − R λ f . (6.14)Now u ( x ) = Z − G λ ( x , y ) g ( y ) d y .Thus the solution to (6.14) is, using (6.9) and (6.10) again, v ( x ) = Du ( x ) = Ψ x ( x ) Z x − C − − b Φ ( y ) D f ( y ) d y + Φ x ( x ) Z x C − + Φ ( y ) D f ( y ) d y = f ( x ) − Ψ x ( x ) Z x − C − − b Φ y ( y ) f ( y ) d y − Φ x ( x ) Z x C − + b Ψ y ( y ) f ( y ) d y .This shows that the kernel[ G λ ] x y = ( Ψ x ( x ) C − − b Φ y ( y ), y < x ; Φ x ( x ) C − + b Ψ y ( y ), y > x . (6.15)generates R λ for functions in H . The operator R λ is zero on H , while the inte-gral of [ G λ ] x y against is Z − [ G ( x , y ) λ ] x y d y = Ψ x ( x ) Z x − C − − b Φ y ( y ) d y + Φ x ( x ) Z x C − + b Ψ y ( y ) d y ,which, by (6.10), is the identity matrix. We can compensate by subtracting from [ G λ ] x y ( x , y ) as kernel.Summarizing, 23 heorem 6.4. The kernel for the operator R λ = − λ T ( I − λ T ) − isK λ ( x , y ) = [ G λ ] x y ( x , y ) − . (6.16)We can now relate this to the kernel b K λ defined using the formal series (5.16).We have shown that they each define the same operator R λ in H . This meansthat they coincide as elements of the L space H (2) [*], so we may choose toidentify them at each point ( x , y ). Theorem 6.5.
The kernels K λ and b K λ are identical. For later use we define here the
Weyl function for the beam Dirichlet prob-lem to be W ( λ ) = λ D Φ (1, λ ) Φ (1, λ ) − , (6.17)The representation (6.16) shows that W ( λ ) = λ K λ (1, 1) + λ . (6.18)It follows that W has a pole at the origin with residue D Φ (1, 0) Φ (1, 0) − = . (6.19)Then the representation (5.16) shows that, formally at least, W ( λ ) = λ + X ν λ − λ ν 〈 f ν , f ν 〉 · f ν ,1 (1) f ν ,2 (1) f ν ,1 (1) f ν ,1 (1) f ν ,2 (1) f ν ,2 (1) f ν ,2 (1) f ν ,1 (1) ¸ . (6.20)(We omit a detailed justification of (6.20) in the general case, since the only usewe shall make is to the case when { λ ν } is finite.)We assume here that the f ν are chosen to be real. It will be useful to under-stand the signs of the entries of the summands. Lemma 6.6. If ν > , then f ν ,1 and f ν ,2 have the same sign.Proof. Because of the relationship between eigenfunctions for ± λ ν , this is equiv-alent to the statement that if ν < f ν ,1 and f ν ,2 have opposite signs. Withour choice of indexing, ν < λ ν >
0. The corresponding φ ν is Ψ v forsome fixed 2-vector v . Then Ψ ( − λ ν ) v = . But λ ν > Ψ ( − λ ν ) are positive (this is the dual of the argument for Proposition 4.2) so v v <
0. Then D φ ν (1) = Ψ x (1, λ ) v = − v . Since f ν is a multiple of v , its entrieshave opposite signs. 24o simplify the notation in (6.20), let α ν = f ν ,1 (1) p 〈 f ν , f ν 〉 , β ν = f ν ,2 (1) p 〈 f ν , f ν 〉 , for ν >
0. (6.21)By Lemma 6.6 we may take α ν and β ν positive. Taking into account the relationbetween f ν and f − ν and between 〈 f ν , f ν 〉 and 〈 f − ν , f − ν 〉 it follows that W ( λ ) = λ + X ν < λ − λ ν · α ν β ν − α ν − β ν α ν β ν ¸ + X ν > λ − λ ν · α ν β ν α ν β ν α ν β ν ¸ , (6.22)where we set α − ν = α ν , β − ν = β ν . The discrete beam is characterized by measures m and n that are supported ondiscrete points − < x < x < . . . < x d − < x d < m j , n j . For convenience we also define x = − x d + = l j = x j + − x j , M = .Here conditions (4.5) and (5.17) both reduce to the assumption that m j n j > j =
1, . . . d .The partial fundamental solution Φ ( x , λ ) satisfies D Φ = x j ,so it is piecewise linear in x , and the derivative D Φ is piecewise constant. Thusfor any given λ the function Φ is characterized by its values Φ j = Φ j ( λ ) = Φ ( x j , λ ), j =
0, . . . d +
1. (7.1)Similarly, D Φ = Φ ′ j is characterized by its one-sided values Φ ′ j = D Φ j ( λ ) = D Φ ( x j − , λ ), j =
1, . . . d +
1. (7.2)The beam equation D Φ = λ M Φ , with initial conditions Φ ( − λ ) = , D Φ ( − λ ) = ,translates to the conditions Φ = , Φ j + = Φ j + l j Φ ′ j + , Φ ′ = , Φ ′ j + = Φ ′ j + λ M j Φ j .25hese relations can be put in two forms: · Φ j + Φ ′ j + ¸ = · l j
10 1 ¸ · Φ j Φ ′ j + ¸ (7.3)and · Φ j + Φ ′ j + ¸ = · + λ l j M j l j λ M j ¸ " Φ ′ j + Φ ′ j = T j · Φ j Φ ′ j ¸ . (7.4) Lemma 7.1.
Each of Φ j and Φ ′ j is a polynomial of degree j − ; the even part isdiagonal and the odd part is off-diagonal. The Dirichlet spectrum has d ele-ments.Proof. The first statement follows by induction from the recursion relations(7.4). A consequence is that the determinant ∆ ( λ ) = det Φ d + is a polynomialof degree d in λ , so the eigenvalues come in d pairs.As shown in the general case, the eigenvalues are distinct and real.The Weyl function (6.17) in the discrete case is W ( λ ) = λ Φ ′ d + Φ − d + . (7.5)The recursion relations (7.3) imply that W ( λ ) has a continued fraction expan-sion involving non-commuting coefficients [21, 23]. Proposition 7.2. W ( λ ) = λ l d + M d + λ l d − + M d − + . . . + λ l (7.6) Proof.
Let W j = λ − Φ ′ j Φ − j . Note that, for λ large enough, all Φ ′ j and Φ j areinvertible. The relations Φ d + = Φ d + l d Φ ′ d + , Φ ′ d + = Φ ′ d + λ M d { Φ d ,imply that Φ d + ( Φ ′ d + ) − = ( λ M d + Φ ′ d Φ − d ) − + l d ,26ence W − d + = λ l d + ( M d + W d ) − .Inverting this expression we obtain W d + = [ λ l d + ( M d + W d ) − ] − .Iterating down to W = ( λ l ) − concludes the proof.We want to reverse this procedure and recover the data { l j } and { M j } fromthe function W . We follow the procedure of Stieltjes [21], starting with the de-termination of certain Padé approximants of W . At step zero, let P = Q = ,so Q W = P + O ( λ − ); W = Q − P + O ( λ − ). (7.7)To proceed, we note that T − j = · − l j − λ M j + λ l j M j ¸ . (7.8)Therefore the identity (7.4) implies that T − d · Φ d + Φ ′ d + ¸ = · Φ d + − l d Φ ′ d + − λ M d Φ d + + ( + λ l d M d ) Φ ′ d + ¸ = · Φ d Φ ′ d ¸ .Multiplying each (block) row on the right by Φ − d + , we obtain · − λ d ( λ W ) − λ M d + (1 + λ l d M d ) λ W ¸ = · Φ d Φ − d + Φ ′ d Φ − d + ¸ .The two equations for W can be rewritten as λ l d W = − Φ d Φ − d + = + O ( λ − ); (7.9)( + λ l d M d ) W = M d − λ − Φ ′ d Φ − d + = M d + O ( λ − ). (7.10)Set P = , Q = λ l d ; P = M d , Q = λ l d M d + . (7.11)Then Q − P and Q − P are Padé approximants to W on the left: W = Q − P + O ( λ − ), W = Q − P + O ( λ − ). (7.12)These two approximates are uniquely determined by the conditions P (0) = , Q (0) = , respectively; see the next section.27his process can be continued. We have[ T d T d − · · · T d − j + ] − · Φ d + Φ ′ d + ¸ = · Φ d − j + Φ ′ d − j + ¸ . (7.13)Let us write, in a temporary notation for this section only,[ T d . . . T d − j + ] − = · a j ( λ ) − b j ( λ ) − c j ( λ ) d j ( λ ) ¸ , 1 ≤ j ≤ d . (7.14) Lemma 7.3. (a) The polynomials a j and b j have degree j − ; the polynomialsc j and d j have degree j , ≤ j ≤ d . (b) For each 1 ≤ j ≤ d , a j (0) = d j (0) = , b j (0) = c j (0) = .(c) The coefficients of even powers in a j , b j , c j and d j are diagonal and thecoefficients of odd powers are off-diagonal. Proof.
Note that each of these statements is true at j = · a − b − c d ¸ = T − d = · − l d − λ M d + λ l d M d . ¸ Note that · a j + − b j + − c j + d j + ¸ = T − d − j · a j − b j − c j d j ¸ (7.15) = · a j + l d − j c j − b j − l d − j d j − λ M d − j a j − ( + λ l d − j M d − j ) c j λ M d − j b j + ( + λ l d − j M d − j ) d j ¸ .The assertion (a) follows by easily by induction. Each T j (0) = , which implies(b). Assertion (c) follows from the fact that multiplication by any entry of T − j preserves these properties.In analogy with the computations that led to (7.9) and (7.10), we multiplyeach (block) row of the identity · Φ d − j + Φ ′ d − j + ¸ = · a j − b j − c j d j ¸ · Φ d + Φ ′ d + ¸ on the right by Φ − d + and obtain the equations λ b j W = a j − Φ d − j + Φ − d + ; d j W = λ − c j + λ − Φ ′ d − j + Φ − d + .28ccordingly, and consistent with previous definitions for j = P j − = a j , Q j − = λ b j , 1 ≤ j ≤ d ; (7.16) P j = λ − c j , Q j = d j , 0 ≤ j ≤ d . (7.17)Note that since c j (0) =
0, each of the P k , Q k is a polynomial.In view of (7.7) (7.9), (7.10), and Lemma 7.3, we have Proposition 7.4.
The polynomials P k , Q k , ≤ k ≤ d , have the properties(a) Q j − and Q j have degree j , P j − and P j have degree j − ;(b) The coefficient of odd powers of Q j − are diagonal, and the coefficients ofeven powers are off-diagonal;(c) The coefficient of even powers of Q j are diagonal, and the coefficients of oddpowers are off-diagonal.Moreover Q j − W = P j − + O ( λ − j ); (7.18) Q j W = P j + O ( λ − j − ). (7.19)In the next section we treat the inverse problem : the problem of recoveringthe beam data { l j }, { M j } from W . The final step of the process described thereuses the fact that the data can be recovered from the leading coefficients of thepolynomials { Q k }. Proposition 7.5.
Let 〈 Q k 〉 denote the leading coefficient of Q k . Then for ≤ j ≤ d , 〈 Q j − 〉〈 Q j − 〉 − = l d − j + ; (7.20) 〈 Q j 〉〈 Q j − 〉 − = M d − j + . (7.21) Proof.
Let 〈 a j 〉 , 〈 b j 〉 , 〈 c j 〉 , 〈 d j 〉 denote the leading coefficients of a j , b j , c j , d j .Because of Lemma 7.3 (a) and (7.15), it follows that the recursion for the matrixof principal coefficients is given by · 〈 a j 〉 −〈 b j 〉−〈 c j 〉 〈 d j 〉 ¸ = · l d − j + 〈 c j − 〉 − l d − j + 〈 d j − 〉− l d − j + 〈 d j − 〉 l d − j + M d − j + 〈 d j 〉 ¸ . (7.22)At the first step, Q = and 〈 Q 〉 = l d , so 〈 Q 〉〈 Q 〉 − = 〈 Q 〉 = l d .29t each subsequent step, (7.22) implies that 〈 Q j − 〉 = 〈 b j 〉 = M − d − j + 〈 d j 〉 = M − d − j + £ l d − j + M d − j + 〈 d j − 〉 ¤ = l d − j + 〈 d j − 〉 = l d − j + 〈 Q j − 〉 ,which proves (7.20). Similarly, at each step (7.22) implies that 〈 Q j 〉 = 〈 d j 〉 = M d − j + 〈 b j 〉 = M d − j + 〈 Q j − 〉 ,which proves (7.21). We shall show that the Weyl function W has an asymptotic expansion W ( λ ) = λ C + λ C + . . . 1 λ n + C n + O µ λ n + ¶ as λ → ∞ . (8.1)The denominators Q k of the Padé approximants to W can be recoveredfrom this asymptotic expansion of W . For example, subsitute the expansion(8.1) for W in (7.18) and expand. Since Q j − has no constant term and the con-stant term of P j − is , the term of order 0 in the expansion is and the termsof order −
1, . . . , 1 − j in the expansion are zero. Writing Q j − = λ j Q ( j − ) j + · · · + λ Q ( j − )2 + λ Q ( j − )1 , (8.2)the resulting system of equations can be written h Q ( j − )1 Q ( j − )2 . . . Q ( j − ) j i C C . . . C j − C C . . . C j . . . C j − C j . . . C j − = £ ¤ . (8.3)Write Q j = λ j Q ( j + ) j + · · · + λ Q ( j + )1 + . (8.4)Since Q j (0) = , the same argument leads to the system h Q ( j + )1 Q ( j + )2 . . . Q ( j + ) j i C C . . . C j C C . . . C j + . . . C j C j + . . . C j − = − £ C C . . . C j − ¤ . (8.5)30n principle, the matrix equations (8.3) and (8.5), considered as scalar equa-tions, consist of 4 j linear equations in 4 j unknowns. However we know thateach of the coefficients of Q ( j ± ) is either a diagonal or an off-diagonal matrix,so there are only 2 d unknowns. Moreover, as we shall show, the same is trueof each of the matrices C k , so the associated 2 j × j matrix for these equationshas only 2 · j non-zero entries. As we shall show, each system (8.3) and (8.5)decomposes easily into two uncoupled systems of j equations in j unknowns,permitting simple formulas for the leading coefficients.To understand the C k , we return to the formula (6.22) for W : W ( λ ) = λ + − X ν =− d λ − λ ν · α ν β ν − α ν − β ν α ν β ν ¸ + d X ν = λ − λ ν · α ν β ν α ν β ν α ν β ν ¸ ,where α ν and β ν are positive.For large | λ | , ( λ − λ ν ) − = P ∞ n = λ k ν / λ k + , so C = " + d X ν = α ν β ν , (8.6)and C k = d X ν = ½ λ k − ν · α ν β ν − α ν − β ν α ν β ν ¸ + λ k ν · α ν β ν α ν β ν α ν β ν ¸¾ , k ≥ ν and λ ν have opposite signs, so C k = P d ν = | λ ν | k " − α ν − β ν , k odd; P d ν = | λ ν | k " α ν β ν α ν β ν , k even, k ≥ C k = · a k a k ¸ , k even; C k = · b k c k ¸ , k odd, (8.7)where a = + d X ν = α ν β ν ; a k = d X ν = | λ ν | k α ν β ν , k even, k ≥ b k = − X ν > | λ ν | k α ν ; c k = − X ν > | λ ν | k β ν , k odd.31et us consider the systems (8.3) and (8.5) for j = h Q (2 − )1 Q (2 − )2 i · C C C C ¸ = [ ]; h Q (2 + )1 Q (2 + )2 i · C C C C ¸ = − [ C C ]. (8.8)The key structural fact here is that each row or column consists of one diago-nal matrix and one off-diagonal matrix. For larger values of j there is a similarstructure, with diagonal matrices and off-diagonal matrices alternating. Fillingin the entries, the first of the systems (8.8) is · x x x x ¸ a b a c b a c a = · ¸ , (8.9)where x k , x k are the non-zero elements in the first and second rows of thecoefficient Q (2 − ) k , respectively.Because of the way that the positions of zero and non-zero elements in therows and columns either match or complement each other, there are cancella-tions. For example, the product of the first row of the matrix on the left with thesecond or third columns of the matrix on the right is zero. Therefore the fourequations associated to the first row reduce to two, which can be written as asystem £ x x ¤ · a b c a ¸ = £ ¤ . (8.10)Similarly, the four equations associated with the second row reduce to £ x x ¤ · a c b a ¸ = £ ¤ . (8.11)A second way to organize this is by a suitable permutation of rows and columns,so that rows with the same pattern of zero entries are juxtaposed, and the samefor columns. Then the original system of 8 equations becomes · x x x x ¸ a b c a a c b a = · ¸ . (8.12)The same procedure applies in general to the equations for the coefficients Q ( j − ) k of Q j − , yielding an equivalent form in which the original 2 j × j matrixis reduced to a diagonal form with two j × j matrices, adjoints of each other, onthe diagonal. We write this explicitly below.32 similar analysis of the second of the systems (8.8) yields a different formof canonical reduction. Here the system has the form · x x x x ¸ b a c a a b a c = − · a b a c ¸ (8.13)where x k and x k are the non-zero entries of the first and second rows of thecoefficient Q (2 + ) k , respectively. Again the positioning of the zeros in the rowsand columns tells us that these equations reduce to two uncoupled systems £ x x ¤ · c a a b ¸ = − £ a b ¤ ; (8.14) £ x x ¤ · b a a c ¸ = − £ a c ¤ . (8.15)As in the case of (8.10), (8.11), the system (8.13) can be rearranged to the form · x x x x ¸ c a a b b a a c = − · a b a c ¸ . (8.16)Let us pass to the general case for the coefficients Q (2 j ± ) k of Q j − and Q j .We start with the (2 d + × (2 d +
2) Hankel matrix H = C C C . . . C d C C C . . . C d + C C C . . . C d + . . . C d C d + C d + . . . C d . (8.17)Writing out the 2 × H = a b a b . . .0 a c a c b a b a c a c a . . . a b a b . . .0 a c a c b a b a c a c c . . .. . . . . . (8.18)33n a notation that is best explained by (8.12) we introduce two ( d + × ( d + H : H NW = a b a b . . . c a c a . . . a b a b . . . c a c a . . .. . . ; H SE = a c a c . . . b a b a . . . a c a c . . . b a b a . . .. . . .We denote the j × j principal minors of H NW and H SE by H NWj and H SEj , re-spectively. Note that they are transposes of each other:[ H NWj ] t = H SEj .Therefore they have the same determinantdet H NWj = det H SEj = ∆ j . (8.19)Following the same procedure as for Q , the equations for the coefficients of Q j − are £ x x . . . x j ¤ H NWj = £ ¤ ; (8.20) £ x x . . . x j ¤ H SEj = £ ¤ , (8.21)where x k and x k are the non-zero elements in the first and second rows of thecoefficient of λ k in Q j − .As remarked in (7.20) and (7.21), we can reconstruct the beam data { l j }, { M j }from the leading coefficients 〈 Q k 〉 of the polynomials { Q k }. For Q j − , we wantto compute x j and x j in (8.20). By Cramer’s rule, x j can be obtained by re-placing the last row of the matrix in (8.20) by the right-hand side of (8.20) andcomputing the determinant. The same procedure for (8.21) gives x j = ( − j − ∆ NWj ∆ j , x j = ( − j − ∆ SEj ∆ j , (8.22)where ∆ NWj is the determinant of H NWj , ∆ NWj is the determinant of H NWj withthe first column and last row eliminated, and similarly for ∆ SEj and ∆ SEj . ByProposition 7.4, since Q j − has degree j , the leading coefficient 〈 Q j − 〉 is di-agonal if j is odd and off-diagonal if j is even. Thus we have Proposition 8.1.
The leading coefficient of Q j − is 〈 Q j − 〉 = ∆ NWj ∆ j ∆ SEj ∆ j (8.23)34 f j is odd, 〈 Q j − 〉 = − ∆ NWj ∆ j ∆ SEj ∆ j (8.24) if j is even. We turn now to consideration of the coefficients of Q j . In line with (8.16),we introduce two d × d matrices that are obtained by reorganizing H after re-moving the first two columns and last two rows: H NE = c a c a . . . a b a b . . . c a c a . . . a b a b . . .. . . ; H SW = b a b a . . . a c a c . . . b a b a . . . a c a c . . .. . . .Let H NEj and H SWj be the j × j principal minors of H NE and H SW , respectively.The equations for the coefficients of Q j are £ x x . . . x j ¤ H NEj = − £ a b . . . a j − ¤ ; (8.25) £ x x . . . x j ¤ H SWj = − £ a c . . . a j − ¤ . (8.26)Here x k and x k are the non-zero entries in the first and second rows of thecoefficient of λ k in Q j . Replacing the last row of H NEJ by the negative of theright-hand side of (8.25), then moving that to be the first row, gives H NWj . Ap-plying the same reasoning to (8.26), we obtain x j = ( − j ∆ j ∆ NEj ; x j = ( − j ∆ j ∆ SWj . (8.27)By Lemma 7.4, since Q j has degree j , the top coefficient is off-diagonal if j isodd and diagonal if j is even. Therefore Proposition 8.2.
The leading coefficient of Q j is 〈 Q j 〉 = − ∆ j ∆ NEj ∆ j ∆ SWj (8.28) if j is odd, 〈 Q j 〉 = ∆ j ∆ NEj ∆ j ∆ SWj (8.29) if j is even.
35e are now in a position to compute the data { l k }, { M k }, via (7.20), (7.21).Note that H NWj = H SWj ; H SEj = H NEj − (8.30)We use these identities to rewrite (8.23) and (8.24).If j is odd, we have l d − j + = 〈 Q j − 〉〈 Q j − 〉 − = ∆ SWj − ∆ j ∆ NEj − ∆ j ∆ j − ∆ NEj − ∆ j − ∆ SWj − − = ∆ SWj − ∆ NEj − ∆ j ∆ j − .If j is even, we have l d − j + = − ∆ NEj − ∆ j ∆ SWj − ∆ j ∆ j − ∆ NEj − ∆ j − ∆ SWj − − = ∆ SWj − ∆ NEj − ∆ j ∆ j − .If j is odd, we have M d − j + = 〈 Q j 〉〈 Q j − 〉 − = − ∆ j ∆ NEj ∆ j ∆ SWj ∆ SWj − ∆ j ∆ NEj − ∆ j − = − ∆ j ∆ NEj ∆ NEj − ∆ j ∆ SWj ∆ SWj − .If j is even, we have M d − j + = ∆ j ∆ NEj ∆ j ∆ SWj ∆ SWj − ∆ j ∆ NEj − ∆ j − = − ∆ j ∆ NEj ∆ NEj − ∆ j ∆ SWj ∆ SWj − . Under an additional assumption of smoothness, the compatibility conditionsthat relate D x Φ = ( + λ M ) Φ , M = · nm ¸ . (A.1)and D t Φ = [ bD x + a ] Φ , (A.2)36amely D t D x Φ = D x D t Φ lead to λ M t Φ = { b xx + a x + λ [ b , M ]} D x Φ + { a xx + b x + λ ( bM ) x + λ b x M + λ [ a , M ]} Φ . (A.3)At a given value of t , this is a differential equation for Φ of order at most one.We are assuming that Φ is a solution of a nontrivial second–order equation. Weassume that the differential operator in (A.3) trivializes:0 = b xx + a x + λ [ b , M ], (A.4) λ M t = a xx + b x + λ ( bM ) x + λ b x M + λ [ a , M ], (A.5)since otherwise the system is degenerate.As in Section 2 we suppose that a = a + λ − a , b = b + λ − b ,and that a j and b j are bounded, j =
0, 1. Each equation in (A.4), (A.5) leads tothree equations, for the coefficients of the powers λ k , k = −
1, 0, 1.For k = − a ) xx + b ) x = b ) xx + a ) x =
0. (A.6)Thus ( a ) x = − ( b ) xx b ) xxx − b ) x =
0. (A.7)The second equation implies that b = C e x + C e − x + C and since b isbounded, b is a constant matrix, and by the first equation so is a .For k = = [ b , M ] ; M t = ( b M ) x + ( b ) x M + [ a , M ]. (A.8)We assume that m n , so first equation in (A.8) implies that the diagonal partof b is a multiple of the identity matrix, and the off-diagonal part is a multipleof M : b = u I + p M . (A.9)Therefore the diagonal part of the second equation in (A.8) gives0 = ( pmn ) x + ( pn ) x m + ( a ) m − ( a ) n ;0 = ( pmn ) x + ( pm ) x n + ( a ) n − ( a ) m . (A.10)37dding these two equations gives0 = p x ( mn ) + p ( mn ) x .Multiplying by p ( mn ) gives 0 = [ p ( mn ) ] x , so p ( mn ) is constant. If p a priori relationship between m and n . Thereforewe assume p =
0. With this assumption, equations (A.10) imply that the off-diagonal part of a is proportional to M . We can write b = u , a = · w ( x ) + v ( x ) 00 w ( x ) − v ( x ) ¸ + q M . (A.11)The remaining information from equations (A.4), (A.5) is contained in the equa-tions for the k = = ( b ) xx + a ) x + [ b , M ] ; (A.12)0 = ( a ) xx + b ) x + b M x + [ a , M ]., (A.13)since a , b are constant. Write a = · α α α α ¸ ; b = · β β β β ¸ .Looking at the diagonal terms, then the off-diagonal terms, in (A.12), we find u xx + w x = v x = β n − β m ; (A.14)2( qn ) x + ( β − β ) n = = qm ) x + ( β − β ) m . (A.15)Multiply the left side of (A.15) by n , the right side by m , and add, to obtain:0 = £ ( qn ) x m + ( qm ) x n ¤ = £ q x ( mn ) + q ( mn ) x ¤ .As above, unless q = a priori relation q ( mn ) = constant. Tak-ing q =
0, (A.15) implies β = β .Looking at the off-diagonal terms in (A.13), we obtain equations for n andfor m with constant coefficients: β n x = ( α − α ) n ; β m x = ( α − α ) m .In order to avoid trivial cases, we must assume that β = α = α . Com-puting the diagonal part of (A.13), taking into account (A.14) gives0 = − u xxx + u x + ( β m + β n ) x ] ; (A.16)0 = α m − α n . (A.17)38o avoid a trivial linear relation between m and n we need the off–diagonalterms α , α of a to vanish.Summing up to this point: a = · γ − u x + v γ − u x − v ¸ + λ · α α ¸ ; b = · u u ¸ + λ · β β ¸ ,where γ , α , β , and β are constant.Keeping in mind the obvious symmetry between ( ϕ , m ) on one hand and( ϕ , n ) on the other, we symmetrize by taking β = β = β . Moreover, the firstLax equation (A.1) has an additional gauge symmetry Φ → ω ( t , λ ) Φ . Under thisgauge transformation a → ω t ω − + a ,and thus, by choosing ω to satisfy ω t + ( γ + αλ ) ω =
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