University of Richmond UR Scholarship Repository Math and Computer Science Faculty Publications Math and Computer Science 7-1998 Isospectral Sets for Fourth-Order Ordinary Differential Operators Lester Caudill University of Richmond, [email protected] Peter A. Perry Albert W. Schueller Follow this and additional works at: http://scholarship.richmond.edu/mathcs-faculty-publications Part of the Ordinary Differential Equations and Applied Dynamics Commons Recommended Citation Caudill, Lester F., Peter A. Perry, and Albert W. Schueller. "Isospectral Sets for Fourth-Order Ordinary Differential Operators." SIAM Journal on Mathematical Analysis 29, no. 4 (July 1998): 935-66. doi:10.1137/s0036141096311198. This Article is brought to you for free and open access by the Math and Computer Science at UR Scholarship Repository. It has been accepted for inclusion in Math and Computer Science Faculty Publications by an authorized administrator of UR Scholarship Repository. For more information, please contact [email protected]. SIAM J. MATH. ANAL. °c 1998 Society for Industrial and Applied Mathematics Vol. 29, No. 4, pp. 935–966, July 1998 008 ISOSPECTRAL SETS FOR FOURTH-ORDER ORDINARY DIFFERENTIAL OPERATORS∗ LESTER F. CAUDILL, JR.y , PETER A. PERRYz , AND ALBERT W. SCHUELLERx 4 0 0 Abstract. Let L(p)u = D u ¡ (p1u ) + p2u be a fourth-order differential operator acting on 2 2 2 00 L [0; 1] with p ≡ (p1;p2) belonging to LR[0; 1] × LR[0; 1] and boundary conditions u(0) = u (0) = u(1) = u00(1) = 0. We study the isospectral set of L(p) when L(p) has simple spectrum. In particular 2 2 we show that for such p, the isospectral manifold is a real-analytic submanifold of LR[0; 1] × LR[0; 1] which has infinite dimension and codimension. A crucial step in the proof is to show that the gradients of the eigenvalues of L(p) with respect to p are linearly independent: we study them as solutions of a non-self-ajdoint fifth-order system, the Borg system, among whose eigenvectors are the gradients. Key words. inverse spectral problem, ordinary differential equations AMS subject classifications. 34A55, 34L20 PII. S0036141096311198 1. Introduction. This paper initiates a study of isospectral sets of coefficients for self-adjoint, fourth-order ordinary differential operators, in Liouville{Green nor- mal form, on the finite interval [0; 1]. Such operators are labelled by a pair of coef- ficients p =(p1;p2). Our motivation is twofold: first, we would like to understand the inverse spectral problem for fourth-order operators such as the Euler{Bernoulli operator of mechanics; second, we would like to develop techniques of analysis which are systematic in nature and are therefore likely to be useful in the study of other singular and higher order ordinary differential operators. Our goal is to understand the set of coefficients isospectral to a given pair p =(p1;p2) as a Hilbert submanifold of a suitable Hilbert space of coefficients, in analogy to the analysis of the second- order Sturm{Liouville problem carried out by Trubowitz and his collaborators (see [11, 18, 19, 26, 27], and see [10] for more recent results). As in the work of Trubowitz et. al., we use methods of global analysis to study the isospectral manifold as a level set of the direct spectral map from coefficients to spectra. For the class of operators we consider, the gradient gn(x; p) of a given eigenvalue λn(p) is an ordered pair consisting of an eigenfunction square and the square of its derivative, and so the gradient of the mapping from coefficients to spectra is the infinite sequence of all such ordered pairs. A crucial part of the analysis is to show that these ordered pairs form a linearly independent set. Our approach differs from the approach to the Sturm{Liouville problem taken in [11, 18, 19, 26, 27] in two respects. First of all, we use resolvent perturbation ∗Received by the editors October 28, 1997; accepted for publication (in revised form) July 23, 1997; published electronically March 25, 1998. This research was supported by the National Science Foundation and the Commonwealth of Kentucky under its EPSCoR program. A portion of this work was carried out while the first author was visiting the Department of Mathematics, University of Kentucky. http://www.siam.org/journals/sima/29-4/31119.html yDepartment of Mathematics and Computer Science, University of Richmond, Richmond, VA 23173 ([email protected]). zDepartment of Mathematics, University of Kentucky, Lexington, KY 40506-0027 (perry@ ms.uky.edu). xDepartment of Mathematics, Whitman College, Walla Walla, WA 99362 (schuellaw@whitman. edu). 935 936 L. F. CAUDILL, P. A. PERRY, AND A. W. SCHUELLER techniques rather than integral equations and complex analysis to obtain the nec- essary eigenvalue and eigenfunction asymptotics: see the thesis of the third author [29], where these techniques are developed at greater length. Secondly, we study orthogonality properties of the gradients, not using special identities, but rather by studying an associated non-self-adjoint, fifth-order system, the Borg system, among whose eigenvectors are exactly the gradients gn(x; p). Our system is the analogue, for fourth-order differential operators, of a third-order non-self-adjoint eigenvalue prob- lem introduced by Borg [9] in his study of completeness of eigenfunction squares in the Sturm{Liouville problem. We believe this technique to be a powerful one which admits generalization to other inverse spectral problems involving ordinary differential operators. To describe our results in detail, we first specify the class of fourth-order operators which we will study. In order to study the isospectral set as a Hilbert manifold, we wish to study operators L(p) where p ranges over a Hilbert space of coefficients; here L(p) is the operator 4 (1.1) L(p)u = D u ¡ D(p1Du)+p2u 2 on L [0; 1], where p =(p1;p2). In what follows, we will impose \double Dirichlet" boundary conditions u(0) = u00(0) = u(1) = u00(1)=0;although our methods can be used to treat other separated, self-adjoint boundary conditions. 2 2 A natural choice for the Hilbert space of coefficients is E ≡ LR[0; 1] × LR[0; 1], 2 where LR[0; 1] denotes real-valued, square-integrable, measurable functions on [0; 1]. For such singular coefficients it is convenient to define the operator L(p) by the method of sesquilinear forms (see, for example, Kato [17, Chapter 6]). Since we wish to study real analyticity of various maps on E, it will also be convenient to introduce 2 2 EC ≡ LC[0; 1] × LC[0; 1] and define L(p) for p 2 EC. To this end, we introduce the sesquilinear form Z 1 00 00 0 0 (1.2) q(u; v)= u (x)v (x)+p1(x)u(x)v(x)+p2(x)u(x)v(x)dx 0 with the form domain Q(q)=fu2H2[0; 1] : u(0) = u(1)=0g for p 2 EC. It is not difficult to see that the form q with p = 0 is a closed positive form. Using this fact and simple perturbative estimates, one can show that the form q with 0 6= p 2 EC is also closed and sectorial, i.e., that the set fq(u; u):u2Q(q);kukL2[0;1] =1g is contained in a sector of the complex plane of the form <(z) ≥−c,j=(z)|≤(<(z)+c). Here c depends only on kpkE; a complete proof is given in [29, section 5.2]. It follows from the form representation theorem (see, for example, Theorem VI.2.1 of [17]) that there is a unique sectorial operator L(p), i.e., a unique closed operator with numerical range in a sector, associated with the sesquilinear form q. It follows from the same theorem that for all p 2 EC, the domain of L(p) is contained in the H3[0; 1] functions with u(0) = u00(0) = u(1) = u00(1)=0.ThusL(p) is an operator with compact resolvent, and its spectrum consists of an infinite sequence, fλn(p)g, of discrete eigenvalues. Using the form representation theorem, one can also show ISOSPECTRAL SETS FOR DIFFERENTIAL OPERATORS 937 that if p 2 C1([0; 1]; C2), then L(p) is the operator (1.1). For more singular p the action of L(p) may be understood in terms of the quasi-derivatives associated with the operator L(p): see Naimark [25] for the general theory and Schueller [29, section 5.3] for its application to fourth-order operators. We wish to study isospectral sets of L(p) for p =(p1;p2) 2 E. In order to apply techniques of global analysis, we need to realize the direct spectral map from coefficients to spectral data as a map between Hilbert spaces. To this end, we set Z 1 µ0(p)=p1 = p1(x)dx 0 and λ (p)¡λ (0) ¡ n2π2p µ (p)= n n 1 : n n2π2 1 2 We will show that the sequence fµn(p)gn=0 belongs to the Hilbert space F ≡ ` (0[N). The direct spectral map is the mapping µ : E ! F defined by µ(p)=fµn(p)g. The isospectral set M(p) of a given p 2 E is the set of all q 2 E with µ(q)=µ(p). We will say that p 2 E has simple spectrum if the sectorial operator L(p) has only simple eigenvalues. First of all, we will prove the following theorem. Theorem 1.1. The set of p 2 E with simple spectrum is open and dense in E. Denote this set by E. There are physically relevant families of fourth-order prob- lems, such as the Liouville{Green normal forms of the Euler{Bernoulli equation, which are known to have simple spectrum (see, e.g., [14]).