Approximation of isolated eigenvalues of general singular ordinary differential operators

Let A be a self-adjoint operator defined by a general singular ordinary differential expression τ on an interval (a, b), ? ∞ ≤ a < b ≤ ∞. We show that isolated eigenvalues in any gap of the essential spectrum of A are exactly the limits of eigenvalues of suitably chosen self-adjoint realizations A_n of τ on subintervals (a_n, b_n) of (a, b) with a_n → a, b_n → b. This means that eigenvalues of singular ordinary differential operators can be approximated by eigenvalues of regular operators. In the course of the proof we extend a result, which is well known for quasiregular differential expressions, to the general case: If the spectrum of A is not the whole real line, then the boundary conditions needed to define A can be given using solutions of (τ ? λ)u = 0, where λ is contained in the regularity domain of the minimal operator corresponding to τ.     

On finite rank perturbations of definitizable operators

It was shown by P. Jonas and H. Langer that a selfadjoint definitizable operator A in a Krein space remains definitizable after a finite rank perturbation in resolvent sense if the perturbed operator B is selfadjoint and the resolvent set ρ(B) is nonempty. It is the aim of this note to prove a more general variant of this perturbation result where the assumption on ρ(B) is dropped. As an application a class of singular ordinary differential operators with indefinite weight functions is studied.     

Maximal Regular Abstract Elliptic Equations and Applications

The oblique derivative problem is addressed for an elliptic operator differential equation with variable coefficients in a smooth domain. Several conditions are obtained, guaranteing the maximal regularity, the Fredholm property, and the positivity of this problem in vector-valued L _p-spaces. The principal part of the corresponding differential operator is nonselfadjoint. We show the discreteness of the spectrum and completeness of the root elements of this differential operator. These results are applied to anisotropic elliptic equations.     

An unusual self-adjoint linear partial differential operator

In an American Mathematical Society Memoir, published in 2003, the authors Everitt and Markus apply their prior theory of symplectic algebra to the study of symmetric linear partial differential expressions, and the generation of self-adjoint differential operators in Sobolev Hilbert spaces. In the case when the differential expression has smooth coefficients on the closure of a bounded open region, in Euclidean space, and when the region has a smooth boundary, this theory leads to the construction of certain self-adjoint partial differential operators which cannot be defined by applying classical or generalized conditions on the boundary of the open region.

This present paper concerns the spectral properties of one of these unusual self-adjoint operators, sometimes called the ``Harmonic' operator.

The boundary value problems considered in the Memoir (see above) and in this paper are called regular in that the cofficients of the differential expression do not have singularities within or on the boundary of the region; also the region is bounded and has a smooth boundary. Under these and some additional technical conditions it is shown in the Memoir, and emphasized in this present paper, that all the self-adjoint operators considered are explicitly determined on their domains by the partial differential expression; this property makes a remarkable comparison with the case of symmetric ordinary differential expressions.

In the regular ordinary case the spectrum of all the self-adjoint operators is discrete in that it consists of a countable number of eigenvalues with no finite point of accumulation, and each eigenvalue is of finite multiplicity. Thus the essential spectrum of all these operators is empty.

This spectral property extends to the present partial differential case for the classical Dirichlet and Neumann operators but not to the Harmonic operator. It is shown in this paper that the Harmonic operator has an eigenvalue of infinite multiplicity at the origin of the complex spectral plane; thus the essential spectrum of this operator is not empty.

Both the weak and strong formulations of the Harmonic boundary value problem are considered; these two formulations are shown to be equivalent.

In the final section of the paper examples are considered which show that the Harmonic operator, defined by the methods of symplectic algebra, has a domain that cannot be determined by applying either classical or generalized local conditions on the boundary of the region.

     

Asymptotics of the Distribution of Eigenvalues of a Selfadjoint Second Order Hyperbolic Differential Operator on the Two-Dimensional Torus

We study the asymptotic properties of the discrete spectrum for general selfadjoint second order hyperbolic operators on the two-dimensional torus. For a broad class of operators with sufficiently smooth coefficients and the principal part coinciding with the wave operator in the light cone coordinates we prove the discreteness of the spectrum and obtain an asymptotic formula for the distribution of eigenvalues. In some cases we can indicate the first two asymptotic terms. We discuss the relations of these questions to analytic number theory and mathematical physics.     

Approximation of approximation numbers by truncation

LetA be a bounded linear operator onsome infinite-dimensional separable Hilbert spaceH and letA _n be the orthogonal compression ofA to the span of the firstn elements of an orthonormal basis ofH. We show that, for eachk1, the approximation numberss _k(A_n) converge to the corresponding approximation numbers _k(A) asn. This observation implies almost at once some well known results on the spectral approximation of bounded selfadjoint operators. For example, it allows us to identify the limits of all upper and lower eigenvalues ofA _n in the case whereA is selfadjoint. These limits give us all points of the spectrum of a selfadjoint operator which lie outside the convex hull of the essential spectrum. Moreover, it follows that the spectrum of a selfadjoint operatorA with a connected essential spectrum can be completely recovered from the eigenvalues ofA _n asn goes to infinity.     

Inverse boundary spectral problem for non–selfadjoint maxwells–s equations with incomplete data

The inverse boundary spectral problem for selfadjoint Maxwell–s equations is to reconstruct unknown coefficient functions in Maxwell– equations from the knowledge of the boundary spectral data, i.e. fromt eh eigenvalues and the boudnary value of the eigenfunctions. Since the spectrum of non–selfadjoint Maxwell–s operator consists of normal eigenvalues and an interval, the complete boundary spectral data can be defind only in a very complicated way. In this article we show that the coefficients can be reconstructed from incomplete data, that is, from the large eigenvalues and the boundary values of the generalized eigenfunctions. Particularly, we do not need the nfinit–dimensional data corresponding to the non–discrete spectrum.     

Spectral analysis of singular ordinary differential operators with indefinite weights

In this paper we develop a perturbation approach to investigate spectral problems for singular ordinary differential operators with indefinite weight functions. We prove a general perturbation result on the local spectral properties of selfadjoint operators in Krein spaces which differ only by finitely many dimensions from the orthogonal sum of a fundamentally reducible operator and an operator with finitely many negative squares. This result is applied to singular indefinite Sturm-Liouville operators and higher order singular ordinary differential operators with indefinite weight functions.     

Almost-Normed Spaces of Functions with Given Asymptotics, Lagrangian Asymptotics, and Their Application to Ordinary Differential Equations

The concept of almost-normed spaces is introduced. It is proved that the space of sufficiently smooth functions asymptotically approximating to polynomials (of degrees no higher than a given one) as their argument tends to infinity is an almost-normed space. It is demonstrated that this space is a complete metric space with respect to the metrics generated by the almost-norm introduced. The space of functions strongly asymptotically approximating to polynomials is defined, and its embedding into the space of functions asymptotically approximating to polynomials is proved. The results obtained give a new approach to studying boundary-value problems with asymptotic initial value data at singular points of ordinary differential equations.     

Über das wesentliche Spektrum zum Eigenwertproblem der automorphen Formen

The eigenvalue problem of automorphic forms is the problem to determine the spectral behaviour of a certain second order elliptic partial differential operator having its domain of definition in an appropriate Hilbert space of square integrable functions defined in the upper halfplane. This Hilbert space depends on a discontinuous group G, a real parameter k, and a multiplier system v on G of weight 2k. We prove: The essential spectrum of the self-adjoint linear operator under consideration contains the interval \([\frac{1}{4}, \infty \rangle\) if G has a singular cusp or if G is a Fuchsian group of the second kind. If G has a sufficiently large fundamental domain, the essential spectrum of our differential operator is non-empty and unbounded above.     

The splitting index for linear differential systems

Consider the linear differential operator associated with an n-dimensional first order linear system of time varying ordinary differential equations. Conditions are given on the system for time near plus and minus infinity which guarantee that the operator is Fredholm. The splitting index is introduced and it is shown to be the negative of the ordinary index of a Fredholm operator. The splitting index is shown to be invariant under appropriate perturbations and is computable in terms of the asymptotic properties of the coefficient matrix for a wide class of systems. The asymptotic conditions on the system are discussed in various function space topologies and a new concept of admissibility of a pair of Banach spaces is introduced whereby a pair is admissible with respect to all operators whose coefficients lie in a given function space.     

Strict inequality of Robin eigenvalues for elliptic differential operators on Lipschitz domains

On a bounded Lipschitz domain we consider two selfadjoint operator realizations of the same second order elliptic differential expression subject to Robin boundary conditions, where the coefficients in the boundary conditions are functions. We prove that inequality between these functions on the boundary implies strict inequality between the eigenvalues of the two operators, provided that the inequality of the functions in the boundary conditions is strict on an arbitrarily small nonempty, open set.     

Bounds for the spectrum of a matrix differential operator with a damping term

We consider a second-order matrix ordinary regular differential nonselfadjoint operator with a damping term and selfadjoint boundary conditions. An estimate for the resolvent and bounds for the spectrum are established. An operator is said to be stable if its spectrum lies in the right half-plane. By the obtained bounds, stability and instability conditions are established.     

Bounds for the spectrum of a matrix differential operator with a damping term

We consider a second-order matrix ordinary regular differential nonselfadjoint operator with a damping term and selfadjoint boundary conditions. An estimate for the resolvent and bounds for the spectrum are established. An operator is said to be stable if its spectrum lies in the right half-plane. By the obtained bounds, stability and instability conditions are established.     

Algebraic Equivalence and Homology Classes of Real Algebraic Cycles

In this note we study the spectral properties of a multiplication operator in the space L_p(X)^m which is given by an m by m matrix of measurable functions. Our particular interest is directed to the eigenvalues and the isolated spectral points which turn out to be eigenvalues. We apply these results in order to investigate the spectrum of an ordinary differential operator with so called “floating singularities”.     

Lp – Lq Decay Estimates fo the Solutions of Strictly Hyperbolic Equations of Second Order with Increasing in Time Coefficients

One of the most important questions in the theory of nonlinear wave equations is that for global existence of solutions. An essential tool is the Strichartz inequality for special solutions of the wave equation.In the last time different results were proved generalizing the classical one of Strichartz. In the present paper L_p – L_q estimates are proved for the solutions of strictly hyperbolic equations of second order with time dependent coefficients where these are unbounded at infinity. In the first step the WKB method is applied to the construction of a fundamental system of solutions for ordinary differential equations depending on a parameter. In a second step the method of stationary phase yields the asymptotical behaviour of Fourier multipliers with nonstandard phase functions depending on a parameter.     

Essential self-adjointness of Schrödinger-type operators

The essential self-adjointness of the strongly elliptic operator L = ∑_j,k=1ⁿ (?_j ? ib_j(x)) a_jk(x)(?_k ? ib_k(x)) + q(x) acting on C₀^∞(Rⁿ) is considered, where the matrix (a_jk) is real and symmetric, b_j and q are real, a_jk and b_j are sufficiently smooth, and q?L_loc². It has been shown by Ural'ceva and also Laptev that if q is bounded below and n ? 3 the minimal operator may not be self-adjoint if the principal coefficients rise too rapidly. Thus a theorem of Weyl for ordinary differential operators does not extend to partial differential operators. In this paper it is shown that if q is bounded below and if the principal coefficients are “well behaved” within a sequence of closed shells which go to infinity, then the minimal operator is self-adjoint. It is also shown that a number of results which were known to be true when q is sufficiently smooth may be extended to include the case where q?L_loc². The principal tools used are a distribution inequality due to Tosio Kato and a general maximum principle due to Walter Littman.     

Supersymmetric Dirichlet Operators, Spectral Gaps, and Correlations

In this article we construct a supersymmetric extension of the Dirichlet operator associated with a tempered Gibbs measure on $$ \user2{\mathbb{R}}^{\user2{\mathbb{Z}}^d }. $$ Under fairly general assumptions on the interaction potentials we show that the Dirichlet operator (resp. its supersymmetric extension) is essentially selfadjoint on the set of smooth, bounded cylinder functions (resp. differential forms), for all inverse temperatures. Assuming that the on-site potentials have a non-degenerated minimum and no other critical point we prove that, for sufficiently large inverse temperatures, one observes a number of subsequent gaps in the spectrum of the Dirichlet operator. For translation invariant systems with a sufficiently weak (but in general infinite range) ferromagnetic interaction, we prove the validity of a formula for the leading asymptotics of the correlation of two spin variables, as their distance and the inverse temperature tend to infinity, which has originally been derived by J. Sj?strand for finite-dimensional systems. Communicated by Bernard Helffer. Submitted: 14/05/ 2005 Accepted: 22/05/2005     

On the stabilizability problem in Banach space

The model is a linear system defined on Banach (state and control) spaces, with the operator acting on the state only the infinitesimal generator of a strongly continuous semigroup. The stabilizability problem of expressing the control through a bounded operator acting on the state as to make the resulting feedback system globally asymptotically stable is considered. On the negative side, and in contrast with the finite dimensional theory, a few counter examples are given of systems which are densely controllable in the space and yet are not stabilizable, even if some further “nice properties” hold. Use is made of the notion of essential spectrum and its stability under relatively compact perturbations. On the positive side, it is shown, however, that for large classes of systems of physical interest (classical selfadjoint boundary value problems, delay equations, etc.) controllability on a suitable finite dimensional subspace still yields stabilizability on the whole space.