Spectral theory of discontinuous functions of self-adjoint operators and scattering theory

In the smooth scattering theory framework, we consider a pair of self-adjoint operators H₀, H and discuss the spectral projections of these operators corresponding to the interval (−∞,λ). The purpose of the paper is to study the spectral properties of the difference D(λ) of these spectral projections. We completely describe the absolutely continuous spectrum of the operator D(λ) in terms of the eigenvalues of the scattering matrix S(λ) for the operators H₀ and H. We also prove that the singular continuous spectrum of the operator D(λ) is empty and that its eigenvalues may accumulate only at “thresholds” in the absolutely continuous spectrum.     

The structurally stable linear systems on the half-line are those with exponential dichotomies

Let A be a selft-adjoint operator on the Hilbert space L²Ω, ?) = {u ε L_loc²(Ω)|∫_Ω|² ?(x)dx < + ∞} defined by means of a closed, semibounded, sesquilinear form a(·, ·). We obtain a necessary and sufficuents condition for the spectrum of A to be discrete. We apply this result to a Sturm-Liouville problem for an elliptic operator with discontinuous coefficients defined on an unbounded domain and to the study of the spectrum of a Hamiltonian defined by a pseudodifferential operator.     

Spectral analysis of nonselfadjoint Schrödinger operators with a matrix potential

Dissipative Schrödinger operators with a matrix potential are studied in L₂((0,∞);E)(dimE=n<∞) which are extension of a minimal symmetric operator L₀ with defect index (n,n). A selfadjoint dilation of a dissipative operator is constructed, using the Lax-Phillips scattering theory, the spectral analysis of a dilation is carried out, and the scattering matrix of a dilation is founded. A functional model of the dissipative operator is constructed and its characteristic function's analytic properties are determined, theorems on the completeness of eigenvectors and associated vectors of a dissipative Schrödinger operator are proved.     

On the independence of Heegner points in the function field case

Let ∞ be a fixed place of a global function field k. Let E be an elliptic curve defined over k which has split multiplicative reduction at ∞ and fix a modular parametrization ΦE:X₀(N)→E. Let be Heegner points associated to the rings of integers of distinct quadratic “imaginary” fields K₁,…,Kr over (k,∞). We prove that if the “prime-to-2p” part of the ideal class numbers of ring of integers of K₁,…,Kr are larger than a constant C=C(E,ΦE) depending only on E and ΦE, then the points P₁,…,Pr are independent in . Moreover, when k is rational, we show that there are infinitely many imaginary quadratic fields for which the prime-to-2p part of the class numbers are larger than C.     

On a vector space analogue of Kneser’s theorem

In a previous paper [X. Hou, K.H. Leung, Q. Xiang, A generalization of an addition theorem of Kneser, J. Number Theory 97 (2002) 1-9], the following result was established: let E⊂K be fields such that the algebraic closure of E in K is separable over E. Let A,B be E-subspaces of K such that 0<dim_EA<∞ and 0<dim_EB<∞. Then dim_EAB?dim_EA+dim_EB-dim_EH(AB), where AB is the E-space generated by {ab:a∈A,b∈B} and H(AB)={x∈K:xAB⊂AB}. The separability assumption was essential in the proof of this result. However, even without the separability assumption, no counterexample is known. The present paper shows that no counterexample can be found if dim_EA?5.     

Commutators, C0-semigroups and resolvent estimates

We study the existence and the continuity properties of the boundary values on the real axis of the resolvent of a self-adjoint operator H in the framework of the conjugate operator method initiated by Mourre. We allow the conjugate operator A to be the generator of a C₀-semigroup (finer estimates require A to be maximal symmetric) and we consider situations where the first commutator [H,iA] is not comparable to H. The applications include the spectral theory of zero mass quantum field models.     

Bundle convergence of sequences of pairwise uncorrelated operators in von Neumann algebras and vectors in their L2-spaces

Let H be a separable complex Hilbert space, A a von Neumann algebra in ?(H),a faithful, normal state on A. We prove that if a sequence (X_n: n ≥ 1) of uncorrelated operators in A is bundle convergent to some operator X in A and Σ^∞_n=1n⁻² Var(X_n) log²(n + 1) < ∞, then X is proportional to the identity operator on H. We also prove an analogous theorem for certain uncorrelated vectors in the completion L₂=L₂(A,φ) of A given by the Gelfand-Naimark-Segal representation theorem. Both theorems were motivated by a recent one proved by Etemadi and Lenzhen in the classical commutative setting.     

“Localized” self-adjointness of Schrödinger type operators on Riemannian manifolds

We prove self-adjointness of the Schrödinger type operator , where ∇ is a Hermitian connection on a Hermitian vector bundle E over a complete Riemannian manifold M with positive smooth measure dμ which is fixed independently of the metric, and V∈L_loc¹(EndE) is a Hermitian bundle endomorphism. Self-adjointness of H_V is deduced from the self-adjointness of the corresponding “localized” operator. This is an extension of a result by Cycon. The proof uses the scheme of Cycon, but requires a refined integration by parts technique as well as the use of a family of cut-off functions which are constructed by a non-trivial smoothing procedure due to Karcher.     

Operator spaces with prescribed sets of completely bounded maps

Suppose A is a dual Banach algebra, and a representation π:A→B(?₂) is unital, weak* continuous, and contractive. We use a “Hilbert-Schmidt version” of Arveson distance formula to construct an operator space X, isometric to ?₂⊗?₂, such that the space of completely bounded maps on X consists of Hilbert-Schmidt perturbations of π(A)⊗I_?2. This allows us to establish the existence of operator spaces with various interesting properties. For instance, we construct an operator space X for which the group K₁(CB(X)) contains Z₂ as a subgroup, and a completely indecomposable operator space containing an infinite dimensional homogeneous Hilbertian subspace.     

On oscillatory solutions of certain forced Emden-Fowler like equations

We give a constructive proof of existence to oscillatory solutions for the differential equations x^″(t)+a(t)λ|x(t)|sign[x(t)]=e(t), where t?t₀?1 and λ>1, that decay to 0 when t→+∞ as O(t−μ) for μ>0 as close as desired to the “critical quantity” . For this class of equations, we have limt→+∞E(t)=0, where E(t)<0 and E^″(t)=e(t) throughout [t₀,+∞). We also establish that for any μ>μ^? and any negative-valued E(t)=o(t−μ) as t→+∞ the differential equation has a negative-valued solution decaying to 0 at + ∞ as o(t−μ). In this way, we are not in the reach of any of the developments from the recent paper [C.H. Ou, J.S.W. Wong, Forced oscillation of nth-order functional differential equations, J. Math. Anal. Appl. 262 (2001) 722-732].     

Linear systems and determinantal random point fields

In random matrix theory, determinantal random point fields describe the distribution of eigenvalues of self-adjoint matrices from the generalized unitary ensemble. This paper considers symmetric Hamiltonian systems and determines the properties of kernels and associated determinantal random point fields that arise from them; this extends work of Tracy and Widom. The inverse spectral problem for self-adjoint Hankel operators gives sufficient conditions for a self-adjoint operator to be the Hankel operator on L²(0,∞) from a linear system in continuous time; thus this paper expresses certain kernels as squares of Hankel operators. For suitable linear systems (−A,B,C) with one-dimensional input and output spaces, there exists a Hankel operator Γ with kernel ?(x)(s+t)=Ce−(2x+s+t)AB such that gx(z)=det(I+(z−1)ΓΓ†) is the generating function of a determinantal random point field on (0,∞). The inverse scattering transform for the Zakharov-Shabat system involves a Gelfand-Levitan integral equation such that the trace of the diagonal of the solution gives . When A?0 is a finite matrix and B=C†, there exists a determinantal random point field such that the largest point has a generalised logistic distribution.     

The local determination of Jordan bases for H-self-adjoint operators

Let H be an invertible self-adjoint operator on a finite dimensional Hilbert space $\text{X}$. A linear operator A is said to be H-self-adjoint (or self-adjoint relative to H) if HA = A¹H. Let σ(A) denote, as usual, the spectrum of A. If A is H-self-adjoint, then A is similar to A¹ and λ ∈ σ(A) implies λ&#x0304; ∈ σ (A), so that the spectrum of A issymmetric with respect to the real axis. Given spectral information for A at an eigenvalue λ₀ (≠ λ&#x0304;₀), we investigate the corresponding information at λ&#x0304;₀ and, in particular, the unique pairing of Jordan bases for the root subspaces at λ₀ and λ&#x0304;₀.     

Green’s matrices

Our goal is to identify and understand matrices A that share essential properties of the unitary Hessenberg matrices M that are fundamental for Szegö’s orthogonal polynomials. Those properties include: (i) Recurrence relations connect characteristic polynomials {r_k(x)} of principal minors of A. (ii) A is determined by generators (parameters generalizing reflection coefficients of unitary Hessenberg theory). (iii) Polynomials {r_k(x)} correspond not only to A but also to a certain “CMV-like” five-diagonal matrix. (iv) The five-diagonal matrix factors into a product BC of block diagonal matrices with 2 × 2 blocks. (v) Submatrices above and below the main diagonal of A have rank 1. (vi) A is a multiplication operator in the appropriate basis of Laurent polynomials. (vii) Eigenvectors of A can be expressed in terms of those polynomials.Conditions (v) connects our analysis to the study of quasi-separable matrices. But the factorization requirement (iv) narrows it to the subclass of “Green’s matrices” that share Properties (i)-(vii).The key tool is “twist transformations” that provide 2ⁿ matrices all sharing characteristic polynomials of principal minors with A. One such twist transformation connects unitary Hessenberg to CMV. Another twist transformation explains findings of Fiedler who noticed that companion matrices give examples outside the unitary Hessenberg framework. We mention briefly the further example of a Daubechies wavelet matrix. Infinite matrices are included.     

Every composition operator is (mean) asymptotically Toeplitz

Nazarov and Shapiro recently showed that, while composition operators on the Hardy space H² can only trivially be Toeplitz, or even “Toeplitz plus compact,” it is an interesting problem to determine which of them can be “asymptotically Toeplitz.” I show here that if “asymptotically” is interpreted in, for example, the Cesàro (C,α) sense (α>0), then every composition operator on H² becomes asymptotically Toeplitz.     

A remark about the interpolation of spaces of continuous, vector-valued functions

If A is a sectorial operator on a Banach space X, then the space C([0,1];(X,D(A))_θ,∞) is a subspace of the interpolation space (C([0,1];X),C([0,1];D(A)))_θ,∞. The inclusion is strict in general.     

Essential spectra of quasi-parabolic composition operators on Hardy spaces of analytic functions

In this work we study the essential spectra of composition operators on Hardy spaces of analytic functions which might be termed as “quasi-parabolic.” This is the class of composition operators on H² with symbols whose conjugate with the Cayley transform on the upper half-plane are of the form φ(z)=z+ψ(z), where ψ∈H^∞(H) and ℑ(ψ(z))>?>0. We especially examine the case where ψ is discontinuous at infinity. A new method is devised to show that this type of composition operator fall in a C*-algebra of Toeplitz operators and Fourier multipliers. This method enables us to provide new examples of essentially normal composition operators and to calculate their essential spectra.     

Organizing the arithmetic of elliptic curves

Suppose that E is an elliptic curve defined over a number field K, p is a rational prime, and K_∞ is the maximal Z_p-power extension of K. In previous work [B. Mazur, K. Rubin, Elliptic curves and class field theory, in: Ta Tsien Li (Ed.), Proceedings of the International Congress of Mathematicians, ICM 2002, vol. II, Higher Education Press, Beijing, 2002, pp. 185-195; B. Mazur, K. Rubin, Pairings in the arithmetic of elliptic curves, in: J. Cremona et al. (Eds.), Modular Curves and Abelian Varieties, Progress in Mathematics, vol. 224, 2004, pp. 151-163] we discussed the possibility that much of the arithmetic of E over K_∞ (i.e., the Mordell-Weil groups and their p-adic height pairings, the Shafarevich-Tate groups and their Cassels pairings, over all finite extensions of K in K_∞) can be described efficiently in terms of a single skew-Hermitian matrix with entries drawn from the Iwasawa algebra of K_∞/K.In this paper, using work of Nekovár? [J. Nekovár?, Selmer complexes. Preprint available at 〈http://www.math.jussieu.fr/∼nekovar/pu/〉], we show that under not-too-stringent conditions such an “organizing” matrix does in fact exist. We also work out an assortment of numerical instances in which we can describe the organizing matrix explicitly.     

New oscillation criteria for linear matrix Hamiltonian systems

Some new oscillation criteria are established for the matrix linear Hamiltonian system X′=A(t)X+B(t)Y, Y′=C(t)X−A∗(t)Y under the hypothesis: A(t), B(t)=B∗(t)>0, and C(t)=C∗(t) are n×n real continuous matrix functions on the interval [t₀,∞), (−∞<t₀). These results are sharper than some previous results even for self-adjoint second order matrix differential systems.     

Asymptotic observers for a class of evolution equations. A nonlinear approach

We consider in a Hilbert space H the system (E_u) = x = uAx+B(x); y = 〈x. c〉_H, where the control u ε L^∞([0, + ∞[, ℝ⁺) multiplies a possibly unbounded m-dissipative linear operator A. The operator B is nonlinear dissipative, and y stands for the output of the system. We prove, in this nonlinear framework, the existence of a suitable Luenberger-like observer. For this purpose, we show that the usual notions of regularly persistent inputs proposed in [7] or [4] are the appropriate concepts that allow one to generalize the main results of [9] and [8] or [7] for bilinear systems to our nonlinear general system: For each regularly persistent input, the estimation error of the observer converges weakly to zero. If in addition A generates a compact semigroup, the estimation error converges strongly to zero. A prototype of such a system is the heat exchanger system described in [9] or [8].