Singular continuous Floquet operator for systems with increasing gaps

Consider the Floquet operator of a time-independent quantum system, periodically perturbed by a rank one kick, acting on a separable Hilbert space: e^−iH₀Te^{−iκT|φ〉〈φ|}, where T and κ are the period and the coupling constant, respectively. Assume the spectrum of the self-adjoint operator H₀ is pure point, simple, bounded from below and the gaps between the eigenvalues (λ_n) grow like λ_n+1−λ_n∼Cn^d with d?2. Under some hypotheses on the arithmetical nature of the eigenvalues and the vector φ, cyclic for H₀, we prove the Floquet operator of the perturbed system has purely singular continuous spectrum.     

Spectral concentration near embedded eigenvalues

Techniques of Rayleigh-Schrödinger perturbation theory usually employed for perturbation of isolated eigenvalues are used to obtain theorems on spectral concentration near eigenvalues which are not assumed to be isolated. If {H_ϰ} is a family of self adjoint operators convergent in the strong resolvent sense to a self adjoint operator H₀ and λ₀ is an eigenvalue of finite multiplicity of H₀, then the spectrum of {H_κ} is concentrated near λ₀. Moreover, conditions under which concentration still occurs near λ₀ without the assumption of finite multiplicity are obtained in the semibounded case.     

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.     

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.     

Commutator estimates in von Neumann algebras

Let M be a von Neumann algebra. For every self-adjoint locally measurable operator a, there exists a central self-adjoint locally measurable operator c ₀ such that, given any ? > 0, |[a, u _ε ]| ? (1 ? ε)|aε ? c ₀| for some unitary operator u _ε ∈ M. In particular, every derivation δ: M → I (where I is an ideal in M) is inner, and δ = δ _a for a ∈ I.     

Eigenvalue Asymptotics for a Schrödinger Operator with Non-Constant Magnetic Field Along One Direction

We consider the discrete spectrum of the two-dimensional Hamiltonian H = H ₀ + V, where H ₀ is a Schrödinger operator with a non-constant magnetic field B that depends only on one of the spatial variables, and V is an electric potential that decays at infinity. We study the accumulation rate of the eigenvalues of H in the gaps of its essential spectrum. First, under certain general conditions on B and V, we introduce effective Hamiltonians that govern the main asymptotic term of the eigenvalue counting function. Further, we use the effective Hamiltonians to find the asymptotic behavior of the eigenvalues in the case where the potential V is a power-like decaying function and in the case where it is a compactly supported function, showing a semiclassical behavior of the eigenvalues in the first case and a non-semiclassical behavior in the second one. We also provide a criterion for the finiteness of the number of eigenvalues in the gaps of the essential spectrum of H.     

The extraordinary spectral properties of radially periodic Schrödinger operators

Since it became clear that the band structure of the spectrum of periodic Sturm-Liouville operatorst = - (d²/dr²) +q(r) does not survive a spherically symmetric extension to Schrödinger operatorsT =- Δ+ V with V(x) =q(¦x¦) for x ∈ ?^d,d ∈ ? 1, a wealth of detailed information about the spectrum of such operators has been acquired. The observation of eigenvalues embedded in the essential spectrum [μ₀, ∞[ ofT with exponentially decaying eigenfunctions provided evidence for the existence of intervals of dense point spectrum, eventually proved by spherical separation into perturbed Sturm-Liouville operatorst _c = t +(c/r ²). Subsequently, a numerical approach was employed to investigate the distribution of eigenvalues ofT more closely. An eigenvalue was discovered below the essential spectrum in the cased = 2, and it turned out that there are in fact infinitely many, accumulating at μ₀. Moreover, a method based on oscillation theory made it possible to count eigenvalues oft _c contributing to an interval of dense point spectrum ofT. We gained evidence that an asymptotic formula, valid forc → ∞, does in fact produce correct numbers even for small values of the coupling constant, such that a rather precise picture of the spectrum of radially periodic Schrödinger operators has now been obtained.     

Spectral gaps for periodic Schrödinger operators with hypersurface magnetic wells: Analysis near the bottom

We consider a periodic magnetic Schrödinger operator Hh, depending on the semiclassical parameter h>0, on a noncompact Riemannian manifold M such that H¹(M,R)=0 endowed with a properly discontinuous cocompact isometric action of a discrete group. We assume that there is no electric field and that the magnetic field has a periodic set of compact magnetic wells. We suppose that the magnetic field vanishes regularly on a hypersurface S. First, we prove upper and lower estimates for the bottom λ₀(Hh) of the spectrum of the operator Hh in L²(M). Then, assuming the existence of non-degenerate miniwells for the reduced spectral problem on S, we prove the existence of an arbitrarily large number of spectral gaps for the operator Hh in the region close to λ₀(Hh), as h→0. In this case, we also obtain upper estimates for the eigenvalues of the one-well problem.     

On spectral properties of a new operator over sequence spaces c and c0

In this work, we classify and calculate spectra such as point spectrum, continuous spectrum and residual spectrum over sequences spaces ?_∞, c${?}_{\infty },\hspace{0.17em}c$ and c₀ according to a new matrix operator W which is obtained by matrix product.     

Hypercyclicity and unimodular point spectrum

We show that an operator on a separable complex Banach space with sufficiently many eigenvectors associated to eigenvalues of modulus 1 is hypercyclic. We apply this result to construct hypercyclic operators with prescribed K_σ unimodular point spectrum. We show how eigenvectors associated to unimodular eigenvalues can be used to exhibit common hypercyclic vectors for uncountable families of operators, and prove that the family of composition operators C_? on H²(D), where ? is a disk automorphism having +1 as attractive fixed point, has a residual set of common hypercyclic vectors.     

Quantum field theoretic properties of a model of Nelson: Domain and eigenvector stability for perturbed linear operators

The Yukawa-like interaction of a nonrelativistic “nucleon” field with a relativistic “meson” field is studied. E. Nelson defined a self-adjoint Hamiltonian for this model, using an approximate dressing transformation to transform the standard high-momentum cutoff Hamiltonian H_κ into an operator H₀ + A_κ + E_κ, where E_κ is the divergent nucleon self-interaction part which is removed and A_κ is controled κ-uniformly as a sesquilinear form perturbation of H₀; so H_ren is defined to transform to H₀ + A_∞. In the present work it is proved that the model has some of the main properties that are familiar in the axiomatic approach to quantum fields. Vacuum expectation values are proved to exist and satisfy the axioms of Wightman, except for Lorentz covariance. In the case of a small coupling constant, “physical nucleon” one-particle states are constructed and the “one-body problem” of Haag is solved. These field theoretic properties are translated into domain and spectral stability properties for the perturbed operator H₀ + A_∞. Stability theory for perturbations by sesquilinear forms is presented in an Appendix, with a number of new techniques.     

Spectral asymptotic behavior of elliptic systems

One investigates a first-order elliptic self-adjoint pseudodifferential operator A (x,D) acting in sections of a Hermitian vector bundle over a compactn-dimensional manifold x. It is assumed that the principal symbol A(x, ξ) of the operator is locally diagonalizable and that its eigenvaluesaj(x, ξ) have a variable multiplicity and that {a _i,a _k}≠0 whenevera _i=a _k. Under indicated conditions one constructs an expansion of the fundamental solution of the hyperbolic system \(i\frac{{\partial u}}{{\partial t}} = A(x,D)u\) and one investigates the asymptotic properties of the spectrum of the operator A (x,D). For the distribution functionN(λ) of the eigenvalues one establishes that . Under further assumptions on the properties of the bicharacteristic of the symbolsaj(x, ξ) one establishes a stronger estimate of Duistermaat-Guillemin type:N(λ)=Cλ ⁿ +C′λ ^n?i +0(γ ^n?1 )     

Singular continuous Floquet operator for periodic quantum systems

Consider the Floquet operator of a time independent quantum system, acting on a separable Hilbert space, periodically perturbed by a rank one kick: e−iH₀Te−iκT|?〉〈?| where T is the period, κ the coupling constant, and H₀ is a pure point self-adjoint operator, bounded from below. Under some hypotheses on the vector ?, cyclic w.r.t. H₀ we prove the following:

•

If the gaps between the eigenvalues (λn) are such that λn+1−λn?Cn−γ for some γ∈]0,1[ and C>0, then the Floquet operator of the perturbed system is purely singular continuous T-a.e.

•

If H₀ is the Hamiltonian of the one-dimensional rotator on L²(R/T₀Z) and the ratio is irrational, then the Floquet operator is purely singular continuous as soon as κT≠0(2π).

We also establish an integral formula for the family .     

Relations between the half turns of the hyperbolic plane

Let R_a denote the half turn about the point a of the hyperbolic plane H. If the points a, b, c, d lie on the same line and the pair (c, d) is obtained from the pair (a, b) by a translation, then we have R_aR_b = R_cR_d. We study the group G whose generating set is {R_a:a∈H} and whose defining relations are the ones mentioned above together with the relations R²_a = 1. We show that G can be made into a Lie group, G has two connected components, and its identity component G₀ is the universal covering group of PSL₂(R). In particular, it follows that all relations between the half turns in PSL₂(R) follow from the abovementioned relations and a single additional relation of length five.     

Gauge-periodic point perturbations on the Lobachevsky plane

We study periodic point perturbations of the Shrödinger operator with a uniform magnetic field on the Lobachevsky plane. We prove that the spectrum gaps of the perturbed operator are labeled by the elements of the K₀ group of aC ^* algebra associated with the operator. In particular, if theC ^* algebra has the Kadison property, then the operator spectrum has a band structure.     

A spectral Bernstein theorem

We study the spectrum of the Laplace operator of a complete minimal properly immersed hypersurface M in \({\mathbb{R}^{n+1}}\). (1) Under a volume growth condition on extrinsic balls and a condition on the unit normal at infinity, we prove that M has only essential spectrum consisting of the half line [0, +∞). This is the case when \({{\rm lim}_{\tilde{r}\to +\infty}\,\tilde{r}\kappa_i=0}\), where \({\tilde{r}}\) is the extrinsic distance to a point of M and κ _i are the principal curvatures. (2) If the κ _i satisfy the decay conditions \({|\kappa_i|\leq 1/\tilde{r}}\) and strict inequality is achieved at some point \({y\in M}\), then there are no eigenvalues. We apply these results to minimal graphic and multigraphic hypersurfaces.     

Points d’évaluation pour les opérateurs cycliques ayant la propriété de Bishop (β)

Let T be a linear bounded cyclic operator in a separable complex Hilbert space H. Let B(T) and B_a(T) denote, respectively, the set of bounded point evaluation and the set of analytic point evaluation of T. We show that if T has the Bishop property (β), then B_a(T)=B(T)?σ_ap(T), where σ_ap(T) is the approximate spectrum of T. In the particular case when T is an operator of multiplication by z in a Hardy space this was proved by Trent (Pacific J. Math. 80 (1979) 279). On the other hand, using the generalized and the local spectral theory we obtain sufficient conditions on B_a(T) under which the spectrum of T and the local spectrum of T at any y≠0 in H coincide. At the end results involving the spectral picture of quasi-similar cyclic operators are given.     

On constant discrete programming problems

Let H be a subset of the set S_n of all permutations

$\text{1}\text{2}\text{???}\text{n}\text{s(1)}\text{s(2)}\text{???}\text{s(n)}$

C=6c_ij6 a real n?n matrix L_c(s)=c_1s(1)+c_2s(2)+???+c_ns(n) for s ? H. A pair (H, C) is the existencee of reals a₁,b₁,a₂,b₂,…a_n,b_n, for which c_ij=a₁+b_j if (i,j)?D(H), where D(H)={(i,j):(?h?H)(j=h(i))}.For a pair (H,C) the specifity of it is proved in the case, when H is either a special cyclic class of permutations or a special union of cyclic classes. Specific pairs with minimal sets H are in some sense described.     

Error bounds in the isometric Arnoldi process

Error bounds for the eigenvalues computed in the isometric Arnoldi method are derived. The Arnoldi method applied to a unitary matrix U successively computes a sequence of unitary upper Hessenberg matrices H_k, k = 1,2,… The eigenvalues of the H_k's are increasingly better approximations to eigenvalues of U. An upper bound for the distance of the spectrum of H_k from the spectrum of U, and an upper bound for the distance between each individual eigenvalue of H_k and one of U are given. Between two eigenvalues of H_k on the unit circle, there is guaranteed to lie an eigenvalue of U. The results are applied to a problem in signal processing.