Bootstrap Multiscale Analysis and Localization¶in Random Media

We introduce an enhanced multiscale analysis that yields subexponentially decaying probabilities for bad events. For quantum and classical waves in random media, we obtain exponential decay for the resolvent of the corresponding random operators in boxes of side L with probability higher than 1 − e ^{− L ζ}, for any 0<ζ<1. The starting hypothesis for the enhanced multiscale analysis only requires the verification of polynomial decay of the finite volume resolvent, at some sufficiently large scale, with probability bigger than 1 − (d is the dimension). Note that from the same starting hypothesis we get conclusions that are valid for any 0 < ζ < 1. This is achieved by the repeated use of a bootstrap argument. As an application, we use a generalized eigenfunction expansion to obtain strong dynamical localization of any order in the Hilbert–Schmidt norm, and better estimates on the behavior of the eigenfunctions. Received: 29 November 2000 / Accepted: 21 June 2001     

Riemann–Hilbert Approach to a Generalised Sine Kernel and Applications

We investigate the asymptotic behaviour of a generalised sine kernel acting on a finite size interval [−q ; q]. We determine its asymptotic resolvent as well as the first terms in the asymptotic expansion of its Fredholm determinant. Further, we apply our results to build the resolvent of truncated Wiener–Hopf operators generated by holomorphic symbols. Finally, the leading asymptotics of the Fredholm determinant allows us to establish the asymptotic estimates of certain oscillatory multidimensional coupled integrals that appear in the study of correlation functions of quantum integrable models.     

Trotter–Kato Product Formula and Operator-Norm Convergence

We find necessary and sufficient conditions for the operator-norm convergence of the Trotter–Kato product formula. Using them we prove that this convergence takes place: (i) if the resolvent of one of the involved operators is compact, either (ii) if one operator is relatively compact with respect to another one, or (iii) if the product of resolvents of the involved operators is compact. Received: 19 October 1998 / Accepted: 22 February 1999     

On the Stability of the Kerr Metric

The reduced (in the angular coordinate ϕ) wave equation and Klein–Gordon equation are considered on a Kerr background and in the framework of C ⁰-semigroup theory. Each equation is shown to have a well-posed initial value problem, i.e., to have a unique solution depending continuously on the data. Further, it is shown that the spectrum of the semigroup's generator coincides with the spectrum of an operator polynomial whose coefficients can be read off from the equation. In this way the problem of deciding stability is reduced to a spectral problem and a mathematical basis is provided for mode considerations. For the wave equation it is shown that the resolvent of the semigroup's generator and the corresponding Green's functions can be computed using spheroidal functions. It is to be expected that, analogous to the case of a Schwarzschild background, the quasinormal frequencies of the Kerr black hole appear as resonances, i.e., poles of the analytic continuation of this resolvent. Finally, stability of the solutions of the reduced Klein–Gordon equation is proven for large enough masses. Received: 28 August 2000 / Accepted: 4 April 2001     

Application of the group foliation method to the complex Monge-Ampère equation

We apply the method of group foliation to the complex Monge-Ampère equation (CMA ₂) to establish a regular framework for finding its non-invariant solutions. We employ an infinite symmetry subgroup ofCMA ₂ to produce a foliation of the solution space into orbits of solutions with respect to this group and a corresponding splitting ofCMA ₂ into an automorphic system and a resolvent system. We propose a new approach to group foliation which is based on the commutator algebra of operators of invariant differentiation. This algebra together with its Jacobi identities provides the commutator representation of the resolvent system. Presented by M.B. Sheftel at the DI-CRM Workshop held in Prague, 18–21 June 2000.     

Eigenvector Localization for Random Band Matrices with Power Law Band Width

It is shown that certain ensembles of random matrices with entries that vanish outside a band around the diagonal satisfy a localization condition on the resolvent which guarantees that eigenvectors have strong overlap with a vanishing fraction of standard basis vectors, provided the band width W raised to a power μ remains smaller than the matrix size N. For a Gaussian band ensemble, with matrix elements given by i.i.d. centered Gaussians within a band of width W, the estimate μ ≤ 8 holds.     

Numerical diagonalization analysis of the ground-state superfluid-localization transition in two dimensions

Ground state of the two-dimensional hard-core-boson system in the presence of the quenched random chemical potential is investigated by means of the exact-diagonalization method for the system sizes up to L=5. The criticality and the DC conductivity at the superfluid-localization transition have been controversial so far. We estimate, with the finite-size scaling analysis, the correlation-length and the dynamical critical exponents as and z=2, respectively. The AC conductivity is computed with the Gagliano-Balseiro formula, with which the resolvent (dynamical response function) is expressed in terms of the continued-fraction form consisting of Lanczos tri-diagonal elements. Thereby, we estimate the universal DC conductivity as . Received 19 August 1998     

Operator expansion method and beyond

Nuclear structure problems ofββ decay are discussed, focusing on methods to deal with a number of nuclear intermediate states, the operator expansion method and the resolvent operator expansion based on Lanczos algorithm, and on extensions of quasiparticle RPA toward a self-consistent formulation. Also, preliminary results are shown for electron-inducedββ transitions which might be feasible for investigations of second-order weak processes. Presented at Workshop on calculation of double-beta-decay matrix elements (MEDEX’97), Prague, May 27–31, 1997. Numerical calculations were performed by using the VP2100 computer system at Institute for Nuclear Study, University of Tokyo.     

How to measure the pomeron phasein diffractive dipion photoproduction

The study of charge asymmetry of pions in the high-energy process ep → eπ⁺π^-p (γp → π⁺π^-p) at very small dipion momenta offers a method to measure the phase of the forward hadronic (quasi-) elastic amplitude γp → ρp. We estimate the potential of such measurements at HERA. Received: 19 February 2004, Revised: 1 August 2005, Published online: 3 November 2005     

On Asymptotic Expansions and Scales of Spectral Universality in Band Random Matrix Ensembles

 We consider real random symmetric N × N matrices H of the band-type form with characteristic length b. The matrix entries are independent Gaussian random variables and have the variance proportional to , where u(t) vanishes at infinity. We study the resolvent in the limit and obtain the explicit expression for the leading term of the first correlation function of the normalized trace . We examine on the local scale and show that its asymptotic behavior is determined by the rate of decay of u(t). In particular, if u(t) decays exponentially, then . This expression is universal in the sense that the particular form of u determines the value of C > 0 only. Our results agree with those detected in both numerical and theoretical physics studies of spectra of band random matrices. Received: 8 April 2000 / Accepted: 7 June 2002 Published online: 21 October 2002 RID="*" ID="*" Present address: Département de Mathématiques, Université de Versailles Saint-Quentin, 78035 Versailles, France.     

Spectral theory of elliptic operators in exterior domains

Diverse closed (and selfadjoint) realizations of elliptic differential expressions A = Σ_{0⩽|α|,|β|⩽m} (−1) ^α D ^α a _α,β (x)D ^β , a _α,β (·) ∈ C ^∞($ \bar \Omega $ \bar \Omega ) on smooth (bounded or unbounded) domains Ω in ℝ ⁿ with compact boundary ∂Ω are considered. Trace-ideal properties of powers of resolvent differences for these closed realizations of A are proved by using the concept of boundary triples and operator-valued Weyl-Titchmarsh functions, and estimates for negative eigenvalues of certain selfadjoint extensions of the nonnegative minimal operator are derived. Our results extend classical theorems due to Vishik, Povzner, Birman, and Grubb.     

New Approach to Quantum Scattering Near the Lowest Landau Threshold for a Schrödinger Operator with a Constant Magnetic Field

 For a fixed magnetic quantum number m results on spectral properties and scattering theory are given for the three-dimensional Schr?dinger operator with a constant magnetic field and an axisymmetrical electric potential V. Asymptotic expansions for the resolvent of the Hamiltonian H _m  = H _om  + V are deduced as the spectral parameter tends to the lowest Landau threshold E ₀. In particular it is shown that E ₀ can be an eigenvalue of H _m . Furthermore, asymptotic expansions of the scattering matrix associated with the pair (H _m , H _om ) are derived as the energy parameter tends to E ₀. Received December 11, 2000; accepted in final form June 16, 2001 Published online June 10, 2002     

Unraveling the f<Subscript>0</Subscript> nature by connecting KLOE and BABAR data through analyticity

We define a general procedure, based on analyticity and dispersion relations, to estimate low-energy amplitudes for processes like: φ→e ⁺ e ^- M and φ→γM, starting from cross-section data on e ⁺ e ^-→φM, where M is a generic light scalar or pseudoscalar meson. In particular this procedure is constructed to obtain predictions on the radiative decay rate which are crucially linked on the assumed quark structure for the meson M under consideration. Three cases are analyzed: M = η, M = f ₀(qˉ) and M = f ₀(qqˉ). While in the η case the estimate of the branching fraction for the radiative decay φ→ηγ is in agreement with the data, in the case of f ₀, such agreement is obtained only under the hypothesis of a tetraquark scalar meson.     

Dirac Operators on Quantum Projective Spaces

We construct a family of self-adjoint operators D _N , ${N\in{\mathbb Z}}We construct a family of self-adjoint operators D _N , N ? \mathbb Z{N\in{\mathbb Z}} , which have compact resolvent and bounded commutators with the coordinate algebra of the quantum projective space \mathbb CP^l_q{{\mathbb C}{\rm P}^{\ell}_q} , for any ℓ ≥ 2 and 0 < q < 1. They provide 0⁺-dimensional equivariant even spectral triples. If ℓ is odd and N=\frac12(l+1){N=\frac{1}{2}(\ell+1)} , the spectral triple is real with KO-dimension 2ℓ mod 8.     

Liouville operator resolvent method applied to the hubbard dimer

Summary The problem of two hydrogenlike atomic sites (s-shells only) coupled via a hopping matrix is considered. We employ a new formalism, the Liouville operator resolvent method, to evaluate the single-particle thermodynamic Green’s function for this system. Details of the method are also discussed. To speed up publication, the authors of this paper have agreed to not receive the proofs for correction.     

The Lp-Theory of the Spectral Shift Function,¶the Wegner Estimate, and the Integrated Density¶of States for Some Random Operators

We develop the L ^p -theory of the spectral shift function, for p≥ 1, applicable to pairs of self-adjoint operators whose difference is in the trace ideal ℐ _p , for $0 < p≤ 1. This result is a key ingredient of a new, short proof of the Wegner estimate applicable to a wide variety of additive and multiplicative random perturbations of deterministic background operators. The proof yields the correct volume dependence of the upper bound. This implies the local H?lder continuity of the integrated density of states at energies in the unperturbed spectral gap. Under an additional condition of the single-site potential, local H?lder continuity is proved at all energies. This new Wegner estimate, together with other, standard results, establishes exponential localization for a new family of models for additive and multiplicative perturbations. Received: 27 July 2000 / Accepted: 1 November 2000     

Smoothing Property for Schrödinger Equations¶with Potential Superquadratic at Infinity

We prove a smoothing property for one dimensional time dependent Schr?dinger equations with potentials which satisfy at infinity, k≥ 2. As an application, we show that the initial value problem for certain nonlinear Schr?dinger equations with such potentials is L ₂ well-posed. We also prove a sharp asymptotic estimate of the L ^p -norm of the normalized eigenfunctions of H=−Δ+V for large energy. Dedicated to Jean-Michel Combes on the occasion of his Sixtieth Birthday Received: 10 October 2000 / Accepted: 29 March 2001     

Scaling of Linking and Writhing Numbers for Spherically Confined and Topologically Equilibrated Flexible Polymers

Scaling laws for Gauss linking number Ca and writhing number Wr for spherically confined flexible polymers with thermally fluctuating topology are analyzed. For ideal (phantom) polymers each of N segments of length unity confined to a spherical pore of radius R there are two scaling regimes: for sufficiently weak confinement (R≫N ^1/3) each chain has |Wr|≈N ^1/2, and each pair of chains has average |Ca|≈N/R ^3/2; alternately for sufficiently tight confinement (N ^1/3≫R), |Wr|≈|Ca|≈N/R ^3/2. Adding segment-segment avoidance modifies this result: for n chains with excluded volume interactions |Ca|≈(N/n)^1/2 f(φ) where f is a scaling function that depends approximately linearly on the segment concentration φ=nN/R ³. Scaling results for writhe are used to estimate the maximum writhe of a polymer; this is demonstrated to be realizable through a writhing instability that occurs for a polymer which is able to change knotting topology and which is subject to an applied torque. Finally, scaling results for linking are used to estimate bounds on the entanglement complexity of long chromosomal DNA molecules inside cells, and to show how “lengthwise” chromosome condensation can suppress DNA entanglement.     

Time Quasi-Periodic Unbounded Perturbations¶of Schrödinger Operators and KAM Methods

We eliminate by KAM methods the time dependence in a class of linear differential equations in ℓ² subject to an unbounded, quasi-periodic forcing. This entails the pure-point nature of the Floquet spectrum of the operator H ₀+εP(ωt) for ε small. Here H ₀ is the one-dimensional Schr?dinger operator p ²+V, V(x)∼|x|^α, α <2 for |x|→∞, the time quasi-periodic perturbation P may grow as |x|^β, β <(α−2)/2, and the frequency vector ω is non resonant. The proof extends to infinite dimensional spaces the result valid for quasiperiodically forced linear differential equations and is based on Kuksin's estimate of solutions of homological equations with non-constant coefficients. Received: 3 October 2000 / Accepted: 20 December 2000     

The quantum Sturm-Liouville equation,quantum resolvent,quantum integrals,and quantum KdV: The fast decrease case

We construct quantum operators solving the quantum versions of the Sturm-Liouville equation and the resolvent equation, and show the existence of conserved currents. The construction depends on the following input data: the basic quantum field O(k) and the regularization.