Noncrossing Normal Ordering for Functions of Boson Operators

Normally ordered forms of functions of boson operators are important in many contexts in particular concerning Quantum Field Theory and Quantum Optics. Beginning with the seminal work of Katriel (Lett. Nuovo Cimento 10(13):565–567, 1974), in the last few years, normally ordered forms have been shown to have a rich combinatorial structure, mainly in virtue of a link with the theory of partitions. In this paper, we attempt to enrich this link. By considering linear representations of noncrossing partitions, we define the notion of noncrossing normal ordering. Given the growing interest in noncrossing partitions, because of their many unexpected connections (like, for example, with free probability), noncrossing normal ordering appears to be an intriguing notion. We explicitly give the noncrossing normally ordered form of the functions (a ^r (a ^† ) ^s ) ⁿ ) and (a ^r +(a ^† ) ^s ) ⁿ , plus various special cases. We are able to establish for the first time bijections between noncrossing contractions of these functions, k-ary trees and sets of lattice paths.     

Resolvent Expansion and Time Decay of the Wave Functions for Two-Dimensional Magnetic Schrödinger Operators

We consider two-dimensional Schrödinger operators H(B, V) given by Eq. (1.1) below. We prove that, under certain regularity and decay assumptions on B and V, the character of the expansion for the resolvent (H(B, V) ? λ)^?1 as λ → 0 is determined by the flux of the magnetic field B through \({\mathbb{R}^2}\) . Subsequently, we derive the leading term of the asymptotic expansion of the unitary group e ^{?i t H(B, V)} as t → ∞ and show how the magnetic field improves its decay in t with respect to the decay of the unitary group e ^{?i t H(0, V)}.     

A Geometric Approach to Time Evolution Operators of Lie Quantum Systems

Lie systems in Quantum Mechanics are studied from a geometric point of view. In particular, we develop methods to obtain time evolution operators of time-dependent Schrödinger equations of Lie type and we show how these methods explain certain ad hoc methods used in previous papers in order to obtain exact solutions. Finally, several instances of time-dependent quadratic Hamiltonian are solved.     

Multi-Line Geometry of Qubit–Qutrit and Higher-Order Pauli Operators

The commutation relations of the generalized Pauli operators of a qubit–qutrit system are discussed in the newly established graph-theoretic and finite-geometrical settings. The dual of the Pauli graph of this system is found to be isomorphic to the projective line over the product ring . A “peculiar” feature in comparison with two-qubits is that two distinct points/operators can be joined by more than one line. The multi-line property is shown to be also present in the graphs/geometries characterizing two-qutrit and three-qubit Pauli operators’ space and surmised to be exhibited by any other higher-level quantum system. This work was partially supported by the Science and Technology Assistance Agency under the contract # APVT–51–012704, the VEGA grant agency projects # 2/6070/26 and # 7012 (all from Slovak Republic), the trans-national ECO-NET project # 12651NJ “Geometries over Finite Rings and the Properties of Mutually Unbiased Bases” (France) and by the CNRS–SAV Project # 20246 “Projective and Related Geometries for Quantum Information” (France/Slovakia).     

Brownian Approximation and Monte Carlo Simulation of the Non-Cutoff Kac Equation

The non-cutoff Boltzmann equation can be simulated using the DSMC method, by a truncation of the collision term. However, even for computing stationary solutions this may be very time consuming, in particular in situations far from equilibrium. By adding an appropriate diffusion, to the DSMC-method, the rate of convergence when the truncation is removed, may be greatly improved. We illustrate the technique on a toy model, the Kac equation, as well as on the full Boltzmann equation in a special case.     

Absolutely Continuous Spectrum of Schrödinger Operators with Slowly Decaying and Oscillating Potentials

The aim of this paper is to extend a class of potentials for which the absolutely continuous spectrum of the corresponding multidimensional Schrödinger operator is essentially supported by [0,). Our main theorem states that this property is preserved for slowly decaying potentials provided that there are some oscillations with respect to one of the variables.Acknowledgement A.L and O.S. are grateful for the partial support of the ESF European programme SPECT. S.N. would like to thank the Gustafsson Foundation which has allowed him to spend one month at the Royal Institute of Technology in Stockholm. This research was also partly supported by the KBN grant 5, PO3A/026/21. g1925l.     

The Absolutely Continuous Spectrum of One-dimensional Schrödinger Operators

This paper deals with general structural properties of one-dimensional Schrödinger operators with some absolutely continuous spectrum. The basic result says that the ω limit points of the potential under the shift map are reflectionless on the support of the absolutely continuous part of the spectral measure. This implies an Oracle Theorem for such potentials and Denisov-Rakhmanov type theorems. In the discrete case, for Jacobi operators, these issues were discussed in my recent paper (Remling, The absolutely continuous spectrum of Jacobi matrices, http://arxiv.org/abs/0706.1101, 2007). The treatment of the continuous case in the present paper depends on the same basic ideas.     

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     

Time Evolution of Spontaneous Emission Rate of Two Undistinguished Radiators

The behavior of the system of radiators at short and long time intervals in comparison with the retardation between them is studied. The entanglement behavior of atomic states in the process of spontaneous emission is determined. It is demonstrated that at a short time interval the rate of spontaneous emission in an oscillatory manner tends to the exponential law of spontaneous emission. The simple kinetic equation, which describes this stage of system evolution, is obtained.     

Correlation Functions of Harish-Chandra Integrals over the Orthogonal and the Symplectic Groups

The Harish-Chandra correlation functions, i.e. integrals over compact groups of invariant monomials with the weight exp tr (X Ω Y Ω ^† ) are computed for the orthogonal and symplectic groups. We proceed in two steps. First, the integral over the compact group is recast into a Gaussian integral over strictly upper triangular complex matrices (with some additional symmetries), supplemented by a summation over the Weyl group. This result follows from the study of loop equations in an associated two-matrix integral and may be viewed as the adequate version of Duistermaat–Heckman’s theorem for our correlation function integrals. Secondly, the Gaussian integration over triangular matrices is carried out and leads to compact determinantal expressions.     

Fluctuations of Matrix Elements of Regular Functions of Gaussian Random Matrices

We find the limit of the variance and prove the Central Limit Theorem (CLT) for the matrix elements φ _jk (M), j,k=1,…,n of a regular function φ of the Gaussian matrix M (GOE and GUE) as its size n tends to infinity. We show that unlike the linear eigenvalue statistics Tr φ(M), a traditional object of random matrix theory, whose variance is bounded as n→∞ and the CLT is valid for Tr φ(M)−E{Tr φ(M)}, the variance of φ _jk (M) is O(1/n), and the CLT is valid for . This shows the role of eigenvectors in the forming of the asymptotic regime of various functions (statistics) of random matrices. Our proof is based on the use of the Fourier transform as a basic characteristic function, unlike the Stieltjes transform and moments, used in majority of works of the field. We also comment on the validity of analogous results for other random matrices.     

Dynamics of Many-Body Correlation Green's Functions Ⅱ. Equal Time Limit

Under equal time limit,it is shown that the dynamic equations of the n-body correlation Green's functions G(n)c leads to the set of equations for time evolution of the n-body correlations c_n and the many-body correlation Green's functions and their equations of motion are independent of the time order.     

An Example of Embedded Singular Continuous Spectrum for One-Dimensional Schrödinger Operators

We present a new example of a potential such that the corresponding Schrödinger operator in the halfaxis has singular continuous spectrum embedded in the absolutely continuous spectrum. The singular part is supported in an essentialy dense set. This generalizes a result of C. Remling [3].Mathematics Subject Classification: 34L40, 81Q10.     

The Matrix Algebras of Continuous Groups with Antilinear Operations

The corepresentation theory of continuous groups is presented without the assumption that the subgroup G of the group with antilinear operations is unitary. Continuous groups of the form: G+a ₀ G are defined, where G denotes a linear Lie group and a ₀ denotes an antilinear operation which fulfils the condition a²₀=±1a^{2}_{0}=\pm1. The matrix algebras connected with the groups G+a ₀ G are defined. The structural constants of these algebras fulfill the conditions following from the Jacobi identities. Applications are presented to the groups G=SU(d), d=1,2,… , for a ₀=K, the complex conjugation operation, and to the group SL(2,C) for a ₀=K or Θ, the time-reversal operation.     

Generic Continuous Spectrum for Ergodic Schrödinger Operators

We consider families of discrete Schrödinger operators on the line with potentials generated by a homeomorphism on a compact metric space and a continuous sampling function. We introduce the concepts of topological and metric repetition property. Assuming that the underlying dynamical system satisfies one of these repetition properties, we show using Gordon’s Lemma that for a generic continuous sampling function, the set of elements in the associated family of Schrödinger operators that have no eigenvalues is large in a topological or metric sense, respectively. We present a number of applications, particularly to shifts and skew-shifts on the torus.     

Schrödinger Operators with Empty Singularly Continuous Spectra

Let H be a semibounded perturbation of the Laplacian H ₀ in L ²( ^d ). For an admissible function sufficient conditions are given for the completeness of the scattering system (H), (H ₀). If is the exponential function and if e^{– H} is an integral operator we denote the kernel of the difference D = e^{– H} – e^{– H} ₀ by D (x, y), > 0. The singularly continuous spectrum of H is empty if^d dx ^d dy |D(x,y)| (1 + |y|²)< for some > 1. This result is applied to potential perturbations and to perturbations by imposing Dirichlet boundary conditions.     

Orbital Electron Capture Rate and Time Modulation of H- and He-like Ions in a Storage-Ring

We have studied in a heavy ion storage ring the orbital electron capture decays of H- and He-like ¹⁴⁰Pr and ¹⁴²Pm ions and found that the H-like ions with one electron in the K-shell decay 1.49(8) and 1.44(6) times faster, than the corresponding He-like ions with two electrons in the K-shell. This result is explained by spin statistics due to the hyperfine structure of the H-like ions.     

Stieltjes Integrals of Hölder Continuous Functions with Applications to Fractional Brownian Motion

We give a new estimate on Stieltjes integrals of Hölder continuous functions and use it to prove an existence-uniqueness theorem for solutions of ordinary differential equations with Hölder continuous forcing. We construct stochastic integrals with respect to fractional Brownian motion, and establish sufficient conditions for its existence. We prove that stochastic differential equations with fractional Brownian motion have a unique solution with probability 1 in certain classes of Hölder-continuous functions. We give tail estimates of the maximum of stochastic integrals from tail estimates of the Hölder coefficient of fractional Brownian motion. In addition we apply the techniques used for ordinary Brownian motion to construct stochastic integrals of deterministic functions with respect to fractional Brownian motion and give tail estimates of its maximum.