scholarly journals Spectral expansions of non-self-adjoint generalized Laguerre semigroups

2021 ◽  
Vol 272 (1336) ◽  
Author(s):  
Pierre Patie ◽  
Mladen Savov
Keyword(s):  
Hilbert Space ◽  
Laguerre Polynomials ◽  
Spectral Expansion ◽  
Convergence To Equilibrium ◽  
Markov Semigroup ◽  
New Developments ◽  
Spectral Expansions ◽  
Original Analysis ◽  
Non Local ◽  
Spectral Projections

We provide the spectral expansion in a weighted Hilbert space of a substantial class of invariant non-self-adjoint and non-local Markov operators which appear in limit theorems for positive-valued Markov processes. We show that this class is in bijection with a subset of negative definite functions and we name it the class of generalized Laguerre semigroups. Our approach, which goes beyond the framework of perturbation theory, is based on an in-depth and original analysis of an intertwining relation that we establish between this class and a self-adjoint Markov semigroup, whose spectral expansion is expressed in terms of the classical Laguerre polynomials. As a by-product, we derive smoothness properties for the solution to the associated Cauchy problem as well as for the heat kernel. Our methodology also reveals a variety of possible decays, including the hypocoercivity type phenomena, for the speed of convergence to equilibrium for this class and enables us to provide an interpretation of these in terms of the rate of growth of the weighted Hilbert space norms of the spectral projections. Depending on the analytic properties of the aforementioned negative definite functions, we are led to implement several strategies, which require new developments in a variety of contexts, to derive precise upper bounds for these norms.

Farley Buneman instabilities in the Auroral region: Continuous kinetic and hybrid simulations.

2021 ◽  
Author(s):  
Enrique Rojas ◽  
David Hysell
Keyword(s):  
Computational Cost ◽  
Phase Speed ◽  
Auroral Region ◽  
Comprehensive Description ◽  
New Developments ◽  
Particle In Cell ◽  
New Methods ◽  
E Region ◽  
Non Local ◽  
Radar Measurements

<p>Farley-Buneman instabilities generate a spectrum of field-aligned plasma density irregularities in the E region. Although fully kinetic particle-in-cell simulations offer a comprehensive description of the underlying physics, its computational cost for studying non-local phenomena is tremendous. New methods based on hybrid and continuous approaches have to be explored to capture non-local physics.</p><p>In this work, we present new developments on a continuous solver of Farley-Buneman waves. We compare the performance of fully kinetic (continuous), hybrid, and fluid models. Furthermore, we investigate phase speed saturation, wave turning effects, and dominant wavelengths and assess how well these correspond to radar measurements. Finally, we describe some initial attempts at constructing simple surrogate models to capture the dominant microphysics of these simulations.</p>

MORE ON FINITE TEMPERATURE ANYON SUPERCONDUCTIVITY

1992 ◽  
Vol 06 (26) ◽  
pp. 1623-1637 ◽  
Author(s):  
Y. LEBLANC ◽  
J.C. WALLET
Keyword(s):  
High Temperature ◽  
Finite Temperature ◽  
Linear Response ◽  
Bessel Functions ◽  
Laguerre Polynomials ◽  
Linear Response Theory ◽  
Threshold Temperature ◽  
Type I ◽  
Type Ii ◽  
Non Local

In the framework of finite temperature linear response theory, we analyze to a greater extent the nature of anyonic superconductivity. Using identities among Laguerre polynomials and Bessel functions, we provide simple and useful expressions for the response function in the form of high temperature expansions. The physical penetration depth as well as the Landau-Ginzburg coherence length are calculated for all temperatures. We find that for statistics restricted by n≪450, anyon superconductors are type II local (London) superconductors at low temperature and type I non-local (Pippard) superconductors at high temperature. The threshold temperature is also obtained.

scholarly journals Coherent modification of entanglement: benefits due to extended Hilbert space

2016 ◽  
Vol 16 (11&12) ◽  
pp. 954-968
Author(s):  
Dmitry Solenov
Keyword(s):  
Hilbert Space ◽  
Continuous Time ◽  
Computing System ◽  
Local Network ◽  
Toffoli Gate ◽  
Quantum Random Walks ◽  
Physical Systems ◽  
Time Quantum ◽  
Speed Up ◽  
Non Local

A quantum computing system is typically represented by a set of non-interacting (local) two-state systems—qubits. Many physical systems can naturally have more accessible states, both local and non-local. We show that the resulting non-local network of states connecting qubits can be efficiently addressed via continuous time quantum random walks, leading to substantial speed-up of multiqubit entanglement manipulations. We discuss a three-qubit Toffoli gate and a system of superconducting qubits as an illustration.

scholarly journals On non-local ergodic Jacobi semigroups: spectral theory, convergence-to-equilibrium and contractivity

2021 ◽  
Vol 8 ◽  
pp. 331-378
Author(s):  
Patrick Cheridito ◽  
Pierre Patie ◽  
Anna Srapionyan ◽  
Aditya Vaidyanathan
Keyword(s):  
Spectral Theory ◽  
Convergence To Equilibrium ◽  
Non Local

scholarly journals Type II superstring field theory with cyclic $L_\infty$ structure

2020 ◽  
Vol 2020 (3) ◽  
Author(s):  
H Kunimoto ◽  
T Sugimoto
Keyword(s):  
Field Theory ◽  
Hilbert Space ◽  
String Field Theory ◽  
Symplectic Form ◽  
Closed String ◽  
Type Ii ◽  
Gauge Invariant ◽  
Non Local ◽  
String Amplitudes ◽  
Algebraic Formulation

Abstract We construct a complete type II superstring field theory that includes all the NS–NS, R–NS, NS–R, and R–R sectors. As in the open and heterotic superstring cases, the R–NS, NS–R, and R–R string fields are constrained by using the picture-changing operators. In particular, we use a non-local inverse picture-changing operator for the constraint on the R–R string field, which seems to be inevitable due to the compatibility of the extra constraint with the closed string constraints. The natural symplectic form in the restricted Hilbert space gives a non-local kinetic action for the R–R sector, but it correctly provides the propagator expected from the first-quantized formulation. Extending the prescription previously obtained for the heterotic string field theory, we give a construction of general type II superstring products, which realizes a cyclic $L_\infty$ structure, and thus provides a gauge-invariant action based on the homotopy algebraic formulation. Three typical four-string amplitudes derived from the constructed string field theory are demonstrated to agree with those in the first-quantized formulation. We also give the half-Wess–Zumino–Witten action defined in the medium Hilbert space whose left-moving sector is still restricted to the small Hilbert space.

scholarly journals Multipartite Quantum Systems and Representations of Wreath Products

2020 ◽  
Vol 226 ◽  
pp. 02013
Author(s):  
Vladimir Kornyak
Keyword(s):  
Hilbert Space ◽  
Quantum Correlations ◽  
Wreath Products ◽  
Quantum Systems ◽  
Efficient Computation ◽  
Multipartite Quantum Systems ◽  
Irreducible Components ◽  
Local Correlations ◽  
Non Local ◽  
Multipartite System

The multipartite quantum systems are of particular interest for the study of such phenomena as entanglement and non-local correlations. The symmetry group of the whole multipartite system is the wreath product of the group acting in the “local” Hilbert space and the group of permutations of the constituents. The dimension of the Hilbert space of a multipartite system depends exponentially on the number of constituents, which leads to computational difficulties. We describe an algorithm for decomposing representations of wreath products into irreducible components. The C implementation of the algorithm copes with representations of dimensions in quadrillions. The program, in particular, builds irreducible invariant projectors in the Hilbert space of a multipartite system. The expressions for these projectors are tensor product polynomials. This structure is convenient for efficient computation of quantum correlations in multipartite systems with a large number of constituents.

scholarly journals Tsirelson's bound and supersymmetric entangled states

2014 ◽  
Vol 470 (2170) ◽  
pp. 20140253 ◽  
Author(s):  
L. Borsten ◽  
K. Brádler ◽  
M. J. Duff
Keyword(s):  
Hilbert Space ◽  
Quantum Mechanics ◽  
Transition Probabilities ◽  
Entangled States ◽  
Supersymmetric Extension ◽  
Standard Quantum ◽  
Local Resource ◽  
Non Local ◽  
Minimal Supersymmetric Extension ◽  
Tsirelson Bound

A superqubit, belonging to a (2|1)-dimensional super-Hilbert space, constitutes the minimal supersymmetric extension of the conventional qubit. In order to see whether superqubits are more non-local than ordinary qubits, we construct a class of two-superqubit entangled states as a non-local resource in the CHSH game. Since super Hilbert space amplitudes are Grassmann numbers, the result depends on how we extract real probabilities and we examine three choices of map: (1) DeWitt (2) Trigonometric and (3) Modified Rogers. In cases (1) and (2), the winning probability reaches the Tsirelson bound p win = cos 2 π / 8 ≃ 0.8536 of standard quantum mechanics. Case (3) crosses Tsirelson's bound with p win ≃0.9265. Although all states used in the game involve probabilities lying between 0 and 1, case (3) permits other changes of basis inducing negative transition probabilities.

scholarly journals Absolute and Uniform Convergence of Spectral Expansion in the Eigenfunctions of a Third-Order Ordinary Differential Operator

2020 ◽  
Vol 19 ◽  
Keyword(s):  
Differential Operator ◽  
Uniform Convergence ◽  
Sufficient Conditions ◽  
Spectral Expansion ◽  
Ordinary Differential Operator ◽  
Third Order ◽  
Spectral Expansions

In this paper studied the convergence of spectral expansions of functions of the class W1 1 ( ) G ,G= ( ) 0,1 in eigenfunctions of an ordinary differential operator of third order with integral coefficients. Sufficient conditions for absolute and uniform convergence are obtained and the rate of uniform convergence of these expansions on the interval G is found.

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