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Joao Nuno Prata

Publications and source records attributed to Joao Nuno Prata.

At least 19 recordsLinked to original sources
arXiv2023

An existence and uniqueness result about algebras of Schwartz distributions

We prove that there exists essentially one {\it minimal} differential algebra of distributions $\A$, satisfying all the properties stated in the Schwartz impossibility result [L. Schwartz, Sur l'impossibilité de la multiplication des distributions, 1954], and such that $\C_p^{\infty} \subseteq \A \subseteq \DO' $ (where $\C_p^{\infty}$ is the set of piecewise smooth functions and $\DO'$ is the set of Schwartz distributions over $\RE$). This algebra is endowed with a multiplicative product of distributions, which is a generalization of the product defined in [N.C.Dias, J.N.Prata, A multiplicative product of distributions and a class of ordinary differential equations with distributional coefficients, 2009]. If the algebra is not minimal, but satisfies the previous conditions, is closed under anti-differentiation and the dual product by smooth functions, and the distributional product is continuous at zero then it is necessarily an extension of $\A$.

math.FA↗
arXiv2020

Partial Traces and the Geometry of Entanglement; Sufficient Conditions for the Separability of Gaussian States

The notion of partial trace of a density operator is essential for the understanding of the entanglement and separability properties of quantum states. In this paper we investigate these notions putting an emphasis on the geometrical properties of the covariance ellipsoids of the reduced states. We thereafter focus on Gaussian states and we give new and easily numerically implementable sufficient conditions for the separability of all Gaussian states. Unlike the positive partial transposition criterion, none of these conditions is however necessary.

quant-ph↗
arXiv2020

Ordinary differential equations with singular coefficients: an intrinsic formulation with applications to the Euler-Bernoulli beam equation

We study a class of linear ordinary differential equations (ODE)s with distributional coefficients. These equations are defined using an {\it intrinsic} multiplicative product of Schwartz distributions which is an extension of the Hörmander product of distributions with non-intersecting singular supports [L. Hörmander, The Analysis of Linear Partial Diffe\-rential Operators I, Springer-Verlag, 1983]. We provide a regularization procedure for these ODEs and prove an existence and uniqueness theorem for their solutions. We also determine the conditions for which the solutions are regular and distributional. These results are used to study the Euler-Bernoulli beam equation with discontinuous and singular coefficients. This problem was addressed in the past using intrinsic products (under some restrictive conditions) and the Colombeau formalism (in the general case). Here we present a new intrinsic formulation that is simpler and more general. As an application, the case of a non-uniform static beam displaying structural cracks is discussed in some detail.

math.CA↗
arXiv2019

On Orthogonal Projections of Symplectic balls

We study the orthogonal projections of symplectic balls in $\mathbb{R}^{2n}$ on complex subspaces. In particular we show that these projections are themselves symplectic balls under a certain complexity assumption. Our main result is a refinement of a recent very interesting result of Abbondandolo and Matveyev extending the linear version of Gromov's non-squeezing theorem. We use a conceptually simpler approach where the Schur complement of a matrix plays a central role.

math.SG↗
arXiv2019

Uncertainty relations for a non-canonical phase-space noncommutative algebra

We consider a non-canonical phase-space deformation of the Heisenberg-Weyl algebra that was recently introduced in the context of quantum cosmology. We prove the existence of minimal uncertainties for all pairs of non-commuting variables. We also show that the states which minimize each uncertainty inequality are ground states of certain positive operators. The algebra is shown to be stable and to violate the usual Heisenberg-Pauli-Weyl inequality for position and momentum. The techniques used are potentially interesting in the context of time-frequency analysis.

math-ph↗
arXiv2018

Ordinary differential equations with point interactions: An inverse problem

Given a linear ordinary differential equation (ODE) on $\RE$ and a set of interface conditions at a finite set of points $I \subset \RE$, we consider the problem of determining another differential equation whose {\it global} solutions satisfy the original ODE on $\RE \backslash I $, and the interface conditions at $I $. Using an extension of the product of distributions with non-intersecting singular supports presented in [L. Hörmander, The Analysis of Linear Partial Diffe\-rential Operators I, Springer-Verlag, 1983], we determine an {\it intrinsic} solution of this problem, i.e. a new ODE, satisfying the required conditions, and strictly defined within the space of Schwartz distributions. Using the same formalism, we determine a singular perturbation formulation for the $n$-th order derivative operator with interface conditions.

math.FA↗
arXiv2017

A refinement of the Robertson-Schrödinger uncertainty principle and a Hirschman-Shannon inequality for Wigner distributions

We propose a refinement of the Robertson-Schrodinger uncertainty principle (RSUP) using Wigner distributions. This new principle is stronger than the RSUP. In particular, and unlike the RSUP, which can be saturated by many phase space functions, the refined RSUP can be saturated by pure Gaussian Wigner functions only. Moreover, the new principle is technically as simple as the standard RSUP. In addition, it makes a direct connection with modern harmonic analysis, since it involves the Wigner transform and its symplectic Fourier transform, which is the radar ambiguity function. As a by-product of the refined RSUP, we derive inequalities involving the entropy and the covariance matrix of Wigner distributions. These inequalities refine the Shanon and the Hirschman inequalities for the Wigner distribution of a mixed quantum state $ρ$. We prove sharp estimates which critically depend on the purity of $ρ$ and which are saturated in the Gaussian case.

math-ph↗
arXiv2016

One-dimensional Schrödinger operators with singular potentials: A Schwartz distributional formulation

Using an extension of the Hörmander product of distributions, we obtain an intrinsic formulation of one-dimensional Schrödinger operators with singular potentials. This formulation is entirely defined in terms of standard {\it Schwartz} distributions and does not require (as some previous approaches) the use of more general distributions or generalized functions. We determine, in the new formulation, the action and domain of the Schrödinger operators with arbitrary singular boundary potentials. We also consider the inverse problem, and obtain a general procedure for constructing the singular (pseudo) potential that imposes a specific set of (local) boundary conditions. This procedure is used to determine the boundary operators for the complete four-parameter family of one-dimensional Schrödinger operators with a point interaction. Finally, the $δ$ and $δ'$ potentials are studied in detail, and the corresponding Schrödinger operators are shown to coincide with the norm resolvent limit of specific sequences of Schrödinger operators with regular potentials.

math.SP↗
arXiv2013

Robertson-Schroedinger type formulation of Ozawa's noise-disturbance uncertainty principle

In this work we derive a matrix formulation of a noise-disturbance uncertainty relation, which is akin to the Robertson-Schrödinger uncertainty principle. Our inequality is stronger than Ozawa's uncertainty principle and takes noise-disturbance correlations into account. Moreover, we show that, for certain types of measurement interactions, it is covariant with respect to linear symplectic transformations of the noise and disturbance operators.

quant-ph↗
arXiv2012

Self-adjoint, globally defined Hamiltonian operators for systems with boundaries

For a general self-adjoint Hamiltonian operator $H_0$ on the Hilbert space $L^2(\RE^d)$, we determine the set of all self-adjoint Hamiltonians $H$ on $L^2(\RE^d)$ that dynamically confine the system to an open set $Ω\subset \RE^d$ while reproducing the action of $ H_0$ on an appropriate operator domain. In the case $H_0=-Δ+V$ we construct these Hamiltonians explicitly showing that they can be written in the form $H=H_0+ B$, where $B$ is a singular boundary potential and $H$ is self-adjoint on its maximal domain. An application to the deformation quantization of one-dimensional systems with boundaries is also presented.

math-ph↗
arXiv2009

A multiplicative product of distributions and a class of ordinary differential equations with distributional coefficients

We construct a generalization of the multiplicative product of distributions presented by L. Hörmander in [L. Hörmander, {\it The analysis of linear partial differential operators I} (Springer-Verlag, 1983)]. The new product is defined in the vector space ${\mathcal A}(\bkR)$ of piecewise smooth functions $f: \bkR \to \bkC$ and all their (distributional) derivatives. It is associative, satisfies the Leibniz rule and reproduces the usual pointwise product of functions for regular distributions in ${\mathcal A}(\bkR)$. Endowed with this product, the space ${\mathcal A}(\bkR)$ becomes a differential associative algebra of generalized functions. By working in the new ${\mathcal A}(\bkR)$-setting we determine a method for transforming an ordinary linear differential equation with general solution $ψ$ into another, ordinary linear differential equation, with general solution $χ_Ω ψ$, where $χ_Ω$ is the characteristic function of some prescribed interval $Ω\subset \bkR$.

math.FA↗
arXiv2009

Exact master equation for a noncommutative Brownian particle

We derive the Hu-Paz-Zhang master equation for a Brownian particle linearly coupled to a bath of harmonic oscillators on the plane with spatial noncommutativity. The results obtained are exact to all orders in the noncommutative parameter. As a by-product we derive some miscellaneous results such as the equilibrium Wigner distribution for the reservoir of noncommutative oscillators, the weak coupling limit of the master equation and a set of sufficient conditions for strict purity decrease of the Brownian particle. Finally, we consider a high-temperature Ohmic model and obtain an estimate for the time scale of the transition from noncommutative to ordinary quantum mechanics. This scale is considerably smaller than the decoherence scale.

hep-th↗
arXiv2009

Noncommutative Quantum Mechanics and Quantum Cosmology

We present a phase-space noncommutative version of quantum mechanics and apply this extension to Quantum Cosmology. We motivate this type of noncommutative algebra through the gravitational quantum well (GQW) where the noncommutativity between momenta is shown to be relevant. We also discuss some qualitative features of the GQW such as the Berry phase. In the context of quantum cosmology we consider a Kantowski-Sachs cosmological model and obtain the Wheeler-DeWitt (WDW) equation for the noncommutative system through the ADM formalism and a suitable Seiberg-Witten (SW) map. The WDW equation is explicitly dependent on the noncommutative parameters, $θ$ and $η$. We obtain numerical solutions of the noncommutative WDW equation for different values of the noncommutative parameters. We conclude that the noncommutativity in the momenta sector leads to a damped wave function implying that this type of noncommmutativity can be relevant for a selection of possible initial states for the universe.

hep-th↗
arXiv2008

A multiplicative product of distributions and a global formulation of the confined Schrodinger equation

A concise derivation of a new multiplicative product of Schwartz distributions is presented. The new product $\star$ is defined in the vector space ${\cal A}$ of piecewise smooth functions on $\bkR$ and all their (distributional) derivatives; it is associative, satisfies the Leibnitz rule and reproduces the usual product of functions for regular distributions. The algebra $({\cal A},+,\star)$ yields a sufficiently general setting to address some interesting problems. As an application we consider the problem of deriving a global formulation for quantum confined systems.

math.FA↗
arXiv2006

Deformation Quantization of Confined Systems

The Weyl-Wigner formulation of quantum confined systems poses several interesting problems. The energy stargenvalue equation, as well as the dynamical equation does not display the expected solutions. In this paper we review some previous results in the subject and add some new contributions. We reformulate the confined energy eigenvalue equation by adding to the Hamiltonian a new (distributional) boundary potential. The new Hamiltonian is proved to be globally defined and self-adjoint. Moreover, it yields the correct Weyl-Wigner formulation of the confined system.

quant-ph↗
arXiv2006

Environment-Induced Decoherence in Noncommutative Quantum Mechanics

We address the question of the appearence of ordinary quantum mechanics in the context of noncommutative quantum mechanics. We obtain the noncommutative extension of the Hu-Paz-Zhang master equation for a Brownian particle linearly coupled to a bath of harmonic oscillators. We consider the particular case of an Ohmic regime.

hep-th↗
arXiv2006

Features of Moyal Trajectories

We study the Moyal evolution of the canonical position and momentum variables. We compare it with the classical evolution and show that, contrary to what is commonly found in the literature, the two dynamics do not coincide. We prove that this divergence is quite general by studying Hamiltonians of the form $p^2 /2m + V(q)$. Several alternative formulations of Moyal dynamics are then suggested. We introduce the concept of starfunction and use it to reformulate the Moyal equations in terms of a system of ordinary differential equations on the noncommutative Moyal plane. We then use this formulation to study the semiclassical expansion of Moyal trajectories, which is cast in terms of a (order by order in $\hbar$) recursive hierarchy of i) first order partial differential equations as well as ii) systems of first order ordinary differential equations. The latter formulation is derived independently for analytic Hamiltonians as well as for the more general case of smooth local integrable ones. We present various examples illustrating these results.

quant-ph↗