Juan Manuel Pérez-Pardo

University Carlos III de Madrid | UC3M · Department of Mathematics

A numerical algorithm to solve the spectral problem for arbitrary self-adjoint extensions of 1D regular Schroedinger operators is presented. The construction of all self-adjoint extensions of the symmetric Schroedinger operator on a compact manifold of arbitrary dimension with boundary is discussed. The self-adjoint extensions of such symmetric ope...

We will show how it is possible to generate entangled states out of unentangled ones on a bipartite system by means of dynamical boundary conditions. The auxiliary system is defined by a symmetric but non-self-adjoint Hamiltonian and the space of self--adjoint extensions of the bipartite system is studied. It is shown that only a small set of them...

We construct in this article a class of closed semi-bounded quadratic forms on the space of square integrable functions over a smooth Riemannian manifold with smooth boundary. Each of these quadratic forms specifies a semi-bounded self-adjoint extension of the Laplace-Beltrami operator. These quadratic forms are based on the Lagrange boundary form...

We introduce a relativistic version of the non-self-adjoint operator obtained by a dilation analytic transformation of the quantum harmonic oscillator. While the spectrum is real and discrete, we show that the eigenfunctions do not form a basis and that the pseudospectra are highly non-trivial.

We study the stability of the Schrödinger equation generated by time-dependent Hamil-tonians with constant form domain. That is, we bound the difference between solutions of the Schrödinger equation by the difference of their Hamiltonians. The stability theorem obtained in this article provides a sharper bound than those previously obtained in the...

We study a system composed of a free quantum particle trapped in a box whose walls can change their position. We prove the global approximate controllability of the system: any initial pure state can be driven arbitrarily close to any target pure state in the Hilbert space of the free particle with a predetermined final position of the box. To this...

We provide sufficient conditions for the approximate controllability of infinite-dimensional quantum control systems corresponding to form perturbations of the drift Hamiltonian modulated by a control function. We rely on previous results on controllability of quantum bilinear control systems and obtain a priori L1-bounds of the controls for generi...

We investigate the controllability of an infinite-dimensional quantum system: a quantum particle confined on a Thick Quantum Graph, a generalisation of Quantum Graphs whose edges are allowed to be manifolds of arbitrary dimension with quasi-δ boundary conditions. This is a particular class of self-adjoint boundary conditions compatible with the gra...

We study the stability of the Schrödinger equation generated by time-dependent Hamiltonians with constant form domain. That is, we bound the difference between solutions of the Schrödinger equation by the difference of their Hamiltonians. The stability theorem obtained in this article provides a sharper bound than those previously obtained in the l...

This article presents in a self-contained way A. Uhlmann's celebrated Theorem of monotonicity of the relative entropy under completely positive and trace preserving maps. The Theorem is presented in its more general form and meaningful examples are given.

We study a system composed of a free quantum particle trapped in a box whose walls can change their position. We prove the global approximate controllability of the system. That is, any initial state can be driven arbitrarily close to any target state in the Hilbert space of the free particle with a predetermined final position of the box. To this...

We study two seminal approaches, developed by B. Simon and J. Kisyński, to the well-posedness of the Schrödinger equation with a time-dependent Hamiltonian. In both cases, the Hamiltonian is assumed to be semibounded from below and to have a constant form domain, but a possibly non-constant operator domain. The problem is addressed in the abstract...

We study two seminal approaches, developed by B. Simon and J. Kisyński, to the well-posedness of the Schrödinger equation with a time-dependent Hamiltonian. In both cases the Hamiltonian is assumed to be semibounded from below and to have constant form domain but a possibly non-constant operator domain. The problem is addressed in the abstract sett...

We investigate the controllability of an infinite-dimensional quantum system by modifying the boundary conditions instead of applying external fields. We analyse the existence of solutions of the Schr\"odinger equation for a time-dependent Hamiltonian with time-dependent domain, but constant form domain. A stability theorem for such systems, which...

A representation theorem for non-semibounded Hermitian quadratic forms in terms of a (non-semibounded) self-adjoint operator is proven. The main assumptions are closability of the Hermitian quadratic form, the direct integral structure of the underlying Hilbert space and orthogonal additivity. We apply this result to several examples, including the...

We present Balmaseda, A. scheme for controlling the state of a quantum system Pérez-Pardo, J. M. modifying the boundary conditions. This constitutes an infinite-dimensional control problem. We provide conditions for the existence of solutions of the dynamics and prove that this system is approximately controllable.

An analysis of the invariance properties of self-adjoint extensions of symmetric operators under the action of a group of symmetries is presented. For a given group G, criteria for the existence of G-invariant self-adjoint extensions of the Laplace–Beltrami operator over a Riemannian manifold are illustrated and critically revisited. These criteria...

An analysis of the invariance properties of self-adjoint extensions of symmetric operators under the action of a group of symmetries is presented. For a given group $G$, criteria for the existence of $G$-invariant self-adjoint extensions of the Laplace-Beltrami operator over a Riemannian manifold are illustrated and critically revisited. These crit...

We present a scheme for controlling the state of a quantum system by modifying the boundary conditions. This constitutes an infinite-dimensional control problem. We provide conditions for the existence of solutions of the dynamics and prove that this system is approximately controllable.

Potential functions can be used for generating potentials of relevant geometric structures for a Riemannian manifold such as the Riemannian metric and affine connections. We study whether this procedure can also be applied to tensors of rank four and find a negative answer. We study this from the perspective of solving the inverse problem and also...

In this article we give a representation theorem for non-semibounded Hermitean quadratic forms in terms of a (non-semibounded) self-adjoint operator. The main assumptions are closability of the Hermitean quadratic form, the direct integral structure of the underlying Hilbert space and orthogonal additivity. We apply this result to several examples,...

Potential functions can be used as generating potentials of relevant geometric structures for a Riemannian manifold such as the Riemannian metric and affine connections. We study wether this procedure can also be applied to tensors of rank four and find a negative answer. We study this from the perspective of solving the inverse problem and also fr...

The purpose of this article is to exploit the geometric structure of Quantum Mechanics and of statistical manifolds to study the qualitative effect that the quantum properties have in the statistical description of a system. We show that the end points of geodesics in the classical setting coincide with the probability distributions that min-imise...

A numerical scheme to compute the spectrum of a large class of self-adjoint extensions of the Laplace-Beltrami operator on manifolds with boundary in any dimension is presented. The algorithm is based on the characterisation of a large class of self-adjoint extensions of Laplace-Beltrami operators in terms of their associated quadratic forms. The c...

The search for a potential function $S$ allowing to reconstruct a given metric tensor $g$ and a given symmetric covariant tensor $T$ on a manifold $\mathcal{M}$ is formulated as the Hamilton-Jacobi problem associated with a canonically defined Lagrangian on $T\mathcal{M}$. The connection between this problem, the geometric structure of the space of...

We investigate the validity of Huygens’ principle for forward propagation in the massless Dirac-Weyl equation. The principle holds for odd space dimension n, while it is invalid for even n. We explicitly solve the cases n=1,2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackag...

The problem of self-adjoint extensions of Dirac-type operators in manifolds with boundaries is analysed. The boundaries might be regular or non-regular. The latter situation includes point-like interactions, also called delta-like potentials, in manifolds of dimension higher than one. Self-adjoint boundary conditions for the case of dimension 2 are...

We analyse the effects of Robin boundary conditions on quantum field theories of spin 0, 1 and 1/2. In particular, we show that these conditions always lead to the appearance of edge states that play a significant role in quantum Hall effect and topological insulators. We prove in a rigorous way the existence of spectral lower bounds on the kinetic...

We propose a new point of view regarding the problem of time in quantum mechanics, based on the idea of replacing the usual time operator $\mathbf{T}$ with a suitable real-valued function $T$ on the space of physical states. The proper characterization of the function $T$ relies on a particular relation with the dynamical evolution of the system ra...

We describe how it is possible to describe irreducible actions of the Hodge - de Rham Dirac operator upon the exterior algebra over the quantum spheres ${\rm SU}_q(2)$ equipped with a three dimensional left covariant calculus.

In this article we exploit the geometric structure of statistical manifolds to compare vis-a-vis quantum and classical dynamics. We show that the fixed points of geodesic dynamics in a classical setting coincide with the distributions that minimise the Shannon's entropy. Whereas in a quantum setting this happens only for particular initial conditio...

A numerical scheme to compute a large class of self-adjoint extensions of the Laplace-Beltrami operator on manifolds with boundary in any dimension is presented. The algorithm is based on the characterisation, recently obtained by the authors, of a large class of self-adjoint extensions of Laplace-Beltrami operators in terms of their associated qua...

This is a series of 5 lectures around the common subject of the construction of self-adjoint extensions of symmetric operators and its applications to Quantum Physics. We will try to offer a brief account of some recent ideas in the theory of self-adjoint extensions of symmetric operators on Hilbert spaces and their applications to a few specific p...

We show how to use boundary conditions to drive the evolution on a Quantum Mechanical system. We will see how this problem can be expressed in terms of a time-dependent Schr\"{o}dinger equation. In particular we will need the theory of self-adjoint extensions of differential operators in manifolds with boundary. An introduction of the latter as wel...

Given a unitary representation of a Lie group G on a Hilbert space \({\mathcal H}\), we develop the theory of G-invariant self-adjoint extensions of symmetric operators using both von Neumann’s theorem and the theory of quadratic forms. We also analyze the relation between the reduction theory of the unitary representation and the reduction of the...

The dynamics of the magnetic field in a superconducting phase is described by an effective massive bosonic field theory. If the superconductor is confined in a domain M with boundary \partial M, the boundary conditions of the electromagnetic fields are predetermined by physics. They are time-reversal and also parity invariant for adapted geometry....

The main objective of this dissertation is to analyse thoroughly the construction of self-adjoint extensions of the Laplace-Beltrami operator defined on a compact Riemannian manifold with boundary and the role that quadratic forms play to describe them. Moreover, we want to emphasise the role that quadratic forms can play in the description of quan...

In the context of the geometric formulation of quantum mechanics the observables are characterised by the quadratic forms associated to the self- adjoint operators that describe the corresponding observables in the standard for- mulation. If the self-adjoint operators are bounded, it can be shown, that their associated quadratic forms are in one-to...

A new notion of controllability for quantum systems that takes advantage of the linear superposition of quantum states is introduced. We call such notion von Neumann controllabilty and it is shown that it is strictly weaker than the usual notion of pure state and operator controlability. We provide a simple and effective characterization of it by u...