Lie systems and Schr"odinger equations (original) (raw)

A rigorous derivation of the Hamiltonian structure for the nonlinear Schrödinger equation

Advances in Mathematics

We consider the cubic nonlinear Schrödinger equation (NLS) in any spatial dimension, which is a well-known example of an infinite-dimensional Hamiltonian system. Inspired by the knowledge that the NLS is an effective equation for a system of interacting bosons as the particle number tends to infinity, we provide a derivation of the Hamiltonian structure, which is comprised of both a Hamiltonian functional and a weak symplectic structure, for the nonlinear Schrödinger equation from quantum manybody systems. Our geometric constructions are based on a quantized version of the Poisson structure introduced by Marsden, Morrison and Weinstein [19] for a system describing the evolution of finitely many indistinguishable classical particles. Contents 1. Introduction 2. Statements of main results and blueprint of proofs 2.1. Construction of the Lie algebra G N and Lie-Poisson manifold G * N 2.2. Derivation of the Lie algebra G ∞ and Lie-Poisson manifold G * ∞ 2.3. The connection with the NLS 2.4. Organization of the paper 3. Notation 3.1. Index of notation 4. Preliminaries 4.1. Weak Poisson structures and Hamiltonian systems 4.2. Some Lie algebra facts 4.3. Bosonic functions, operators and tensor products 5. Geometric structure for the N-body problem 5.1. Lie algebra G N of finite hierarchies quantum observables 5.2. Lie-Poisson manifold G * N of finite hierachies of density matrices 5.3. Density matrix maps as Poisson morphisms 6. Geometric structure for infinity hierarchies 6.1. The limit of G N as N → ∞ 6.2. The Lie algebra G ∞ of observable ∞-hierarchies 6.3. Lie-Poisson manifold G * ∞ of density matrix ∞-hierarchies 6.4. The Poisson morphism ι : S(R) → G * ∞ 7. GP Hamiltonian flows 7.1. BBGKY Hamiltonian Flow 7.2. GP Hamiltonian Flow

Lie-Scheffers systems in Physics

We recall the Theorem by Lie and Scheffers concerning the characterization of systems of differential equations admitting a superposition function, i.e. those whose general solution can be written in terms of some particular solutions and constants. Each of these systems is related with a Lie algebra, specified by the own Theorem. We expose some recently developed Lie theoretic and geometric techniques, useful for treating such systems, as a reduction property and a generalization of the Wei-Norman method. We illustrate the theory with some applications, which are mainly inspired in physical problems.

Quantum quasi-Lie systems: properties and applications

The European Physical Journal Plus

A Lie system is a non-autonomous system of ordinary differential equations describing the integral curves of a t-dependent vector field that is equivalent to a t-dependent family of vector fields within a finite-dimensional Lie algebra of vector fields. Lie systems have been generalised in the literature to deal with t-dependent Schrödinger equations determined by a particular class of t-dependent Hamiltonian operators, the quantum Lie systems, and other systems of differential equations through the so-called quasi-Lie schemes. This work extends quasi-Lie schemes and quantum Lie systems to cope with t-dependent Schrödinger equations associated with the here-called quantum quasi-Lie systems. To illustrate our methods, we propose and study a quantum analogue of the classical nonlinear oscillator searched by Perelomov, and we analyse a quantum one-dimensional fluid in a trapping potential along with quantum t-dependent Smorodinsky–Winternitz oscillators.

Lie groups and quantum mechanics

Journal of Mathematical Analysis and Applications, 2013

Mathematical modeling should present a consistent description of physical phenomena. We illustrate an inconsistency with two Hamiltonians -the standard Hamiltonian and an example found in Goldstein -for the simple harmonic oscillator and its quantisation. Both descriptions are rich in Lie point symmetries and so one can calculate many Jacobi Last Multipliers and therefore Lagrangians. The Last Multiplier provides the route to the resolution of this problem and indicates that the great debate about the quantisation of dissipative systems should never have occurred.

Elements of a Theory of Symplectic Covariant Schrödinger Equations in Phase Space

2005

A classical theorem of Stone and von Neumann says that the Schrödinger representation is, up to unitary equivalences, the only irreducible representation of the Heisenberg group on the Hilbert space of square-integrable functions on configuration space. Using the Wigner-Moyal transform we construct an irreducible representation of the Heisenberg group on a certain Hilbert space of square-integrable functions defined on phase space. This allows us to extend the usual Weyl calculus into a phase-space calculus and leads us to a quantum mechanics in phase space, equivalent to standard quantum mechanics. We also briefly discuss the extension of metaplectic operators to phase space and the probabilistic interpretation of the solutions of the phase space Schrödinger equation. 1

Lie–Hamilton systems on the plane: applications and superposition rules

Journal of Physics A: Mathematical and Theoretical, 2015

A Lie-Hamilton system is a nonautonomous system of first-order ordinary differential equations describing the integral curves of a t-dependent vector field taking values in a finitedimensional real Lie algebra of Hamiltonian vector fields with respect to a Poisson structure. We provide new algebraic/geometric techniques to easily determine the properties of such Lie algebras on the plane, e.g., their associated Poisson bivectors. We study new and known Lie-Hamilton systems on R 2 with physical, biological and mathematical applications. New results cover Cayley-Klein Riccati equations, the here defined planar diffusion Riccati systems, complex Bernoulli differential equations and projective Schrödinger equations. Constants of motion for planar Lie-Hamilton systems are explicitly obtained which, in turn, allow us to derive superposition rules through a coalgebra approach.

Symplectic Structures and Quantum Mechanics

Modern Physics Letters B, 1996

Canonical coordinates for the Schrödinger equation are introduced, making more transparent its Hamiltonian structure. It is shown that the Schrödinger equation, considered as a classical field theory, shares with Liouville completely integrable field theories the existence of a recursion operator which allows for the infinitely many conserved functionals pairwise commuting with respect to the corresponding Poisson bracket.

Non-Linear Semi-Quantum Hamiltonians and Its Associated Lie Algebras

Applied Mathematics, 2014

We show that the non-linear semi-quantum Hamiltonians which may be expressed as (where ρ is the Maximum Entropy Principle density operator) and, so, these kind of Hamiltonians always have associated dynamic invariants which are expressed in terms of the quantum degrees of freedom's mean values. Those invariants are useful to characterize the kind of dynamics (regular or irregular) the system displays given that they can be fixed by means of the initial conditions imposed on the semi-quantum non-linear system.