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A symplectic approach to Schrödinger equations in the infinite-dimensional unbounded setting

Javier de Lucas, Julia Lange, Xavier Rivas

TL;DR

This work develops a rigorous symplectic-geometric treatment of $t$-dependent Schrödinger equations on separable Hilbert spaces driven by unbounded self-adjoint Hamiltonians with a common domain of analytic vectors. By encoding quantum dynamics as Hamiltonian flows on normed/Banach manifolds, it clarifies how to handle unbounded observables and constructs a coherent framework using $(oldsymbol{ ext{H}},oldsymbol{ ext{ω}})$ on a domain, with the Schrödinger equation given by $ rac{d}{dt}oldsymbol{\psi}=-iH(t)oldsymbol{\psi}$ corresponding to the Hamiltonian vector field of $ frac{1}{2}igra oldsymbol{\psi}ig|H(t)oldsymbol{\psi}igra$. The Marsden--Weinstein reduction is then employed to project these dynamics onto the projective Hilbert space $oldsymbol{ ext{PH}}$, yielding a reduced symplectic form $oldsymbol{ ext{ω}}_oldsymbol{\mu}$ and reduced Hamiltonians $f_eta^{oldsymbol{\\mu}}$, governing the rays that encode physical states. These contributions provide a rigorous bridge between infinite-dimensional quantum mechanics and symplectic reduction, with potential for extending to Hamilton–Jacobi theory and broader reductive techniques in quantum dynamics.

Abstract

By using the theory of analytic vectors and manifolds modelled on normed spaces, we provide a rigorous symplectic differential geometric approach to $t$-dependent Schrödinger equations on separable (possibly infinite-dimensional) Hilbert spaces determined by unbounded $t$-dependent self-adjoint Hamiltonians satisfying a technical condition. As an application, the Marsden--Weinstein reduction procedure is employed to map above-mentioned $t$-dependent Schrödinger equations onto their projective spaces. Other applications of physical and mathematical relevance are also analysed.

A symplectic approach to Schrödinger equations in the infinite-dimensional unbounded setting

TL;DR

This work develops a rigorous symplectic-geometric treatment of -dependent Schrödinger equations on separable Hilbert spaces driven by unbounded self-adjoint Hamiltonians with a common domain of analytic vectors. By encoding quantum dynamics as Hamiltonian flows on normed/Banach manifolds, it clarifies how to handle unbounded observables and constructs a coherent framework using on a domain, with the Schrödinger equation given by corresponding to the Hamiltonian vector field of . The Marsden--Weinstein reduction is then employed to project these dynamics onto the projective Hilbert space , yielding a reduced symplectic form and reduced Hamiltonians , governing the rays that encode physical states. These contributions provide a rigorous bridge between infinite-dimensional quantum mechanics and symplectic reduction, with potential for extending to Hamilton–Jacobi theory and broader reductive techniques in quantum dynamics.

Abstract

By using the theory of analytic vectors and manifolds modelled on normed spaces, we provide a rigorous symplectic differential geometric approach to -dependent Schrödinger equations on separable (possibly infinite-dimensional) Hilbert spaces determined by unbounded -dependent self-adjoint Hamiltonians satisfying a technical condition. As an application, the Marsden--Weinstein reduction procedure is employed to map above-mentioned -dependent Schrödinger equations onto their projective spaces. Other applications of physical and mathematical relevance are also analysed.
Paper Structure (13 sections, 19 theorems, 60 equations)

This paper contains 13 sections, 19 theorems, 60 equations.

Table of Contents

  1. Introduction
  2. Fundamentals on operators
  3. Geometry on normed and Banach manifolds
  4. Geometry on normed and Banach manifolds
  5. Symplectic geometry in the infinite-dimensional context
  6. Symplectic forms on separable Hilbert spaces
  7. Vector fields related to operators
  8. Hamiltonian functions induced by self-adjoint operators
  9. On --dependent Hamiltonian operators
  10. Reduction process for t-dependent Schrödinger equations
  11. The Marsden--Weinstein reduction of the symplectic form on
  12. Marsden--Weinstein reduction of Schrödinger equations
  13. Conclusions and Outlook

Key Result

Theorem 2.3

(Closed graph theorem Hal_13) Let $A: \mathcal{X}\rightarrow \mathcal{Y}$ be an operator between Banach spaces. The operator $A$ is continuous if and only if its graph is closed in $\mathcal{X}\times \mathcal{Y}$.

Theorems & Definitions (54)

  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Definition 3.1
  • Example 3.2
  • Example 3.4
  • Definition 3.5
  • Theorem 3.6
  • Definition 3.7
  • ...and 44 more