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

Javier de Lucas; Julia Lange; Xavier Rivas; Javier de Lucas; Julia Lange; Xavier Rivas

doi:10.3934/math.20241359

AIMS Mathematics

2024, Volume 9, Issue 10: 27998-28043. doi: 10.3934/math.20241359

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

1.
Centre de Recherches Mathématiques, Université de Montréal, Montréal (Québec) H3C3J7, Canada
2.
Department of Mathematical Methods in Physics, University of Warsaw, Warsaw 02093, Poland
3.
Department of Computer Engineering and Mathematics, Universitat Rovira i Virgili, Tarragona 43007, Spain

Received: 04 July 2024 Revised: 11 September 2024 Accepted: 20 September 2024 Published: 27 September 2024
MSC : 17B66, 34A26, 34A34, 53Z05

By using the theory of analytic vectors and manifolds modeled 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 families of unbounded self-adjoint Hamiltonians admitting a common domain of analytic vectors. This allows one to cope with the lack of smoothness of structures appearing in quantum mechanical problems while using differential geometric techniques. Our techniques also allow for the analysis of problems related to unbounded operators that are not self-adjoint. As an application, the Marsden-Weinstein reduction procedure was employed to map the above-mentioned $ t $-dependent Schrödinger equations onto their projective spaces. We also analyzed other physically and mathematically relevant applications, demonstrating the usefulness of our techniques.
- analytic vector,
- infinite-dimensional symplectic manifold,
- Marsden-Weinstein reduction,
- normed space,
- projective Schrödinger equation,
- unbounded operator
Citation: Javier de Lucas, Julia Lange, Xavier Rivas. A symplectic approach to Schrödinger equations in the infinite-dimensional unbounded setting[J]. AIMS Mathematics, 2024, 9(10): 27998-28043. doi: 10.3934/math.20241359

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Abstract

By using the theory of analytic vectors and manifolds modeled 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 families of unbounded self-adjoint Hamiltonians admitting a common domain of analytic vectors. This allows one to cope with the lack of smoothness of structures appearing in quantum mechanical problems while using differential geometric techniques. Our techniques also allow for the analysis of problems related to unbounded operators that are not self-adjoint. As an application, the Marsden-Weinstein reduction procedure was employed to map the above-mentioned $ t $-dependent Schrödinger equations onto their projective spaces. We also analyzed other physically and mathematically relevant applications, demonstrating the usefulness of our techniques.

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