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

  • 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.

    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

    Related Papers:

  • 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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