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

Author

Massachusetts Institute of Technology. Department of Mathematics, Mendelson, Dana, Nahmod, Andrea R, Pavlović, Nataša, Rosenzweig, Matthew, Staffilani, Gigliola, Massachusetts Institute of Technology. Department of Mathematics, Mendelson, Dana, Nahmod, Andrea R, Pavlović, Nataša, Rosenzweig, Matthew, and Staffilani, Gigliola

Abstract

© 2020 Elsevier Inc. 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 many-body systems. Our geometric constructions are based on a quantized version of the Poisson structure introduced by Marsden, Morrison and Weinstein [24] for a system describing the evolution of finitely many indistinguishable classical particles.

Published
2021

55. Colo-ovarian fistula

Author

Rosenzweig, Matthew, primary, Marshall, Jessica, additional, White, Ronald A, additional, Tismenetsky, Misha, additional, and Shembde, Dwarkanath, additional

Published
2017
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57. Tension pneumoperitoneum caused by blunt thoracic trauma.

Author

Webman, Rachel, Rosenzweig, Matthew, Bholat, Omar, Bernstein, Mark, Todd, S Rob, and Frangos, Spiros G

Subjects
*ABDOMINAL diseases, *ABDOMINAL surgery, *BLUNT trauma, *CHEST injuries, *COMPARTMENT syndrome, *SURGICAL decompression, *PNEUMOPERITONEUM, *DISEASE complications
Abstract

Tension pneumoperitoneum is a rare entity that occurs when free air under pressure accumulates in the abdominal cavity compromising visceral function and blood flow. The case of a 23-year-old man whose chest was run over by the wheels of a truck is presented. He arrived with a severely distended abdomen, significant thoracic trauma, hypoxemia, and elevated airway pressures following intubation. Imaging studies revealed massive intraperitoneal free air. He was treated with a decompressive laparotomy but was not found to have a hollow viscus nor diaphragmatic injury. We hypothesize a possible, as yet unpublished, mechanism: secondary to the patient’s rib fractures and significant torso soft tissue shearing, a defect in the parietal pleura allowed air to track from the thorax inferiorly along subcutaneous and fascial planes, eventually entering into the peritoneal cavity through a violated parietal peritoneum. We present a review of the literature on tension pneumoperitoneum. [ABSTRACT FROM AUTHOR]

Published
2014
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58. Old and new perspectives on effective equations : a study of quantum many-body systems

Author

Rosenzweig, Matthew Harry

Subjects
Nonlinear Schrodinger, Quantum many-body systems, Bose-Einstein condensates, Lieb-Liniger, Hamiltonian systems
Abstract

This dissertation focuses on the study of nonlinear-Schrodinger-type equations as partial differentiation equations (PDEs) arising as effective descriptions of systems of finitely many interacting bosons. We approach this topic from two perspectives. The old perspective consists of proving quantitative convergence in an appropriate function space of solutions to the finite problem to a solution of an effective, limiting PDE, as the number of particles tends to infinity. The new perspective consists of proving qualitative convergence of geometric structure, such as the properties of being an integrable and Hamiltonian system. Through these two complementary perspectives, focusing on both quantitative and qualitative convergence, we gain a deeper understanding of how field theories, both classical and quantum, may be deformed to a new field theory, and of how this deformation may be reversed.

Published
2020

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