Student Analysis Seminar

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Quantum theory, uncertainty principles, and contractive estimates (Part II)

Speaker: Omar Abdelghani, NYU Courant

Location: Warren Weaver Hall 517

Date: Tuesday, November 25, 2025, 11 a.m.

Synopsis:

We will complete our discussion of the self-adjointness and stability problems for atomic systems by stating the Kato-Rellich theorem and reformulating Kato's theorem for atomic Hamiltonians in its language. We then change gears and explore ultracontractivity and its relation to the L^infinity Sobolev inequality. We will spend the remainder of the talk discussing the analogous stability and self-adjointness problem for the P(phi)_2 quantum field theory with finite volume cutoff, where the analogue of ultracontractivity/L^infinity-Sobolev is hypercontractivity. We state a self-adjointness and stability theorem for perturbations of generators of hypercontractive semigroups due partially to Nelson, Glimm-Jaffe, and Segal. Finally, we will discuss the birth of the logarithmic Sobolev inequality from these arguments.