Effective equations of quantum mechanics and phase transitions

Abstract: Effective equations of many-body quantum mechanics form the backbone of many fields of modern physics. Notable examples of effective equations include the Hartree-Fock, Kohn-Sham, and Bogoliubov-de Gennes (BdG) equations. Although their physical derivations vary, we will review an unified formal mathematical frame work for their derivations (if time permits). In this frame work, the BdG equations are the most general form of effective equations. Physically, they form a microscopic description of superconductivity. When the temperature T is lower than a certain critical Tc, superconducting solutions emerge. In this talk, we will demonstrate the the existence of solutions to the BdG equations via variational arguments and show energy instability (hence the formation of a superconducting order parameter) when T < Tc.

This is a joint work with I. M. Sigal

analysis of PDEsclassical analysis and ODEsfunctional analysismetric geometry

Audience: researchers in the topic

HA-GMT-PDE Seminar

Series comments: Description: A senior graduate student/postdoc series in harmonic analysis, geometric measure theory, and partial differential equations.

Meetings will be weekly, and usually will occur on Monday, with some exceptions. For each talk, the Zoom link is made available in the website too. Contact Bruno Poggi at poggi008@umn.edu if you would like to subscribe to the seminar mailing list.