Spectral Theory and PDEs in Quantum Mechanics

Project Reference: Proyecto de Generación de Conocimiento PID2024-155550NB-I00, "Spectral Theory and PDEs in Quantum Mechanics"

Host Institution: Faculty of Science and Technology, Department of Mathematics, University of the Basque Country (EHU)

Duration: Sept. 2025-Aug. 2028

Dirac operators, Relativistic Virial identities, Threshold eigenvalues, Keller and Lieb-Thirring inequalities, Hardy-type inequalities, Schrödinger operators, Clustering conditions, Zonal and Axisymmeteric potentials,  Scattering with Aharonov-Bohm fields, Dispersive equations on the Heisenberg group

Spectral Theory and PDEs have been intrinsically linked since their inception, sharing fundamental questions. The development of Quantum Mechanics in the past century further reinforced this connection, leading to the emergence of powerful new analytical tools. A major breakthrough came in the 1960s with T. Kato’s pioneering work, which established a fundamental duality between the eigenvalue problem for certain Hamiltonians and their associated evolution equations described elegantly by the Fourier Transform and the Spectral Theorem.

This project  seeks to further strengthen this connection by addressing key open problems, including spectral stability under external perturbations, localization of point spectrum, analysis of the spectra of Laplace-Beltrami operators and their relation to dispersive inequalities, and large-time scattering for nonlinear dispersive PDEs. Essential tools for these objectives include virial and Morawetz-type identities, as well as uniform resolvent estimates, which will play a crucial role in the proposed research.