Dismantling Dimensional Barriers: The Sliced Spectral Framework for Periodic Elliptic Operators

Vortrag von Prof. Dr. Ruming Zhang

Datum: 25.03.26  Zeit: 16.30 - 18.00  Raum: Y27H35/36

The study of the limiting absorption principle for elliptic equations with periodic structures is very challenging when the dimension is greater than one . The fundamental obstacle lies in the mismatch between directional physical reality and the direction-independent classic spectral analysis, resulting in intricate geometry of band structures and Fermi surfaces in high dimensions.

In this paper, we introduce the novel Sliced Spectral Analysis (SSA) framework which  introduce the direction into classic spectral analysis. It dismantles the geometrical complexities by reducing the problems  to its analytical essence on the one-dimensional slices. With the new SSA framework, the solution can be formulated in a semi-analytic form, which not only gives an explicit representation, but also reflects the phenomenon in physics.

The new SSA approach resolves the mismatch between mathematics and physics, and also breaks the dimensional barriers. It also opens a door to a lot of further possibilities, ranging from the analysis of solutions and numerical simulations for the solutions.