Undergraduate Research Program: Spring 2026

Research projects

1. Dirac delta-function potentials in the Schroedinger operator

Faculty Mentor: Dr. Sergei Shabanov

Description: Dirac delta-functions potentials are used to model quantum systems with a short-range and high strength potential. However, either scattering or eigenvalue problems for the Schroedinger operator with such a potential in two or higher dimensions requires a regularization of the delta-function (e.g., a Gaussian regularization).  However all observables, such as the discret spectrum of the operator and the scattering matrix, are singular in the limit when the regularization is removed. To obtain finite results for observables a so-called renormalization of the potential strength is applied, a procedure that is similar to renormalization in quantum field theory. Renormalization depends on a regularization of the delta-function. Only for a very limited class of regularizations (in fact, just for a few) it was proved in two spatial dimensions that the observables are regularization-independent. It has recently been proposed that the delta-function potentials can be used in a non-linear Schroedinger equation to model Bose-Einstein condensations, which is a fundamental phenomenon in modern physics. The key mathematical question of regularization-independence of these non-linear and previous linear models remains unaddressed and becomes even more significant. The ultimate project goal is to investigate this problem in the simplest case of the 2D linear Schroedinger operator and either to prove the assertion for any regularization or construct a counter example. Specifically, no regularizations that break the rotational symmetry have ever been studied in this regard, and a technical goal of the project will be to investigate them.

Prerequisites: Linear Algebra (4000-level); PDE (4000-level is sufficient); MAP 6505 (necessary), MAP 6506 (helpful).

Approximate hours per week  students are expected to devote to the project: 4–6 hours.

2. Probabilistic properties of heavy-tailed distributions

Faculty Mentor: Dr.  Arnaud Marsiglietti

Description: Heavy-tailed distributions in probability theory and statistics form an important class of distributions that appear naturally in many areas of applied science. The goal of this project is to study generic probabilistic properties of heavy-tailed distributions, such as deviation inequalities, moment bounds, and entropic estimates. Students will become familiar with the language of probability and will gain experience in problem solving.

Prerequisites: Calculus/intro analysis and probability/statistics courses

Approximate hours per week  students are expected to devote to the project:  4 hours