Postdoctoral Research Associate

Posting Number: S06676P

Job Description:

Looking for a post-doctoral Research Associate to perform full-time mathematical research with the goal of developing a practical yet rigorous mathematical framework for quantum mechanics. The researcher is expected to make contributions in the fields of functional analysis, complex analysis, spectral theory, and partial-differential equations, but also applications in number theory are of interest. Typically, the theory of quantum mechanics relies on the Hilbert space of square-integrable functions (L2) but when the domain space is not compact (e.g. L2 ), or the potential is singular, the eigenfunctions are not contained in the Hilbert space (e.g. e^(ikx)).

In spectral theory, the absence of eigenvectors in the Hilbert space is typically circumvented by using a projection-valued measure. In the field of topological vector spaces, the problem is addressed using the nuclear spectral theorem which is set in the rigged Hilbert space (Gelfand's triple) but the requirements imposed on the Schrodinger operator are non-trivial. Finally, the theory of non self-adjoint operators is very poorly developed.

Current practice in physics is typically to ignore mathematical rigor because an easily accessible mathematical theory is lacking.

Our goal is to develop a mathematically rigorous framework applicable to the Schrodinger equation where eigenvectors are tempered distributions (topological dual of the Schwartz space). The developed framework should be mathematically rigorous yet accessible to the physics community and applicable to existing problems in physics. As applications, open problems in the field of number theory, such as the Dirichlet divisor problem, the Riemann hypothesis (e.g., through the Hilbert-Polya conjecture) are also of interest.

The position would have weekly meetings with prof. William Vandenberghe to discuss research progress and novel research directions.

The research associate can also:

• participate in research in physics, materials science, or electrical engineering relating to electron transport at the nanoscale.
• Supervise Ph.D. students in Materials Science, Physics, and Electrical Engineering
• Participate in teaching mathematics to undergraduate engineering students