arXiv subjects

Brady Zimmerman

Publications and source records attributed to Brady Zimmerman.

2 recordsLinked to original sources

arXiv2025

An analysis of non-selfadjoint first-order differential operators with non-local point interactions

We study the spectra of non-selfadjoint first-order operators on the interval with non-local point interactions, formally given by ${i\partial_x+V+k\langle δ,\cdot\rangle}$. We give precise estimates on the location of the eigenvalues on the complex plane and prove that the root vectors of these operators form Riesz bases of $L^2(0,2π)$. Under the additional assumption that the operator is maximally dissipative, we prove that it can have at most one real eigenvalue, and given any $λ\in\mathbb{R}$, we explicitly construct the unique operator realization such that $λ$ is in its spectrum. We also investigate the time-evolution generated by these maximally dissipative operators.

math.SP↗

arXiv2024

A Model Problem for First Order Mean Field Games with Discrete Initial Data

In this article, we study a simplified version of a density-dependent first-order mean field game, in which the players face a penalization equal to the population density at their final position. We consider the problem of finding an equilibrium when the initial distribution is a discrete measure. We show that the problem becomes finite-dimensional: the final piecewise smooth density is completely determined by the weights and positions of the initial measure. We establish existence and uniqueness of a solution using classical fixed point theorems. Finally, we show that Newton's method provides an effective way to compute the solution. Our numerical simulations provide an illustration of how density penalization in a mean field game tends to the smoothen the initial distribution.

math.OC↗