Quantum Tunneling and Its Absence in Deep Wells and Strong Magnetic Fields

ABSTRACT

In this seminar, we will discuss how in the absence of a magnetic field, a quantum particle always tunnels from one well into a neighboring well through a “classically forbidden” region at a rate which is the reciprocal of the “eigenvalue splitting”. Magnetic systems give rise to entirely new tunneling phenomena. We construct a family of double well potentials containing examples for which quantum tunneling is completely eliminated. On the other hand, magnetic tunneling does occur for typical double-well potentials, and we prove an upper bound on its tunneling rate. This is joint work with C.L. Fefferman and J. Shapiro.

BIOGRAPHY

Michael Weinstein is on the faculty at Columbia University as a Professor of Applied Mathematics in the Department of Applied Physics and Applied Mathematics, and a Professor of Mathematics in the Department of Mathematics. He received his Ph.D. from the Courant Institute at New York University in 1982 under the direction of George C. Papanicolaou, and was a postdoctoral fellow at Stanford University (1982–1984) with Joseph B. Keller. From 1984 to 1988, he was Assistant Professor of Mathematics at Princeton University. In 1988, he joined the mathematics faculty at the University of Michigan-Ann Arbor as an Associate Professor (1988–1992), and then Full Professor (1992–2000). Weinstein was a Member of Technical Staff at Bell Laboratories / Lucent Technologies in the Fundamental Mathematics Research Group from 1998–2004. In 2004, he joined the faculty of Columbia University.

Weinstein is a SIAM Fellow (Society of Industrial and Applied Mathematics) and a Fellow of the American Mathematical Society. He was awarded the Martin Kruskal Prize by SIAM (2018) for his contributions to fundamental and applied aspects of nonlinear waves and coherent structures. He has been a Simons Foundation Math + X Investigator since 2015. In 2025, Weinstein was elected to the American Academy of Arts and Sciences.