for this workshop

## Non-Hermitian quantum mechanics and symplectic geometry

at the

American Institute of Mathematics, San Jose, California

organized by

Eva-Maria Graefe, Michael Hitrik, Roman Schubert, and Alejandro Uribe

This workshop, sponsored by AIM and the NSF, will be devoted to the exploration of new connections between complex symplectic geometry, quantum mechanics with non-hermitian hamiltonians, and spectral theory of non-selfadjoint operators. Symplectic geometry, as the geometry underlying classical hamiltonian mechanics, has been proven to be extremely useful for understanding problems in quantum mechanics and the spectral theory of self-adjoint operators, in the semiclassical limit. This has lead to the areas of microlocal and semiclassical analysis which underpin a large part of the modern theory of PDEs. While quantum mechanics traditionally uses hermitian hamiltonians to describe closed quantum systems, recently there has been a surge of interest in non-hermitian hamiltonians to describe open quantum systems with losses. This leads to complex hamiltonian flows and hence complex symplectic geometry. Recently new connections between these fields have emerged and the aim of the workshop will be to develop a deeper understanding of these connections and to explore if they can be used to attack some of the open problems in the theory of open quantum systems and non self-adjoint operators.

The main topics of the workshop are:

- Complex and symplectic geometry related to the complexification of the group of hamiltonian transformations.
- Semi-classical limit of dynamics generated by non-hermitian hamiltonians, including propagation of generalized coherent states.
- Spectral theory of non-selfadjoint operators, including Bohr-Sommerfeld conditions.
- Non-hermitian quantum mechanics.

The workshop will differ from typical conferences in some regards. Participants will be invited to suggest open problems and questions before the workshop begins, and these will be posted on the workshop website. These include specific problems on which there is hope of making some progress during the workshop, as well as more ambitious problems which may influence the future activity of the field. Lectures at the workshop will be focused on familiarizing the participants with the background material leading up to specific problems, and the schedule will include discussion and parallel working sessions.

The deadline to apply for support to participate in this workshop has passed.

For more information email *workshops@aimath.org*