Joint BCAM-UPV/EHU Analysis and PDE seminar: Implementing Bogoliubov transformations beyond the Shale-Stinespring condition

Date: Thu, Jan 20 2022

Hour: 17:00

Location: BCAM Seminar Room and Online

Speakers: Sascha Lill

LOCATION: BCAM Seminar Room and Online

Quantum many body systems can be mathematically described by vectors in a certain Hilbert space, the so-called Fock space, whose Schr ̈odinger dynamics are generated by a self-adjoint Hamiltonian operator H. Bogoliubov transformations are a convenient way to manipulate H while keeping the physical predictions invariant. They have found widespread use for analyzing the dynamics of quantum many-body systems and justifying simplified models that have been heuristically derived by physicists.

In the 1960s, Shale and Stinespring derived a necessary and sufficient condition for when a Bogoliubov transformation is implementable on Fock space, i.e., for when there exists a unitary operator U such that the manipulated Hamiltonian takes the form U∗HU. However, nonimplementable Bogoliubov transformations appear frequently in the literature for systems of infinite size. In this talk, we therefore construct two extensions of the Fock space on which certain Bogoliubov transformations become implementable, although they violate the Shale-Stinespring condition.

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BCAM and Universität Tubingen

Confirmed speakers:

Sascha Lill