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Matjaž Payrits: Embeddings of Hilbert spaces in superspaces induced by smooth overcomplete bases and their applications to diagonalizing few-mode many-body Hamiltonians
December 5, 2017 @ 2:00 pm - 4:00 pm
The properties of a Hilbert space may sometimes be usefully illuminated by expressing its states with respect to an overcomplete basis parameterized by the points of a smooth manifold. A prime example of the technique is the Segal-Bargmann representation wherein states of a single-mode bosonic Fock space are expanded in terms of the overcomplete basis of coherent states. The Fock space is then found to be isomorphic to the space of holomorphic functions of a certain finite norm. Furthermore, the creation and annihilation operators, and any function of them, can be expressed as functions of the complex position and complex derivative operators.
In this talk I will present the general theory of embedding a Hilbert space in a suitable space of functions over a smooth manifold that parameterizes an overcomplete basis in the original space. In many cases, operators of theoretical interest in the original space may be mapped onto differential operators on the smooth manifold such that the spectrum of the mapped operator contains the spectrum of the original one. This, in particular, allows the use of calculus and geometrical reasoning when diagonalizing Hamiltonians in the original space.
I will demonstrate the technique on a general d-mode many-body system that may be mapped onto a single-particle problem on the (d-1)-sphere. I will finally review some low-d applications of the formalism, which have found utility in the context of few-site tight-binding Hamiltonians and Bose-Einstein condensates of spinful atoms.
* Matjaž Payrits, Imperial College London