Title: A Critical Endpoint in Bargmann-Fock Quantization
Abstract:
Toeplitz quantization on Bargmann--Fock space turns a function into an operator by multiplying by that function and then projecting back to holomorphic states. A basic question is whether boundedness of this quantum operator can be detected by an ordinary bounded function on phase space.
There is a distinguished heat-smoothing scale in this problem. Under the Bargmann transform, the Toeplitz operator becomes a Weyl operator whose symbol is obtained by heat-evolving the original symbol to the critical time t=1/4. Berger and Coburn conjectured in the early 1990s that this gives an exact boundedness test: the Toeplitz operator should be bounded if and only if this critical heat transform is bounded.
I will explain why this natural conjecture fails in both directions. In the process, we also resolve a conjecture of Bauer, Coburn and Isralowitz from 2010. In every dimension, there are symbols for which the Toeplitz operator is bounded, even Hilbert--Schmidt, while the critical heat transform is unbounded; conversely, there are symbols for which the critical heat transform is bounded while the Toeplitz operator is unbounded. The failure is sharp: boundedness of the Toeplitz operator still forces boundedness of the heat transform at every later time t>1/4, but not at the endpoint.
Jared Wunsch
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