# Segal-Bargmann transforms from hyperbolic Hamiltonians

Kirwin WD, Mourão J, Nunes JP, Thiemann T (2021)

**Publication Type:** Journal article

**Publication year:** 2021

### Journal

**Book Volume:** 500

**Article Number:** 125146

**Journal Issue:** 2

**DOI:** 10.1016/j.jmaa.2021.125146

### Abstract

We consider the imaginary time flow of a quadratic hyperbolic Hamiltonian on the symplectic plane, apply it to the Schrödinger polarization and study the corresponding evolution of polarized sections. The flow is periodic in imaginary time and the evolution of polarized sections has interesting features. On the time intervals for which the polarization is real or Kähler, the half–form corrected time evolution of polarized sections is given by unitary operators which turn out to be equivalent to the classical Segal-Bargmann transforms (which are usually associated to the quadratic elliptic Hamiltonian [Formula presented] and to the heat operator). At the right endpoint of these intervals, the evolution of polarized sections is given by the Fourier transform from the Schrödinger to the momentum representation. In the complementary intervals of imaginary time, the polarizations are anti–Kähler and the Hilbert space of polarized sections collapses to H={0}. Hyperbolic quadratic Hamiltonians thus give rise to a new factorization of the Segal-Bargmann transform, which is very different from the usual one, where one first applies a bounded contraction operator (the heat kernel operator), mapping L^{2}–states to real analytic functions with unique analytic continuation, and then one applies analytic continuation. In the factorization induced by an hyperbolic complexifier, both factors are unbounded operators but their composition is, in the Kähler or real sectors, unitary. In another paper [24], we explore the application of the above family of unitary transforms to the definition of new holomorphic fractional Fourier transforms.

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### How to cite

**APA:**

Kirwin, W.D., Mourão, J., Nunes, J.P., & Thiemann, T. (2021). Segal-Bargmann transforms from hyperbolic Hamiltonians. *Journal of Mathematical Analysis and Applications*, *500*(2). https://doi.org/10.1016/j.jmaa.2021.125146

**MLA:**

Kirwin, William D., et al. "Segal-Bargmann transforms from hyperbolic Hamiltonians." *Journal of Mathematical Analysis and Applications* 500.2 (2021).

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