The $\partial $-complex on the Segal–Bargmann space

Friedrich Haslinger Annales Polonici Mathematici MSC: Primary 30H20, 32A36, 32W50; Secondary 47B38. DOI: 10.4064/ap180715-2-11 Published online: 11 April 2019

Abstract

We study certain densely defined unbounded operators on the Segal–Bargmann space. These are the annihilation and creation operators of quantum mechanics. In several complex variables we have the $\partial $-operator and its adjoint $\partial ^*$ acting on $(p,0)$-forms with coefficients in the Segal–Bargmann space. We consider the corresponding $\partial $-complex and study the spectral properties of the corresponding complex Laplacian $\tilde\Box = \partial \partial ^* + \partial ^*\partial $. Finally, we study a more general complex Laplacian $\tilde\Box _D = D D^* + D^* D$, where $D$ is a differential operator of polynomial type, to find the canonical solutions to the inhomogeneous equations $Du=\alpha $ and $D^*v=\beta $.

Authors

  • Friedrich HaslingerFakultät für Mathematik
    Universität Wien
    Oskar-Morgenstern-Platz 1
    A-1090 Wien, Austria
    e-mail

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