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Friedrich Haslinger: Basic estimates and the uncertainty principle

Date of publication: 16. 5. 2024
Complex analysis seminar
Predavalnica 3.06
ID: 930 9885 4181

V torek, 21. maja ob 12. uri in 30 minut, bo v okviru seminarja za kompleksno analizo predaval prof. Friedrich Haslinger z Univerze na Dunaju, Avstrija.

Title: Basic estimates and the uncertainty principle.

Abstract: In the Segal-Bargmann space (also Fock space) A^2(C^n, e^{−|z|^2}) of entire functions the differentiation operators a_ j(f)=\partial f / \partial z_ j (annihilation) and the multiplication operators a_ j * (f) = z_ j f (creation) are unbounded, densely defined adjoint operators with ||f||^2 ≤ ||a_ j (f)||^2 + ||a_ j * (f)||^2 for f \in dom(a_ j), which corresponds to the uncertainty principle. We study certain densely defined unbounded operators on the Segal-Bargmann space, related to the annihilation and creation operators of quantum mechanics. We consider the corresponding D-complex and study properties of the complex Laplacian DD * + D * D, where D is a differential operator of polynomial type, in particular we discuss the corresponding basic estimates, where we express a commutator term as a sum of squared norms. The basic estimates can be seen as a generalization of the uncertainty principle represented in the Segal-Bargmann space. In addition we show that Kahler manifolds (M, h) admitting a real holomorphic vector field, i.e. for which there exists a smooth real-valued function \psi:M \to R such that h^{j\bar k} \partial\psi / \partial \bar z^k \partial / \partial z^j is a holomorphic vector field, have the property that the weighted Bergman space A^2(M, h, e^{−\psi}) exhibits the same duality between differentiation and multiplication as in the Segal-Bargmann space.

Predavanje bo potekalo hibridno, v predavalnici 3.06 na Jadranski 21 in preko aplikacije ZOOM:

Meeting ID: 930 9885 4181

Vljudno vabljeni!

Vodji seminarja

Franc Forstneric in Barbara Drinovec Drnovsek