Abstract. This time I will discuss the observation, going back to the '80s in the one-dimensional case, that a weighted version of the Kohn Laplacian of complex analysis may be viewed as a generalized Schrödinger operator. Then I will present a few original results in several complex variables that crucially exploit this observation, namely:
1. pointwise exponential bounds for the weighted Bergman kernel, whenever the Kohn Laplacian satisfies a coercivity condition,
2. coercivity of the Kohn Laplacian for a class of model weights introduced by A. Nagel,
3. in the case of 2-dimensional model weights, the study of the discreteness of the spectrum of weighted Kohn Laplacians.
References
[1] Dall’Ara, G., Pointwise estimates of weighted Bergman kernels in several complex variables, Advances in Mathematics 285 (2015) 1706-1740.
[2] Dall’Ara, G., Coercivity of Weighted Kohn Laplacians: the case of model monomial weights in $\mathbb C^2$, to appear in Transactions of the American Mathematical Society.
Seminar room 9, 2nd floor, Oskar-Morgenstern-Platz 1
1090 Vienna, Austria