On the dimension of the Fock type spaces

Abstract

We study the weighted Fock spaces in one and several complex variables. We evaluate the dimension of these spaces in terms of the weight function extending and completing earlier results by Rozenblum–Shirokov and Shigekawa.

Introduction

Let ψ be a plurisubharmonic function on ${\mathbb{C}}^n$, $n\ge 1$. The weighted Fock space ${\mathcal{F}}_{\psi }^2$ is the space of entire functions f such that

$${\Vert f\Vert }_{\psi }^2=\underset{{\mathbb{C}}^n}{\int }|f(z){|}^2{e}^{-\psi (z)}dv(z)<\infty ,$$

where dv is the volume measure on ${\mathbb{C}}^n$. Note that ${\mathcal{F}}_{\psi }^2$ is a closed subspace of $L^2({\mathbb{C}}^n,e^{-\psi }dv)$ and hence is a Hilbert space endowed with the inner product

$${〈f,g〉}_{\psi }=\underset{{\mathbb{C}}^n}{\int }f(z)\overline{g(z)}{e}^{-\psi (z)}dv(z),f,g\in {\mathcal{F}}_{\psi }^2.$$

In this paper we study when the space ${\mathcal{F}}_{\psi }^2$ is of finite dimension depending on the weight ψ. This problem (at least for the case $n=1$) is motivated by some quantum mechanics questions, especially by the study of zero modes, eigenfunctions with zero eigenvalues.

In [12, Theorem 3.2], Rozenblum and Shirokov proposed a sufficient condition for the space ${\mathcal{F}}_{\psi }^2$ to be of infinite dimension, when ψ is a subharmonic function.

More precisely, they claimed that if ψ is a finite subharmonic function on the complex plane such that the measure $\mu =\mathrm{\Delta }\psi$ is of infinite mass:

$$\mu (\mathbb{C})=\underset{\mathbb{C}}{\int }d\mu (z)=\infty ,$$

then the space ${\mathcal{F}}_{\psi }^2$ has infinite dimension. Let us also mention here that ${\mathcal{F}}_{\psi }^2$ is a reproducing kernel Hilbert space of entire functions. If $\mu =\mathrm{\Delta }\psi$ is a non-trivial doubling measure, then pointwise estimates on the reproducing kernel can be used to show that ${\mathcal{F}}_{\psi }^2$ has infinite dimension (see [3], [10], [5, Theorem 11.45]).

We improve and extend somewhat the statement of Rozenblum–Shirokov in our paper, give a necessary and sufficient condition on ψ for the space ${\mathcal{F}}_{\psi }^2$ to be of finite dimension, and calculate this dimension.

The situation is much more complicated in ${\mathbb{C}}^n,n\ge 2$. Shigekawa established in [14] (see also [5, Theorem 11.20] in a book by Haslinger), the following interesting result.

Theorem A

Let $\psi :{\mathbb{C}}^n\to \mathbb{R}$ be a ${\mathcal{C}}^{\infty }$ smooth function and let ${\lambda }_0(z)$ be the smallest eigenvalue of the Levi matrix

$$L_{\psi }(z)=i\partial \overline{\partial }\psi (z)={\left(\frac{{\partial }^2\psi (z)}{\partial z_j\partial \overline{z_k}}\right)}_{j,k=1}^n.$$

Suppose that

$$\underset{|z|\to \infty }{\lim }|z{|}^2{\lambda }_0(z)=\infty .$$

Then $\dim {\mathcal{F}}_{\psi }^2=\infty$.

Note that the condition (1.2) is not necessary. A corresponding example is given in [5, Section 11.5] ($\psi (z,w)=|z{|}^2|w{|}^2+|w{|}^4$). In this paper, we improve Theorem A by presenting a weaker condition for the dimension of the Fock space ${\mathcal{F}}_{\psi }^2$ to be infinite. Furthermore, we give several examples that show how far is our condition from being necessary. Finally, we consider several examples (classes of examples) of weight functions ψ of special form and evaluate the dimension of ${\mathcal{F}}_{\psi }^2$.

We should also mention here the related Wiegerdinck problem (on the dimension of the Bergman spaces on pseudoconvex domains), see [8], [11].

The rest of the paper is organized as follows. The case of dimension one is considered in Section 2, and the case of higher dimension is considered in Section 3.

We thank Friedrich Haslinger, Grigori Rozenblum, Włodzimierz Zwonek, and the anonymous referee for helpful remarks.