On the dimension of the Fock type spaces☆
Introduction
Let ψ be a plurisubharmonic function on , . The weighted Fock space is the space of entire functions f such that where dv is the volume measure on . Note that is a closed subspace of and hence is a Hilbert space endowed with the inner product
In this paper we study when the space is of finite dimension depending on the weight ψ. This problem (at least for the case ) 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 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 is of infinite mass: then the space has infinite dimension. Let us also mention here that is a reproducing kernel Hilbert space of entire functions. If is a non-trivial doubling measure, then pointwise estimates on the reproducing kernel can be used to show that 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 to be of finite dimension, and calculate this dimension.
The situation is much more complicated in . Shigekawa established in [14] (see also [5, Theorem 11.20] in a book by Haslinger), the following interesting result.
Theorem A Let be a smooth function and let be the smallest eigenvalue of the Levi matrix Suppose that Then .
Note that the condition (1.2) is not necessary. A corresponding example is given in [5, Section 11.5] (). In this paper, we improve Theorem A by presenting a weaker condition for the dimension of the Fock space 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 .
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.
Section snippets
The case of
Given a subharmonic function denote by the corresponding Riesz measure, . Next, consider the class of the positive σ-finite atomic measures with masses which are integer multiples of 4π. Given a σ-finite measure μ, consider the corresponding atomic measure , In fact, for every atom of μ, has at the point x an atom of size 4π times the integer part of . Denote , .
Denote by the class of the positive σ-finite measures μ
The case of ,
Let denote the n-dimensional complex Euclidean space. Given , we set Denote . Then is the unit ball and is the unit sphere in . Let dσ be the normalized surface measure on .
Theorem 3.1 Let be a smooth function. Given , consider . Suppose that for every , the function is plurisubharmonic outside a compact subset of . Then .
Proof We use the fundamental result of Bedford–Taylor [1] on
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The δ-Neumann Problem and Schrödinger Operators, 2nd edition
2023, The δ-Neumann Problem and Schrödinger Operators, 2nd editionFredholm Toeplitz Operators on Doubling Fock Spaces
2022, Journal of Geometric Analysis
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The results of Section 2 were obtained in the framework of the project 20-61-46016 by the Russian Science Foundation. H. Youssfi was partially supported by the project ANR-18-CE40-0035.