[Paper Review] Bounded Weyl pseudodifferential operators in Fock space

Laurent Amour, Lisette Jager+1 more|arXiv (Cornell University)|Sep 13, 2012

Mathematical Analysis and Transform Methods22 references6 citations

TL;DR

This paper establishes an infinite-dimensional analog of the Calderón-Vaillancourt Theorem for Weyl pseudodifferential operators in Fock space, using a hybrid calculus that combines finite-dimensional Weyl quantization with infinite-dimensional anti-Wick quantization via the Segal-Bargmann transform. The key result is a uniform $ L^2 $-operator norm bound for symbols with controlled derivatives, ensuring convergence of finite-dimensional approximations to a well-defined infinite-dimensional Weyl operator.

ABSTRACT

We aim at constructing an analog of the Weyl calculus in an infinite dimensional setting, in which the usual configuration and phase spaces are ultimately replaced by infinite dimensional measure spaces, the so-called abstract Wiener spaces. The Hilbert space on which the operators act can be seen as a Fock space or, equivalently, as a space of square integrable functions on the configuration space. The construction is not straightforward and needs to split the configuration space into two factors, of which the first one is finite dimensional. Then one defines, for a convenient symbol $F$, a hybrid calculus, acting on the finite dimensional factor as a Weyl operator and on the other one as an anti-Wick operator, defined thanks to an infinite dimensional Segal-Bargmann transformation. One can establish bounds on the hybrid operators. These bounds enable us to prove the convergence of any sequence of hybrid operators associated with an increasing sequence of finite dimensional factors. Their common limit is the Weyl operator $OP_h^{weyl}(F)$, the analog of Calderón-Vaillancourt Theorem being a consequence of the upper mentionned bounds as well.

Motivation & Objective

• To construct a rigorous analog of the Weyl calculus in an infinite-dimensional setting, replacing finite-dimensional phase space with abstract Wiener spaces.
• To define a bounded Weyl pseudodifferential operator on Fock space via a hybrid quantization scheme combining finite-dimensional Weyl and infinite-dimensional anti-Wick operators.
• To establish uniform $ L^2 $-operator norm bounds for such operators under derivative conditions on the symbol, enabling convergence of finite-dimensional approximations.
• To prove that the limit of these approximations coincides with the standard Weyl quantization when the symbol satisfies appropriate decay and smoothness conditions.

Proposed method

• The configuration space is split into a finite-dimensional factor $ \Lambda_n $ and its infinite-dimensional complement $ \Lambda_n^c $, allowing separate treatment via Weyl and anti-Wick quantization.
• A hybrid operator $ \mathrm{Op}_h^{\mathrm{hyb},\Lambda_n}(F) $ is defined as a composition: Weyl quantization on $ \Lambda_n $, anti-Wick quantization on $ \Lambda_n^c $, using the Segal-Bargmann transform.
• The norm of the hybrid operator is bounded using estimates involving $ \varepsilon_j $, with $ \|\mathrm{Op}_h^{\mathrm{hyb},\Lambda_n}(F)\| \leq M \prod_{j=1}^n (1 + 225\pi K_2 \sqrt{h} \varepsilon_j) $ for $ h \leq 1 $, under $ I_2 $-derivative conditions.
• For improved bounds, higher-order derivative conditions up to $ I_4 $ are used, yielding $ \|\mathrm{Op}_h^{\mathrm{hyb},\Lambda_n}(F)\| \leq M \prod_{j=1}^n (1 + 225\pi K_4 h \varepsilon_j^2) $, with $ K_4 = \sup_j \max(1, \varepsilon_j^6) $.
• The convergence of the hybrid operators along an increasing sequence $ \Lambda_n \nearrow \Gamma $ is proven using the Lebesgue Dominated Convergence Theorem on the difference $ \|\mathrm{Op}_h^{\mathrm{hyb},\Lambda_n}(F) - \mathrm{Op}_h^{\mathrm{old\text{-}weyl}}(F)\| \to 0 $.
• The final Weyl operator $ \mathrm{Op}_h^{\mathrm{weyl}}(F) $ is defined as the limit of the hybrid operators, and shown to coincide with the standard Weyl quantization when the symbol satisfies $ H_2(M,\varepsilon) $ and $ (\varepsilon_j)_{j \in \Gamma} \in \ell^1 $.

Experimental results

Research questions

• RQ1 Can a consistent Weyl pseudodifferential calculus be defined in an infinite-dimensional Fock space setting, replacing finite-dimensional phase space?
• RQ2 How can one ensure the boundedness of such operators when the configuration space is infinite-dimensional?
• RQ3 What conditions on the symbol $ F $ guarantee uniform operator norm bounds in the infinite-dimensional limit?
• RQ4 Does the hybrid quantization scheme converge to the standard Weyl quantization in the limit of increasing finite-dimensional approximations?
• RQ5 Under what conditions does the new Weyl operator coincide with the classical one defined via the standard integral formula?

Key findings

• The paper establishes a uniform $ L^2 $-operator norm bound for the hybrid Weyl-anti-Wick operator: $ \|\mathrm{Op}_h^{\mathrm{hyb},\Lambda_n}(F)\| \leq M \prod_{j=1}^n (1 + 225\pi K_2 \sqrt{h} \varepsilon_j) $, where $ K_2 = \sup_j \max(1, \varepsilon_j^3) $, under $ I_2 $-derivative conditions on $ F $.
• Improved bounds are obtained under $ I_4 $-derivative conditions: $ \|\mathrm{Op}_h^{\mathrm{hyb},\Lambda_n}(F)\| \leq M \prod_{j=1}^n (1 + 225\pi K_4 h \varepsilon_j^2) $, with $ K_4 = \sup_j \max(1, \varepsilon_j^6) $.
• The limit of the hybrid operators $ \mathrm{Op}_h^{\mathrm{hyb},\Lambda_n}(F) $ as $ \Lambda_n \nearrow \Gamma $ exists and defines a bounded operator $ \mathrm{Op}_h^{\mathrm{weyl}}(F) $ in the Fock space $ \mathcal{H}(\Gamma) $.
• The constructed operator $ \mathrm{Op}_h^{\mathrm{weyl}}(F) $ coincides with the standard Weyl quantization $ \mathrm{Op}_h^{\mathrm{old\text{-}weyl}}(F) $ when the symbol $ F $ satisfies the $ H_2(M,\varepsilon) $ condition and $ (\varepsilon_j)_{j \in \Gamma} \in \ell^1 $.
• The convergence $ \|\mathrm{Op}_h^{\mathrm{hyb},\Lambda_n}(F) - \mathrm{Op}_h^{\mathrm{old\text{-}weyl}}(F)\| \to 0 $ as $ n \to \infty $ is proven via the Lebesgue Dominated Convergence Theorem on the difference in the integral representation.
• The result provides a rigorous infinite-dimensional analog of the Calderón-Vaillancourt Theorem, extending the classical boundedness result to Fock spaces over abstract Wiener spaces.