On quantum Sobolev spaces consisting of Hilbert-Schmidt operators

TL;DR

This work defines quantum Sobolev spaces consisting of Hilbert-Schmidt operators via a quantum Fourier transform on a locally compact abelian group. By introducing $rak{H}^s_g(G,H)$ with norm $igl( int_{\hat G}(1+g(\xi)^2)^s|\mathcal{F}_U(T)(\xi)|^2 d \xiigr)^{1/2}$, it shows a tight isomorphism to $L^2(\hat G)$ and establishes monotone and functional-analytic embeddings into $ ext{B}(H)$ and Schatten classes under suitable integrability conditions. The paper then applies this framework to operator-valued PDEs, giving explicit solutions for the quantum Poisson-type equation $(oldsymbol{I}-oldsymbol{ riangle})T=S$ and the quantum generalized bosonic string equation $(oldsymbol{ riangle} e^{coldsymbol{ riangle}}-oldsymbol{I})T=S$ via multipliers in quantum Fourier space, with exact norm identities tied to Plancherel. These results provide a rigorous operator-valued generalization of classical Sobolev methods and offer a theoretical basis for applications in quantum physics and quantum harmonic analysis. The approach combines Schatten-class analysis, quantum Fourier analysis, and operator-valued PDE techniques to extend Sobolev-type regularity and solvability to the noncommutative, operator-valued setting.

Abstract

In the present paper, we study quantum Sobolev spaces whose elements are operators of the Hilbert-Schmidt class. We construct these Sobolev spaces from the Fourier transform for operators. Next, we obtain continuous embedding theorems. Finally, we delve into solving partial differential equations where the unknown is an operator. The results have potential applications in quantum physics, providing a new theoretical basis for relevant research.

Theorems & Definitions (22)

• Definition 2.1
• Definition 2.2
• Theorem 2.3
• Theorem 2.4
• Theorem 2.5
• Theorem 2.6
• Definition 3.1
• Remark 3.2
• Theorem 3.3
• proof