What is quantum functional analysis

Alexander Ya. Helemskii

Tomoyuki Yamakami

A. I︠a︡. Khelemskiĭ

Ved Prakash Gupta, Prabha Mandayam, Viakalathur Shankar Sunder

Franz Luef, Yuri I. Manin

Abstract. Representations of the celebrated Heisenberg commutation relations in quantum mechanics (and their exponentiated versions) form the starting point for a number of basic constructions, both in mathematics and mathematical physics (geometric quantization, quantum tori, classical and quantum theta functions) and signal analysis (Gabor analysis). In this paper we will try to bridge the two com-munities, represented by the two co–authors: that of noncommutative geometry and that of signal analysis. After providing a brief comparative dictionary of the two languages, we will show e.g. that the Janssen representation of Gabor frames with generalized Gaussians as Gabor atoms yields in a natural way quantum theta functions, and that the Rieffel scalar product and associativity relations underlie both the functional equations for quantum thetas and the Fundamental Identity of Gabor analysis. §0. Introduction. Gabor analysis is a modern branch of signal analysis with various applications to pseudodifferential operators, harmonic analysis, function spaces, approximation

Tomoyuki Yamakami

This paper initiates a systematic study of quantum functions, which are (partial) functions defined in terms of quantum mechanical computations. Of all quantum functions, we focus on resource-bounded quantum functions whose inputs are classical bit strings. We prove complexity-theoretical properties and unique characteristics of these quantum functions by recent techniques developed for the analysis of quantum computations. We also discuss relativized quantum functions that make adaptive and nonadaptive oracle queries.