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Viktor Maslov

Publications and source records attributed to Viktor Maslov.

7 recordsLinked to original sources
arXiv2016

The Complex Lagrangian Germ and the Canonical Operator

We give a manifestly invariant definition of the Lagrangian complex germ with the minimal degree of accuracy required to define the canonical operator. The equivalence with the traditional definition is proved, and the canonical operator is constructed in new terms. A new form of the quantization condition is given, in which the volume form is assumed to be defined on the universal covering of the Lagrangian manifold rather than on the manifold itself. This allows one to solve a wider class of eigenvalue problems.

math-ph↗
arXiv2002

Idempotent (Asymptotic) Mathematics and the Representation Theory

A brief survey of some basic ideas of the so-called Idempotent Mathematics is presented; an "idempotent" version of the representation theory is discussed. The Idempotent Mathematics can be treated as a result of a dequantization of the traditional mathematics over numerical fields in the limit of the vanishing "imaginary Planck constant"; there is a correspondence, in the spirit of N. Bohr's correspondence principle, between constructions and results in traditional mathematics over the fields of real and complex numbers and similar constructions and results over idempotent semirings. In particular, there is an "idempotent" version of the theory of linear representations of groups. Some basic concepts and results of the "idempotent" representation theory are presented. In the framework of this theory the well-known Legendre transform can be treated as an idempotent version of the traditional Fourier transform. Some unexpected versions of the Engel theorem are given.

math.RT↗
arXiv2002

Nondigital implementation of the arithmetic of real numbers by means of quantum computer media

In the framework of a model for quantum computer media, a nondigital implementation of the arithmetic of the real numbers is described. For this model, an elementary storage "cell" is an ensemble of qubits (quantum bits). It is found that to store an arbitrary real number it is sufficient to use four of these ensembles and the arithmetic operations can be implemented by fixed quantum circuits.

quant-ph↗
arXiv2001

Idempotent functional analysis: an algebraic approach

In this paper we consider Idempotent Functional Analysis, an `abstract' version of Idempotent Analysis developed by V. P. Maslov and his collaborators. We give a review of the basic ideas of Idempotent Analysis. The correspondence between concepts and theorems of the traditional Functional Analysis and its idempotent version is discussed; this correspondence is similar to N. Bohr's correspondence principle in quantum theory. We present an algebraical approach to Idempotent Functional Analysis. Basic notions and results are formulated in algebraical terms; the essential point is that the operation of idempotent addition can be defined for arbitrary infinite sets of summands. We study idempotent analogs of the main theorems of linear functional analysis and results concerning the general form of a linear functional and scalar products in idempotent spaces.

math.FA↗