arXiv subjects

Yurii V. Brezhnev

Publications and source records attributed to Yurii V. Brezhnev.

11 recordsLinked to original sources

arXiv2022

Linear superposition as a core theorem of quantum empiricism

Clarifying the nature of the quantum state $|Ψ\rangle$ is at the root of the problems with insight into counter-intuitive quantum postulates. We provide a direct-and math-axiom free-empirical derivation of this object as an element of a vector space. Establishing the linearity of this structure--quantum superposition--is based on a set-theoretic creation of ensemble formations and invokes the following three principia: ($\textsf{I}$) quantum statics, ($\textsf{II}$) doctrine of the number in the physical theory, and ($\textsf{III}$) mathematization of matching the two observations with each other (quantum covariance). All of the constructs rest upon a formalization of the minimal experimental entity--the registered micro-event, detector click. This is sufficient for producing the $\mathbb C$-numbers, axioms of linear vector space (superposition principle), statistical mixtures of states, eigenstates and their spectra, and non-commutativity of observables. No use is required of the spatio-temporal concepts. As a result, the foundations of theory are liberated to a significant extent from the issues associated with physical interpretations, philosophical exegeses, and mathematical reconstruction of the entire quantum edifice.

quant-ph↗

arXiv2019

The Born rule

We deduce the Born rule. No use is required of quantum postulates. One exploits only rudimentary quantum mathematics--a linear, not Hilbert', vector space--and empirical notion of the statistical length of a state. Its statistical nature comes from the experimental detector-clicks being formalized into the abstract quantum micro-events. We also comment on that it is not only that the use has not been made of some quantum axioms when deriving the rule but, in a sense, their invoking would be inconsistent.

quant-ph↗

arXiv2015

The sixth Painleve transcendent and uniformization of algebraic curves

We exhibit a remarkable connection between sixth equation of Painleve list and infinite families of explicitly uniformizable algebraic curves. Fuchsian equations, congruences for group transformations, differential calculus of functions and differentials on corresponding Riemann surfaces, Abelian integrals, analytic connections (generalizations of Chazy's equations), and other attributes of uniformization can be obtained for these curves. As byproducts of the theory, we establish relations between Picard-Hitchin's curves, hyperelliptic curves, punctured tori, Heun's equations, and the famous differential equation which Apery used to prove the irrationality of Riemann's zeta(3).

math.CA↗

arXiv2015

Analytic connections on Riemann surfaces and orbifolds

We give a differentially closed description of the uniformizing representation to the analytical apparatus on Riemann surfaces and orbifolds of finite analytic type. Apart from well-known automorphic functions and Abelian differentials it involves construction of the connection objects. Like functions and differentials, the connection, being also the fundamental object, is described by algorithmically derivable ODEs. Automorphic properties of all of the objects are associated to different discrete groups, among which are excessive ones. We show, in an example of the hyperelliptic curves, how can the connection be explicitly constructed. We study also a relation between classical/traditional `linearly differential' viewpoint (principal Fuchsian equation) and uniformizing $τ$-representation of the theory. The latter is shown to be supplemented with the second (to the principal) Fuchsian equation.

math.CA↗

arXiv2015

On a Quantization of the Classical $θ$-Functions

The Jacobi theta-functions admit a definition through the autonomous differential equations (dynamical system); not only through the famous Fourier theta-series. We study this system in the framework of Hamiltonian dynamics and find corresponding Poisson brackets. Availability of these ingredients allows us to state the problem of a canonical quantization to these equations and disclose some important problems. In a particular case the problem is completely solvable in the sense that spectrum of the Hamiltonian can be found. The spectrum is continuous, has a band structure with infinite number of lacunae, and is determined by the Mathieu equation: the Schrödinger equation with a periodic cos-type potential.

math-ph↗

arXiv2013

Non-canonical extension of theta-functions and modular integrability of theta-constants

This is an extended (factor 2.5) version of arXiv:math/0601371 and arXiv:0808.3486. We present new results in the theory of the classical $θ$-functions of Jacobi: series expansions and defining ordinary differential equations (\odes). The proposed dynamical systems turn out to be Hamiltonian and define fundamental differential properties of theta-functions; they also yield an exponential quadratic extension of the canonical $θ$-series. An integrability condition of these \odes\ explains appearance of the modular $\vartheta$-constants and differential properties thereof. General solutions to all the \odes\ are given. For completeness, we also solve the Weierstrassian elliptic modular inversion problem and consider its consequences. As a nontrivial application, we apply proposed techni\-que to the Hitchin case of the sixth Painlevé equation.

math.CA↗

arXiv2012

Spectral/quadrature duality: Picard-Vessiot theory and finite-gap potentials

In the framework of differential Galois theory we treat the classical spectral problem $Ψ"-u(x)Ψ=λΨ$ and its finite-gap potentials as exactly solvable in quadratures by Picard--Vessiot without involving special functions; the ideology goes back to the 1919 works by J. Drach. We show that duality between spectral and quadrature approaches is realized through the Weierstrass permutation theorem for a logarithmic Abelian integral. From this standpoint we inspect known facts and obtain new ones: an important formula for the $Ψ$-function and $Θ$-function extensions of Picard--Vessiot fields. In particular, extensions by Jacobi's $θ$-functions lead to the (quadrature) algebraically integrable equations for the $θ$-functions themselves.

math.CA↗

arXiv2010

A tau-function solution to the sixth Painleve transcendent

We represent and analyze the general solution of the sixth Painleve transcendent in the Picard-Hitchin-Okamoto class in the Painleve form as the logarithmic derivative of the ratio of certain $τ$-functions. These functions are expressible explicitly in terms of the elliptic Legendre integrals and Jacobi $θ$-functions, for which we write the general differentiation rules. We also establish a relation between the P6-equation and the uniformization of algebraic curves and present examples.

math.CA↗

arXiv2008

On the uniformization of algebraic curves

Based on Burnside's parametrization of the algebraic curve $y^2=x^5-x$ we provide remaining attributes of its uniformization: Fuchsian equations and their solutions, accessory parameters, monodromies, conformal maps, fundamental polygons, etc. As a generalization, we construct the zero genus uniformization of arbitrary curves. For hyperelliptic curves all the objects of the theory are explicitly described. We consider a large number of examples and, briefly, applications: Abelian integrals, metrics of Poincare, differential equations of the Jacobi--Chazy and Picard--Fuchs type, and others.

math.CA↗

arXiv2002

Elliptic Solitons and Groebner Bases

We consider the solution of spectral problems with elliptic coefficients in the framework of the Hermite ansatz. We show that the search for exactly solvable potentials and their spectral characteristics is reduced to a system of polynomial equations solvable by the Gröbner bases method and others. New integrable potentials and corresponding solutions of the Sawada-Kotera, Kaup-Kupershmidt, Boussinesq equations and others are found.

nlin.SI↗

arXiv2001

On the Dubrovin Equations for the Finite-gap Potentials

The general technique of derivation of Dubrovin's equation for the arbitrary operator pencils is suggested. The question of unique recovering of the finite-gap potential by coordinates of zeroes of the Psi-function is discussed. The crucial result of the paper is an autonomous form of Dubrovin's equations and new trace-formulas for nontrivial spectral problems of the 3-rd order with trigonal algebraic curve. We show only demonstrative examples, the method is spread into arbitrary spectral problem including matrix ones.

nlin.SI↗