30 research outputs found
On a Quantization of the Classical -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 Schr\"odinger equation
with a periodic cos-type potential
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 -representation of the theory. The latter is
shown to be supplemented with the second (to the principal) Fuchsian equation.Comment: Final version. LaTeX, 16 pages. No figure
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
-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
-series. An integrability condition of these \odes\ explains appearance
of the modular -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 Painlev\'e equation.Comment: Final version; 47 pages, 1 figure, LaTe
Linear superposition as a core theorem of quantum empiricism
Clarifying the nature of the quantum state is at the root of
the problems with insight into (counterintuitive) 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: quantum statics,
doctrine of a number in the physical theory, and
mathematization of matching the two observations with each
other; quantum invariance.
All of the constructs rest upon a formalization of the minimal experimental
entity: observed micro-event, detector click. This is sufficient for producing
the -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 concept of time. 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.Comment: No figures. 64 pages; 68 pages(+4), overall substantial improvements;
70 pages(+2), further improvement
A note on Chudnovsky's Fuchsian equations
We show that four exceptional Fuchsian equations, each determined by the four
parabolic singularities, known as the Chudnovsky equations, are transformed
into each other by algebraic transformations. We describe equivalence of these
equations and their counterparts on tori. The latter are the Fuchsian equations
on elliptic curves and their equivalence is characterized by transcendental
transformations which are represented explicitly in terms of elliptic and theta
functions.Comment: Final version; LaTeX, 27 pages, 1 table, no figure
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).Comment: Final version. Numerous improvements; English, 49 pages, 1 table, no
figures, LaTe