arXiv subjects

Sergei Artamoshin

Publications and source records attributed to Sergei Artamoshin.

5 recordsLinked to original sources

arXiv2019

At Most Two Radii Theorem For A Real Eigenvalue Of The Hyperbolic Laplacian

We study a $(k+1)$-dimensional hyperbolic space of a negative constant sectional curvature $κ=-1/ρ^2$. Let $λ$ be a real eigenvalue and $f_λ (x)$ be an eigenfunction of the hyperbolic Laplacian assuming a non-zero value at $x_0$. Then the average value of $f_λ(x)$ over any sphere centered at $x_0$ allows to identify the corresponding eigenvalue $λ$ uniquely as long as that average value is large enough. Otherwise, to identify the corresponding eigenvalue uniquely, we need to make sure that the computed average value is not zero and then we need to compute an additional average value of $f_λ(x)$ over a small enough sphere centered at the same point $x_0$.

math.DG↗

arXiv2014

One Radius Theorem For A Radial Eigenfunction Of A Hyperbolic Laplacian

Let us fix two different radial eigenfunctions of a hyperbolic Laplacian and assume that both of them have the same value at the origin. Both eigenvalues can be complex numbers. The main goal of this paper is to estimate the lower bound for the interval (0,T], where these two eigenfunctions must assume different values at every point. We shall see that T is a function of two different eigenvalues corresponding to the given pair of radial eigenfunctions. On the other hand, we shall see that at every fixed point and for the value already assumed by a radial eigenfunction at the fixed point, there are infinitely many other radial eigenfunctions, assuming the same value at the fixed point and satisfying the same initial condition.

math.DG↗

arXiv2014

The Spherical Ratio of Two Points and its Integral Properties

At the end of 19-th century, 1874, Hermann Schwartz found that for every point inside a planar disk, a two-dimensional Poisson Kernel can be written as a ratio of two segments, which he called as the geometric interpretation of that Kernel \cite{Schwarz}. About 90 years later, Lars Ahlfors, in his textbook, called this ratio as interesting \cite{Ahlfors}. We shall see here that every two different points, in a multidimensional space also define a similar ratio of two segments. The main goal of this paper is to study some integral properties of the ratio as well as introduce and proof One Radius Theorem for Spherical Ratio of two points. In particular, we introduce a new integration technique involving the ratio of two segments and a non-trivial integral equivalence leading to an integral relationship between Newtonian Potential and Poisson Kernel in any multi-dimensional space.

math.CV↗

arXiv2014

Lower Bounds For The First Eigenvalue Of The Laplacian With Dirichlet Boundary Conditions In A Hyperbolic Space Of A Negative Constant Curvature

In this paper we consider a domain in a space of negative constant sectional curvature. Such assumption about the sectional curvature let us develop a new technique and improve existing lower bounds of eigenvalues from Dirichlet eigenvalue problem, obtained by Alessandro Savo in 2009.

math.DG↗

arXiv2009

Geometric Interpretation of the two dimensional Poisson Kernel and its applications

Hermann Schwarz, while studying complex analysis, introduced the geometric interpretation for the Poisson kernel in 1890. We shall see here that the geometric interpretation can be useful to develop a new approach to some old classical problems as well as to obtain several new results, mostly related to hyperbolic geometry. For example, we obtain One Radius Theorem saying that any two radial eigenfunctions of a Hyperbolic Laplacian assuming the value 1 at the origin can not assume any other common value within some interval [0, p], where the length of this interval depends only on the location of the eigenvalues on the complex plane and does not depend on the distance between them.

math.DG↗