София Румянцева выступила с докладами на двух международных конференциях в Австрии и Германии

Стажер-исследователь Лаборатории геометрической алгебры и приложений София Румянцева выступила с докладами на двух международных конференциях в сентябре 2025:

1. Международная конференция Spectral Theory and Differential Operators (TU Graz, Австрия, 15-19 сентября 2025)

Сайт конференции: https://www.math.tugraz.at/diffop2025/index.html

Программа: https://www.math.tugraz.at/diffop2025/program.html

Доклад: Sofia Rumyantseva, Dmitry Shirokov, "Geometric Algebra Approach to Exploring the Multidimensional Dirac Equation", 16 сентября 2025

Аннотация доклада: In the talk, we will introduce the multidimensional Dirac–Hestenes equation, a real-valued analogue of the Dirac equation formulated in the geometric algebra Cl(1,n), and discuss some of its fundamental properties. It is known that the classical four-dimensional Dirac equation is equivalent to the Dirac–Hestenes equation in geometric algebra Cl(1,3). It means that one might obtain a solution to the Dirac–Hestenes equation using a solution to the Dirac equation and vice versa. The Dirac–Hestenes equation may provide a deeper understanding of geometry in various tasks, as the considering wave function is entirely real. We will show that the theory is extended to the multidimensional case. Since the matrix representation of complex geometric algebra depends on the parity of n, the cases of even and odd n are analyzed separately. In the even-dimensional case, there are two types of spinors which are solutions to the Dirac equation: semi-spinors and double spinors. Additionally, we will demonstrate that the multidimensional Dirac–Hestenes equation exhibits both gauge invariance and Lorentz invariance.

2. Международная конференция QMATH16: Mathematical Results in Quantum Theory (Technical University of Munich, Германия, 1-5 сентября 2025)

Сайт конференции: https://sites.google.com/view/qmath16/home

Программа: https://sites.google.com/view/qmath16/schedule/schedule-overview

Доклад: Sofia Rumyantseva, "Oscillating Energy Tunnel Splitting in a Two-Dimensional Quantum System with su(1,1) Symmetry", 4 сентября 2025

Аннотация доклада: In this talk, we will consider in detail a two-dimensional oscillator with small perturbations in L_2 (R^2) with su(1,1) symmetry. After averaging, the perturbation reduces to a one-dimensional quadratic form in su(1,1) generators, which becomes the primary focus of our analysis. Under certain conditions on external parameters, two symmetric periodic classical trajectories emerge, between which tunneling can occur. The spectral problem will be presented in several representations, which greatly simplify analysis. In the space L_2 (R), it takes the form of a fourth-order differential equation, whereas in the specially constructed space of holomorphic functions, it reduces to a second-order differential equation. By using the complex WKB method, we demonstrate that the asymptotic of energy tunnel splitting not only decreases exponentially, as in the case of the one-dimensional Schrodinger operator with a symmetric double-well potential, but also oscillates due to instanton interference.