Proceedings of the International Geometry Center

From Maxwell’s equations to relativistic Schrödinger equations via Schwartz Linear Algebra and Killing frames on the two-sphere

2026-06-25T09:19:55+03:00

In this work, we develop a rigorous framework partially unifying classical electromagnetism and relativistic quantum mechanics through the language of Schwartz Linear Algebra and via continuous symmetries of the 2-sphere. Building upon our recent analysis of Killing vector fields on the two-sphere and the associated orthonormal right-handed frames, we construct explicit correspondences between two fundamental distribution spaces:

V = S'(M₄, C), W = S'(M₄, C³),

where M₄ is Minkowski space-time. The space V hosts scalar tempered wave distributions (the de Broglie basis of relativistic quantum mechanics), while W hosts vector-valued tempered fields with natural Maxwellian structure. For each unit vector e ∈ S², we introduce a canonical orthonormal tangent frame f_e arising from the Killing field K_e on the sphere (canonically associated with e) and we define the associated families of de Broglie waves η^e and Maxwell-de Broglie fields w^e. By these families we construct two continuous and Schwartz-linear operators:

F_e : V_e → W_e : ψ ↦ ∫M₄* (ψ)η w^e, Ψ_e : W_e → V_e : F ↦ ∫M₄* (F)w^e η^e,

which are shown to be inverse isomorphisms of each other. The operator F_e transforms a wave distribution ψ into a Maxwellian field F_ψ, while the projection Ψ_e recovers the wave distribution from an electromagnetic-like field. We prove that these isomorphisms are dynamically compatible: a scalar distribution ψ ∈ V_e solves the relativistic Schrödinger equation

ih ∂_t ψ = H_m₀ ψ,

if and only if its Maxwellian counterpart F_ψ = F_e(ψ) solves the corresponding generalized Maxwell-Schrödinger equation in W_e:

ih ∂_t F_ψ = H^e_m₀ F_ψ.

This duality holds in both the massless (photonic) case --- where H^e₀ reduces to the curl operator multiplied times the Planck's constant and the speed of light --- and in the massive case --- where H^e_m₀ is the spectral principal square root of the Schwartz diagonalizable and (strictly) positive operator

c²(h∇×)² + (m₀c²)² I_W^e.

The construction provides a precise mathematical mechanism to pass from Maxwell theory to the relativistic Schrödinger picture, mediated by Killing frames on the two-sphere. Beyond its intrinsic geometric interest, this approach suggests new perspectives in the analysis of wave propagation, polarization, and spectral synthesis, as well as possible applications to inverse problems in electromagnetism and the inclusion of curvature from general relativity.

On partial preliminary group classification of some class of (1+3)-dimensional Monge-Ampere equations. II. Two-dimensional Lie algebras.

2026-06-25T09:18:54+03:00

We have performed the partial preliminary group classification of some class of (1+3)-dimensional Monge-Ampère equations using non-equivalent functional bases of the first-order differential invariants of two-dimensional nonconjugate subalgebras of the Lie algebra of the Poincaré group P(1,4).

Remarks on equivariant asymptotics for complexified toric eigenfunction

2026-06-25T09:17:48+03:00

The eigenfunctions of the Laplace-Beltrami operator on a real-analytic Riemannian manifold admit a symultaneous holomorphic extension to a sufficiently small Grauert tube. If the manifold is endowed with an isometric action of a compact Lie group, one can decompose each eigenspace into isotypical components associated to the irreducible representations of the group. The local asymptotics of the complexified eigenfunctions in a fixed isotypical component and (heuristically speaking) belonging to a spectral band drifting to infinity have been studied recently in the work "Equivariant scaling asymptotics for Poisson and Szegő kernels on Grauert tube boundaries" by Gallivanone and Paoletti (2024) . In this note, we illustrate these results in the special case where the base manifold is a d-dimensional torus with the standard metric, acted upon by a proper subtorus.

On boundary distortion estimates of homeomorphisms with a fixed point

2026-06-25T09:21:19+03:00

The article is devoted to the study of homeomorphisms that distort the modulus of families of paths according to the Poletsky inequality type. We consider the case when the majorant, corresponding to the distortion of the modulus, has finite mean oscillation at any point or satisfies Lehto integral divergence condition. We have obtained the boundary Hölder continuity of such mappings, provided that the image of at least one inner point is located at a fixed distance from the boundary of the mapped domain. In particular, this condition holds for fixed-point mappings. The manuscript deals with cases of good boundaries and domains with prime ends.

Dual Thurston norm of Euler classes of foliations on negatively curved 3-manifolds

2026-06-25T09:15:22+03:00

In this paper we give an upper bound on the dual Thurston norm of the Euler class of an arbitrary smooth foliation ℱ of codimension one defined on a closed oriented 3-manifold M³ of negative curvature, which depends on the constants bounding the injectivity radius inj(M³), the volume Vol(M³), the sectional curvature of M³ and the modulus of the mean curvature of leaves of ℱ.

Sets of distinct representations of numbers in numeral systems with a natural base and a redundant alphabet

2026-06-25T09:13:41+03:00

In this work, we study a numeral system with a natural base s ≥ 2 and a redundant alphabet A_r = {0, 1, ..., r}, where s ≤ r ≤ 2s - 2. We investigate the topological, metric, and fractal properties of the set of numbers in the interval [0, r/(s-1)] that admit a unique representation x = ∑_n=1^∞ (α_n / sⁿ) ≡ Δ^r_s_{α₁α₂...α_n...}, α_n ∈ A_r. A criterion for the uniqueness of a number's representation is established. We prove that the Hausdorff-Besicovitch dimension of the set of numbers with a unique representation is equal to ln(2s - r - 1) / ln s. An analysis of the quantity of representations for numbers having purely periodic representations with a simple (single-digit) period is carried out. It is proved that the set of numbers that admit a continuum of distinct representations has full Lebesgue measure. Conditions for a number to belong to this set are given in terms of one of its representations.