arXiv subjects

Atsushi Katsuda

Publications and source records attributed to Atsushi Katsuda.

5 recordsLinked to original sources

arXiv2026

Noncommutative Floquet-Bloch Theory for Nilpotent Groups: Representation-Theoretic Foundations

Classical Floquet-Bloch theory decomposes abelian periodic problems over the character torus of the lattice. For nonabelian nilpotent lattices, the non-type I obstruction rules out a comparable parametrization of the full unitary dual. We do not attempt to remove this obstruction. Instead, we construct an exact Bloch-type replacement on the representation-theoretic part of the theory which is visible from rational Kirillov data and from finite-dimensional rational fibers. Let $Γ$ be a torsion-free finitely generated nilpotent group and let $G$ be its Malcev completion. For an irreducible unitary representation $π_l$ of $G$ attached to a rational Kirillov parameter $l\in\mathfrak{g}_{\mathbb Q}^{*}$, we prove an exact restriction theorem for $π_l|_Γ$. The branching is first described by induced representations attached to rational polarizations. On the rational odd locus relevant to finite-dimensional representations, it further decomposes into finite-dimensional irreducible representations of $Γ$. On these finite-dimensional rational fibers we construct a positive finitely additive Plancherel measure. It gives Fourier inversion and normalized trace identities for nilpotent lattices, recovering Pytlik's formula in the discrete Heisenberg case.

math.RT↗

arXiv2026

Heat Kernel and Closed Geodesic Asymptotics for Nilpotent Coverings

We establish all order long-time asymptotic expansions for heat kernels on nilpotent coverings and for prime closed geodesics in fixed central classes of nilpotent quotients of compact hyperbolic surfaces. The exact lattice-side input is the finite-dimensional rational Floquet-Bloch theory of the companion paper: rational Kirillov restrictions give exact finite-dimensional fibers, and a generalized Pytlik functional gives exact Fourier-inversion and normalized-trace identities. At a rational parameter $p/q$ the decomposition is exact, and the fluctuation of the fiber integrand is controlled only by $q$. Hence the large-denominator comparison with the smooth Kirillov or Schrödinger normal form is uniform on the rational support of the Pytlik functional; irrational parameters do not enter the rigorous trace argument. For general nilpotent models, coefficient-weighted spectral sums are justified to every fixed order by positive Rockland estimates, the Plancherel-Mellin formula, and a trace-level order-balance argument. In contrast with approaches which usually give leading terms or integrated Edgeworth-type asymptotics, the method gives genuinely local, pointwise higher-order heat-kernel expansions. The same representation-theoretic quantity governs the leading term in the closed-geodesic asymptotics, producing a nilpotent Chebotarev-type phenomenon. The Heisenberg model is computed to the first correction term, and the Engel model is represented through the resolvent and heat-kernel calculus of the quartic oscillato

math.SP↗

arXiv2025

An extension of the Floquet-Bloch theory to nilpotent groups and its applications

We develop a generalized Floquet-Bloch theory for discrete torsion-free nilpotent groups by exploiting their Malcev completions. Our main result is a branching formula that relates finite-dimensional representations of a discrete nilpotent lattice to infinite-dimensional unitary representations of its simply connected nilpotent Lie group. This generalization enables to extend the following two classical asymptotic problems (i) a Chebotarev density analogue for prime closed geodesics on compact negatively curved manifolds with nilpotent covers, and (ii) long time asymptotic expansions of heat kernels on such coverings to the nilpotent setting. As a by-product, we derive a semi-classical expansion for the Harper operator, presenting an alternative to mathematical justification of Wilkinson's formula by Helffer-Sjöstrand. We conclude by proposing several avenues for future work: extensions to general hyperbolic flows and noncompact manifolds (in particular knot complements and related quasi-morphisms), connections to modified Riemann-Hilbert problems and opers. Furthermore, we give a brief comment on asymptotic behavior of knot invariants and infinite extensions in number theory.

math.DG↗

arXiv2002

Boundary regularity for the Ricci equation, geometric convergence, and Gel'fand's inverse boundary problem

This paper explores and ties together three themes. The first is to establish regularity of a metric tensor, on a manifold with boundary, on which there are given Ricci curvature bounds, on the manifold and its boundary, and a Lipschitz bound on the mean curvature of the boundary. The second is to establish geometric convergence of a (sub)sequence of manifolds with boundary with such geometrical bounds and also an upper bound on the diameter and a lower bound on injectivity and boundary injectivity radius, making use of the first part. The third theme involves the uniqueness and conditional stability of an inverse problem proposed by Gel'fand making essential use of the results of the first two parts.

math.SP↗

arXiv2002

Stability and Reconstruction in Gel'fand Inverse Boundary Spectral Problem

We consider stability and approximate reconstruction of Riemannian manifold when the finite number of eigenvalues of the Laplace-Beltrami operator and the boundary values of the corresponding eigenfunctions are given. The reconstruction can be done in stable way when manifold is a priori known to satisfy natural geometrical conditions related to curvature and other invariant quantities.

math.AP↗