arXiv subjects

Anton Alexa

Publications and source records attributed to Anton Alexa.

8 recordsLinked to original sources

arXiv2026

Time-Scaled Intertwining Cocycles and Identifiability of Multi-Semigroup Mixtures on Hilbert Operator Networks

We study rigidity phenomena for time-scaled intertwining families of dissipative semigroups $\mathcal S_i(t)=e^{-tA_i}$ and prove that a network of bounded injective operators satisfying $K_{ij}\mathcal S_j(t)=\mathcal S_i(\lambda_{ij}t)K_{ij}$ and $K_{ik}=K_{ij}K_{jk}$ necessarily admits a multiplicative gauge representation $\lambda_{ij}=\tau_i/\tau_j$, if and only if the renormalized generators $\{\tau_iA_i\}$ form a common isospectral class with matching eigenspace dimensions; in particular, eigenspaces are transported isomorphically across sectors. The operators $K_{ij}$ define parallel transport in a flat Hilbert bundle over the index network, with flatness derived from the intertwining constraints rather than assumed. As an application, the mixture observable $M(t)=\sum_i w_i\mathcal B_0K_{0i}\mathcal S_i(t)\psi_i$ reduces under finite spectral support to a structured exponential sum. Under spectral separation, the modal parameters are uniquely identifiable, with sector tags determined intrinsically by the operator spectra; under eigenspace observability, active state components are uniquely recovered. Finite-window exact reconstruction holds from $2L$ samples, and the stability bound $\|\widehat\Theta-\Theta_\ast\|_{\mathcal X}\le C_{\mathrm{stab}}\kappa_{\mathrm{exp}}\varepsilon$ follows with constants explicitly controlled by the spectral geometry and observability of the network.

math.FA↗

arXiv2026

Threshold Hierarchy for Packet-Scale Boundary Cancellation of Dirichlet Eigenfunctions

We identify geometry--dependent minimal packet scales required for cancellation of boundary correlations of high--frequency Dirichlet eigenfunctions on smooth strictly convex domains. The main result is a threshold hierarchy: for zero--mean boundary weights, the energy--weighted packet average of boundary correlation coefficients vanishes once the packet length exceeds a scale determined by the vanishing order of curvature moments of the weight. In particular, the threshold $N_k/k^{1-2/d}\to\infty$ suffices when $\int_{\partial\Omega} w,d\sigma=0$, while a strictly weaker threshold applies when additionally $\int_{\partial\Omega} H,w,d\sigma=0$, reducing in dimension $d=3$ to the minimal condition $N_k\to\infty$. The thresholds follow from the boundary local Weyl law. As a structural consequence of the Rellich identity alone, the single--mode share of boundary energy within any sublinear spectral packet is of order $1/N_k$. All estimates are independent of eigenvalue monotonicity and remain stable under eigenvalue crossings.

math.SP↗

arXiv2025

A Dynamical Approach to the Berezin-Li-Yau Inequality

We develop a dynamical method for proving the sharp Berezin-Li-Yau inequality. The approach is based on the volume-preserving mean curvature flow and a new monotonicity principle for the Riesz mean $R_\Lambda(\Omega_t)$. For convex domains we show that $R_\Lambda$ is monotone non-decreasing along the flow. The key input is a geometric correlation inequality between the boundary spectral density $Q_\Lambda$ and the mean curvature $H$, established in all dimensions: in $d=2$ via a near-disk Fourier analysis, and in $d\ge 3$ via the boundary Weyl expansion together with a local spectral rigidity argument near the ball, with a first-zero exclusion principle closing the global step. Since the flow converges smoothly to the ball, the monotonicity implies the sharp Berezin-Li-Yau bound for every smooth convex domain. As an application, we obtain a sharp dynamical Ces\`aro-P\'olya inequality for eigenvalue averages.

math.DG↗

arXiv2025

Rigidity of Spectral Encodings under Weyl Growth Conditions

We prove that the geometric Weyl bulk-density exponent $(d-2)/2$ rigidifies spectral encodings $C=\pi-\phi(\lambda)$ in the O-regularly varying class: the bulk power law forces $\phi\in\mathrm{RV}_1$ (asymptotic linearity). For polynomial-type encodings $C=\pi-\epsilon\lambda^k L(\lambda)$ with $L\in\mathrm{RV}_0$, this yields the unique admissible exponent $k=1$. The affine encoding then gives $N_{\mu_C}(C)\sim\gamma_d\,\epsilon^{-d/2}(\pi-C)^{d/2}$ as $C\to-\infty$, allowing recovery of $d$ and $\gamma_d$ from bulk encoded data. This transfer is stable under perturbations $\delta(\lambda)=o(\lambda)$, with explicit slowly varying error control. We further formalize asymptotic spectral equivalence classes: if $\phi\in\mathrm{RV}_k$, the induced map scales asymptotic spectral dimension as $d_{\mathrm{as}}\mapsto d_{\mathrm{as}}/k$; hence dimension preservation is equivalent to $\phi\in\mathrm{RV}_1$, with strict affine normalization at first order when $L(\lambda)\to1$.

math.SP↗

arXiv2025

Spectral Deformation Flow and Dimension Recovery: Invariant-Based Rigidity for Simply-Connected Closed Manifolds

We study an effective spectral deformation flow for mode amplitudes $C_n(\tau)$, governed by a second-order self-adjoint operator $\hat{C}$ on a compact interval. The flow is encoded in the multi-function $C(v,\tau,n)$ and exhibits global stabilization toward a symmetric spectral attractor. To connect this dynamics with geometry, we introduce a deformation-spectrum encoding of compact Riemannian manifolds through a shifted Laplace--Beltrami spectrum. Within this framework, we analyze energy decay, entropy decay, and the bulk asymptotic spectral density of the encoded manifold spectrum, obtaining an information-theoretic and spectral route to dimension recovery. We further formulate a rigidity criterion showing that, when the deformation spectral invariants coincide with those of the round sphere, the spherical profile is the unique manifold-compatible asymptotic realization within the present framework. In dimension four, this yields a topological conclusion together with a spectral obstruction against exotic smooth structures that produce distinct invariants. The results position the spectral flow as an effective geometric model, rather than as a direct replacement for tensorial geometric flows on arbitrary manifolds.

math.SP↗

arXiv2025

Discrete Spectrum and Spectral Rigidity of a Second-Order Geometric Deformation Operator

We analyze the spectral properties of a self-adjoint second-order differential operator $\hat{C}$, defined on the Hilbert space $L^2([-v_c, v_c])$ with Dirichlet boundary conditions. We derive the discrete spectrum $\{C_n\}$, prove the completeness of the associated eigenfunctions, and establish orthogonality and normalization relations. The analysis follows the classical Sturm--Liouville framework and confirms that the deformation modes $C_n$ form a spectral basis on the compact interval. We further establish a spectral rigidity result: uniform spectral coefficients imply a constant profile $C(v) = \pi$, which does not belong to the Sobolev domain of the operator. These results provide a rigorous foundation for further investigations in spectral geometry and functional analysis.

math.SP↗

arXiv2025

Essential Self-Adjointness of the Geometric Deformation Operator on a Compact Interval

We define a second-order differential operator $\hat{C}$ on the Hilbert space $L^2([-v_c, v_c])$, constructed from a smooth deformation function $C(v)$. The operator is considered on the Sobolev domain $H^2([-v_c, v_c]) \cap H^1_0([-v_c, v_c])$ with Dirichlet boundary conditions. We prove that $\hat{C}$ is essentially self-adjoint by verifying its symmetry and computing von Neumann deficiency indices, which vanish. All steps are carried out explicitly. This result ensures the mathematical consistency of the operator and enables future spectral analysis on compact intervals.

math.SP↗

arXiv2025

A Variational Scalar Conformal Flow for Lorentz-Contracted Geometry: Algebraic Decay and Canonical Normalization

We introduce the scalar function $C(v)=\pi(1-v^2/c^2)$ as a conformal factor associated, within the model, with longitudinal Lorentz contraction. Extending $C(v)$ to a one-parameter family $C(v,\tau)$, we construct a variational scalar conformal flow that drives the factor toward the equilibrium $C=\pi$ without singularities. The main result is an explicit algebraic decay law for the energy functional: $E(\tau)\sim \tau^{-1/2}$ for generic initial data and $E(\tau)\sim \tau^{-5/2}$ for the physical initial condition $C(v,0)=\pi(1-v^2/c^2)$. More generally, if the initial deviation vanishes as $v^n$ near $v=0$, then $E(\tau)\sim \tau^{-(2n+1)/2}$. This behavior is explained by the gapless continuous spectrum of the relaxation operator, whose spectral measure satisfies $d\mu(k)\sim k^{-1/2}dk$ near $k=0$. As an application, within the conformally homogeneous class of compact simply-connected $3$-manifolds with constant positive background curvature, the flow acts as a canonical normalization mechanism selecting $C=\pi$ as the unique conformal representative whose curvature invariants agree with those of the unit $S^3$.

math-ph↗