Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1508.06922

On Agmon metrics and exponential localization for quantum graphs

Evans M. Harrell II·Anna V. Maltsev
View original source ↗Read source document ↗Export metadata ↓

Abstract

We investigate the rate of decrease at infinity of eigenfunctions of quantum graphs by using Agmon's method to prove $L^2$ and $L^\infty$ bounds on the product of an eigenfunction with the exponential of a certain metric. A generic result applicable to all graphs is that the exponential rate of decay is controlled by an adaptation of the standard estimates for a line, which are of classical Liouville-Green (WKB) form. Examples reveal that this estimate can be the best possible, but that a more rapid rate of decay is typical when the graph has additional structure. In order to understand this fact, we present two alternative estimates under more restrictive assumptions on the graph structure that pertain to a more rapid decay. One of these depends on how the eigenfunction is distributed along a particular chosen path, while the other applies to an average of the eigenfunction over edges at a given distance from the root point.

Explore related subjects

math-phmath.MPmath.SP

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Evans M. Harrell II, Anna V. Maltsev. 2015-08-27. On Agmon metrics and exponential localization for quantum graphs. https://arxiv.org/abs/1508.06922

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Field Theory via Higher Geometry II: Thickened Smooth Sets as Synthetic Foundations

This is the second in a series of papers that aim to develop rigorous and most encompassing foundations for field theory, where in the first installment we laid out the natural formulation of bosonic variational field theory via the ``functorial geometry'' of smooth sets, namely in the topos over the site of spaces with smooth maps between them. Here, we extend this to the category ThickenedSmoothSets of {\it infinitesimally thickened smooth sets}. We first describe the Cahiers topos in a simplified, but still fully rigorous, $\mathbb{R}$-algebraic setting -- which should serve as a more accessible introduction to the theory of Synthetic Differential Geometry to both physicists and mathematicians. Then, we formulate local Lagrangian field theory in this setting in which infinitesimal spaces exist and interact correctly with the field-theoretic spaces of infinite jet bundles, off-shell and on-shell spaces of fields etc. This discussion subsumes previous constructions and further recovers all the relevant tangent bundles of traditional (off-shell and on-shell) field theory considerations via the synthetic tangent bundle construction, i.e., as ``infinitesimal curves'' in those spaces, which were previously defined only in an indirect manner. Beyond finally putting such aspects of the theory on a firm foundation, this approach recognizes the variational principle of local Lagrangian field theory, equivalently, as an intersection of thickened smooth sets. Lastly, as we will show in detail, it permits a mathematical formalization (and recovery) of perturbative field theory as the actual restriction to the infinitesimal neighborhood around a field configuration -- a statement which thus far had remained only in the realm of intuition. Crucially, all of these results are obtained with no use of infinite dimensional manifold theory, topology or functional analysis of mapping spaces.

math-ph↗
arXiv2026

Gradual eigenvector ergodization in coupled Ginibre matrices

Non-Hermitian random matrices provide a useful framework for understanding universal characteristics of dissipative quantum chaotic systems with loss or gain. We consider a model of two such systems represented by two independent $N\times N$ complex Ginibre matrices interacting via a deterministic matrix $c{\bf 1}_N$, where $c$ is the complex coupling parameter whose magnitude $|c|$ controls the interaction strength. We characterize quantitatively how the eigenvectors of the whole system, initially localized in one of the individual subsystems for $|c|=0$, eventually spread over the full system with growing interaction strength. The resulting asymptotic formula describing such spread in the limit $N\to \infty$ is very explicit and provides a full picture of the gradual ergodization of eigenvectors as a function of the coupling parameter $|c|$ in the whole transition regime. As a by-product of our method we also compute the mean eigenvalue density for our model at the origin of the spectral bulk $z=0$ in the fully ergodic regime, when the coupling is scaled with the matrix size as $c=\sqrt{N}\tilde{c}$. We find that as $N\to \infty$ the limiting density at the origin vanishes beyond the critical value $|\tilde{c}|=1.$ This is compatible with the expected split of the density support in the complex plane into two disjoint domains.

math-ph↗
arXiv2026

Structure-preserving diffuse-domain accelerated saddle dynamics for wetting transitions

Wetting transitions on textured substrates play a central role in the design of functional surfaces, but resolving their transition mechanisms requires efficient exploration of complex energy landscapes. In particular, saddle points are essential for revealing the connectivity among metastable states and transition pathways. In this work, we develop a diffuse-domain accelerated saddle dynamics (DD-ASD) framework for mass-constrained wetting transitions on complex textured substrates. The diffuse-domain formulation embeds complex solid geometries into a Cartesian grid, avoiding body-fitted mesh construction in repeated saddle-point searches. An orthogonal projection is applied consistently to the phase-field state and unstable-direction dynamics, preserving the prescribed droplet mass at the discrete level. Momentum acceleration is incorporated into the saddle dynamics to improve the efficiency of high-index saddle searches. Under the stated local spectral and exact-eigenspace assumptions, we establish a local convergence rate of $1-\mathcal{O}(1/\sqrtκ)$ on the effective mass-conserving subspace. Numerical experiments demonstrate the effectiveness of the proposed method in resolving wetting solution landscapes, transition pathways, and energy barriers on textured substrates. We further extend the framework to fully three-dimensional wetting-landscape computations.

math-ph↗