arXiv subjects

Evans M. Harrell II

Publications and source records attributed to Evans M. Harrell II.

At least 19 recordsLinked to original sources

arXiv2026

Bounds on eigenvalue ratios of quantum graphs

We study ratios of eigenvalues of the Laplacian on compact metric graphs. Our goals are threefold: First, we prove a sharp Ashbaugh--Benguria-type bound for the ratio of the first two eigenvalues on compact trees with Dirichlet conditions at all leaves, concretely showing that the ratio is maximized when the graph is an interval or an equilateral star. This improves a previous Payne--Pólya--Weinberger-type result due to Nicaise [Bull. Sci. Math., II. Sér. 111 (1987), 401--413]. Second, we extend this bound to a set of inequalities for the ratio of any pair of eigenvalues of such compact Dirichlet trees which respect the Weyl asymptotics up to an absolute constant. Third, we show that on non-trees, on which we also allow any mix of Neumann and Dirichlet conditions at the leaves, it is possible to recover bounds on the eigenvalue ratios depending only on the number of independent cycles and the number of Neumann leaves, in addition to the eigenvalue indices. This complements previously known counterexamples to analogues of the Ashbaugh--Benguria bound for general quantum graphs, by showing that the only way the bound can fail is through cycles and Neumann leaves, and by explicitly quantifying the extent to which it can fail.

math.SP↗

arXiv2024

On the Fundamental Eigenvalues gap of Sturm-Liouville Operators

We use methods of direct optimization as in [9] to find the minimizers of the fundamental gap of Sturm-Liouville operators on an interval, under the constraint that the potential is of single-well form and that the weight function is of single-barrier form, and under similar constraints expressed in terms of convexity.

math.SP↗

arXiv2024

On topological states and secular equations for quantum-graph eigenvalues

Quantum graphs without interaction which contain equilateral cycles possess "topological" bound states which do not correspond to zeroes of one of the two variants of the secular equation for quantum graphs. Instead, their eigenvalues lie in the set of singularities of the vertex-scattering secular matrix. This observation turns out to be representative of a wider phenomenon. We introduce a notion of topological bound states and show that they are linear combinations of functions supported on generators of the fundamental group of the graph (hence the "topological" in the name), including for graphs that have interactions on the edges. Using an Ihara-style theorem, we elucidate the role of such topological bound states in the spectral analysis of quantum graph Hamiltonians using secular matrices. En route we determine the set of the fixed vectors of the bond-scattering matrix. This work is dedicated to E.B. Davies on the occasion of his 80th birthday and in honor of his important contributions to the theory of quantum graphs, e.g., \cite{DaExLi,Da13} and of his broad and influential work on spectral theory, e.g., \cite{Da89,Da95}.

math.SP↗

arXiv2023

Gaps between consecutive eigenvalues for compact metric graphs

On a compact metric graph, we consider the spectrum of the Laplacian defined with a mix of standard and Dirichlet vertex conditions. A Cheeger-type lower bound on the gap $λ_2 - λ_1$ is established, with a constant that depends only on the total length of the graph and minimum edge length. We also prove some improvements of known upper bounds for eigenvalue gaps and ratios for metric trees and extensions to certain other types of graphs.

math.SP↗

arXiv2022

The heat kernel on the diagonal for a compact metric graph

We analyze the heat kernel associated to the Laplacian on a compact metric graph, with standard Kirchoff-Neumann vertex conditions. An explicit formula for the heat kernel as a sum over loops, developed by Roth and Kostrykin, Potthoff, and Schrader, allows for a straightforward analysis of small-time asymptotics. We show that the restriction of the heat kernel to the diagonal satisfies a modified version of the heat equation. This observation leads to an "edge" heat trace formula, expressing the a sum over eigenfunction amplitudes on a single edge as a sum over closed loops containing that edge. The proof of this formula relies on a modified heat equation satisfied by the diagonal restriction of the heat kernel. Further study of this equation leads to explicit formulas for completely symmetric graphs.

math.SP↗

arXiv2019

Complementary asymptotically sharp estimates for eigenvalue means of Laplacians

We present asymptotically sharp inequalities, containing a second term, for the Dirichlet and Neumann eigenvalues of the Laplacian on a domain, which are complementary to the familiar Berezin-Li-Yau and Kröger inequalities in the limit as the eigenvalues tend to infinity. We accomplish this in the framework of the Riesz mean $R_1(z)$ of the eigenvalues by applying the averaged variational principle with families of test functions that have been corrected for boundary behaviour.

math.SP↗

arXiv2018

Optimal bounds on the fundamental spectral gap with single-well potentials

We characterize the potential-energy functions $V(x)$ that minimize the gap $Γ$ between the two lowest Sturm-Liouville eigenvalues for \[ H(p,V) u := -\frac{d}{dx} \left(p(x)\frac{du}{dx}\right)+V(x) u = λu, \quad\quad x\in [0,π], \] where separated self-adjoint boundary conditions are imposed at end points, and $V$ is subject to various assumptions, especially convexity or having a "single-well" form. In the classic case where $p=1$ we recover with different arguments the result of Lavine that $Γ$ is uniquely minimized among convex $V$ by the constant, and in the case of single-well potentials, with no restrictions on the position of the minimum, we obtain a new, sharp bound, that $Γ> 2.04575\dots$.

math.SP↗

arXiv2018

Localization and landscape functions on quantum graphs

We discuss explicit landscape functions for quantum graphs. By a "landscape function" $Υ(x)$ we mean a function that controls the localization properties of normalized eigenfunctions $ψ(x)$ through a pointwise inequality of the form $$ |ψ(x)| \le Υ(x). $$ The ideal $Υ$ is a function that a) responds to the potential energy $V(x)$ and to the structure of the graph in some formulaic way; b) is small in examples where eigenfunctions are suppressed by the tunneling effect, and c) relatively large in regions where eigenfunctions may - or may not - be concentrated, as observed in specific examples. It turns out that the connectedness of a graph can present a barrier to the existence of universal landscape functions in the high-energy régime, as we show with simple examples. We therefore apply different methods in different régimes determined by the values of the potential energy $V(x)$ and the eigenvalue parameter $E$.

math.SP↗

arXiv2016

Two-term, asymptotically sharp estimates for eigenvalue means of the Laplacian

We present asymptotically sharp inequalities for the eigenvalues $μ_k$ of the Laplacian on a domain with Neumann boundary conditions, using the averaged variational principle introduced in \cite{HaSt14}. For the Riesz mean $R_1(z)$ of the eigenvalues we improve the known sharp semiclassical bound in terms of the volume of the domain with a second term with the best possible expected power of $z$. In addition, we obtain two-sided bounds for individual $μ_k$, which are semiclassically sharp. In a final section, we remark upon the Dirichlet case with the same methods.

math.SP↗

arXiv2015

On Agmon metrics and exponential localization for quantum graphs

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.

math-ph↗

arXiv2012

On a transformation of Bohl and its discrete analogue

Fritz Gesztesy's varied and prolific career has produced many transformational contributions to the spectral theory of one-dimensional Schrödinger equations. He has often done this by revisiting the insights of great mathematical analysts of the past, connecting them in new ways, and reinventing them in a thoroughly modern context. In this short note we recall and relate some classic transformations that figure among Fritz Gestesy's favorite tools of spectral theory, and indeed thereby make connections among some of his favorite scholars of the past, Bohl, Darboux, and Green. After doing this in the context of one-dimensional Schrödinger equations on the line, we obtain some novel analogues for discrete one-dimensional Schrödinger equations. \smallskip Dem einzigartigen Fritz gewidmet.

math.SP↗

arXiv2011

On the behavior at infinity of solutions to difference equations in Schroedinger form

We offer several perspectives on the behavior at infinity of solutions of discrete Schroedinger equations. First we study pairs of discrete Schroedinger equations whose potential functions differ by a quantity that can be considered small in a suitable sense as the index n \rightarrow \infty. With simple assumptions on the growth rate of the solutions of the original system, we show that the perturbed system has a fundamental set of solutions with the same behavior at infinity, employing a variation-of-constants scheme to produce a convergent iteration for the solutions of the second equation in terms of those of the original one. We use the relations between the solution sets to derive exponential dichotomy of solutions and elucidate the structure of transfer matrices. Later, we present a sharp discrete analogue of the Liouville-Green (WKB) transformation, making it possible to derive exponential behavior at infinity of a single difference equation, by explicitly constructing a comparison equation to which our perturbation results apply. In addition, we point out an exact relationship connecting the diagonal part of the Green matrix to the asymptotic behavior of solutions. With both of these tools it is possible to identify an Agmon metric, in terms of which, in some situations, any decreasing solution must decrease exponentially. A discussion of the discrete Schroedinger problem and its connection with orthogonal polynomials on the real line is presented in an Appendix.

math.CA↗

arXiv2009

On semiclassical and universal inequalities for eigenvalues of quantum graphs

We study the spectra of quantum graphs with the method of trace identities (sum rules), which are used to derive inequalities of Lieb-Thirring, Payne-Pólya-Weinberger, and Yang types, among others. We show that the sharp constants of these inequalities and even their forms depend on the topology of the graph. Conditions are identified under which the sharp constants are the same as for the classical inequalities; in particular, this is true in the case of trees. We also provide some counterexamples where the classical form of the inequalities is false.

math.SP↗

arXiv2009

Trace identities for commutators, with applications to the distribution of eigenvalues

We prove trace identities for commutators of operators, which are used to derive sum rules and sharp universal bounds for the eigenvalues of periodic Schroedinger operators and Schroedinger operators on immersed manifolds. In particular, we prove bounds on the eigenvalue lambda_{N+1} in terms of the lower spectrum, bounds on ratios of means of eigenvalues, and universal monotonicity properties of eigenvalue moments, which imply sharp versions of Lieb-Thirring inequalities. In the geometric context we derive a version of Reilly's inequality, bounding the eigenvalue lambda_{N+1} of the Laplace-Beltrami operator on an immersed manifold of dimension d by a universal constant times the square of the maximal mean curvature times N^{2/d}.

math.SP↗

arXiv2008

Eigenvalue inequalities for Klein-Gordon Operators

We consider the pseudodifferential operators $H_{m,Ω}$ associated by the prescriptions of quantum mechanics to the Klein-Gordon Hamiltonian $\sqrt{|{\bf P}|^2+m^2}$ when restricted to a compact domain $Ω$ in ${\mathbb R}^d$. When the mass $m$ is 0 the operator $H_{0,Ω}$ coincides with the generator of the Cauchy stochastic process with a killing condition on $\partial Ω$. (The operator $H_{0,Ω}$ is sometimes called the {\it fractional Laplacian} with power 1/2, cf. \cite{Gie}.) We prove several universal inequalities for the eigenvalues $0 < β_1 < β_2 \le >...$ of $H_{m,Ω}$ and their means $\overline{β_k} := \frac{1}{k} \sum_{\ell=1}^k{β_\ell}$. Among the inequalities proved are: {\overline{β_k}} \ge {\rm cst.} (\frac{k}{|Ω|})^{1/d} for an explicit, optimal "semiclassical" constant, and, for any dimension $d \ge 2$ and any $k$: β_{k+1} \le \frac{d+1}{d-1} \overline{β_k}. Furthermore, when $d \ge 2$ and $k \ge 2j$, \frac{\overlineβ_{k}}{\overlineβ_{j}} \leq \frac{d}{2^{1/d}(d-1)}(\frac{k}{j})^{\frac{1}{d}}. Finally, we present some analogous estimates allowing for an external potential energy field, i.e, $H_{m,Ω}+ V(\bf x)$, for $V(\bf x)$ in certain function classes.

math.SP↗

arXiv2008

Universal bounds and semiclassical estimates for eigenvalues of abstract Schroedinger operators

We prove trace inequalities for a self-adjoint operator on an abstract Hilbert space. These inequalities lead to universal bounds on spectral gaps and on moments of eigenvalues lambda_k that are analogous to those known for Schroedinger operators and the Dirichlet Laplacian, on which the operators of interest are modeled. In addition we produce inequalities that are new even in the model case. These include a family of differential inequalities for generalized Riesz means and theorems stating that arithmetic means of lambda_k^p for p <= 3 are universally bounded from above by multiples of the geometric mean of the lambda_k. For Schroedinger operators and the Dirichlet Laplacian these bounds are Weyl-sharp, i.e., saturated by the standard semiclassical estimates for lambda_k at large k.

math.SP↗

arXiv2007

Differential inequalities for Riesz means and Weyl-type bounds for eigenvalues

We derive differential inequalities and difference inequalities for Riesz means of eigenvalues of the Dirichlet Laplacian, R_σ(z) := \sum_k{(z -λ_k)_+^σ}. Here ${λ_k}_{k=1}^{\infty}$ are the ordered eigenvalues of the Laplacian on a bounded domain $Ω\subset \R^d$, and $x_+ := \max(0, x)$ denotes the positive part of the quantity $x$. As corollaries of these inequalities, we derive Weyl-type bounds on $λ_k$, on averages such as $\bar{λ_k} := {\frac 1 k}\sum_{\ell \le k}λ_\ell$, and on the eigenvalue counting function. For example, we prove that for all domains and all $k \ge j \frac{1+\frac d 2}{1+\frac d 4}$, {\bar{λ_{k}}}/{\bar{λ_{j}}} \le 2 (\frac{1+\frac d 4}{1+\frac d 2})^{1+\frac 2 d}({\frac k j})^{\frac 2 d}.

math.SP↗

arXiv2007

On the critical exponent in an isoperimetric inequality for chords

The problem of maximizing the $L^p$ norms of chords connecting points on a closed curve separated by arclength $u$ arises in electrostatic and quantum--mechanical problems. It is known that among all closed curves of fixed length, the unique maximizing shape is the circle for $1 \le p \le 2$, but this is not the case for sufficiently large values of $p$. Here we determine the critical value $p_c(u)$ of $p$ above which the circle is not a local maximizer finding, in particular, that $p_c(\frac12 L)=\frac52$. This corrects a claim made in \cite{EHL}.

math-ph↗