arXiv subjects

Lotfi Hermi

Publications and source records attributed to Lotfi Hermi.

7 recordsLinked to original sources

arXiv2022

On a Nonlocal Integral Operator Commuting with the Laplacian and the Sturm-Liouville Problem I: Low Rank Perturbations of the Operator

We reformulate all general real coupled self-adjoint boundary value problems as integral operators and show that they are all finite rank perturbations of the free space Green's function on the real line. This free space Green's function corresponds to the nonlocal boundary value problem proposed earlier by Saito [N. Saito, Appl. Comput. Harmonic Anal., 25, 68--97 (2008)]. We prove these perturbations to be polynomials of rank up to 4. They encapsulate in a fundamental way the corresponding boundary conditions.

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arXiv2018

Trace formulas applied to the Riemann $ζ$-function

We use a spectral theory perspective to reconsider properties of the Riemann zeta function. In particular, new integral representations are derived and used to present its value at odd positive integers.

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arXiv2010

On Rayleigh-Type Formulas for a Non-local Boundary Value Problem Associated with an Integral Operator Commuting with the Laplacian

In this article we prove the existence, uniqueness, and simplicity of a negative eigenvalue for a class of integral operators whose kernel is of the form $|x-y|^\rho$, $0 < \rho \leq 1$, $x, y \in [-a, a]$. We also provide two different ways of producing recursive formulas for the Rayleigh functions (i.e., recursion formulas for power sums) of the eigenvalues of this integral operator when $\rho=1$, providing means of approximating this negative eigenvalue. These methods offer recursive procedures for dealing with the eigenvalues of a one-dimensional Laplacian with non-local boundary conditions which commutes with an integral operator having a harmonic kernel. The problem emerged in recent work by one of the authors [45]. We also discuss extensions in higher dimensions and links with distance matrices.

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arXiv2007

On Harrell-Stubbe Type Inequalities for the Discrete Spectrum of a Self-Adjoint Operator

We produce a new proof and extend results by Harrell and Stubbe for the discrete spectrum of a self-adjoint operator. An abstract approach--based on commutator algebra, the Rayleigh-Ritz principle, and an ``optimal'' usage of the Cauchy-Schwarz inequality--is used to produce ``parameter-free'', ``projection-free'' versions of their theorems. We also analyze the strength of the various inequalities that ensue. The results contain classical bounds for the eigenvalues. Extensions of a variety of inequalities à la Harrell-Stubbe are illustrated for both geometric and physical problems.

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arXiv2007

On Riesz Means of Eigenvalues

In this article we prove the equivalence of certain inequalities for Riesz means of eigenvalues of the Dirichlet Laplacian with a classical inequality of Kac. Connections are made via integral transforms including those of Laplace, Legendre, Weyl, and Mellin, and the Riemann-Liouville fractional transform. We also prove new universal eigenvalue inequalities and monotonicity principles for Dirichlet Laplacians as well as certain Schrödinger operators. At the heart of these inequalities are calculations of commutators of operators, sum rules, and monotonic properties of Riesz means. In the course of developing these inequalities we prove new bounds for the partition function and the spectral zeta function (cf. Corollaries 3.5-3.7) and conjecture about additional bounds.

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arXiv2007

Two new Weyl-type bounds for the Dirichlet Laplacian

In this paper, we prove two new Weyl-type upper estimates for the eigenvalues of the Dirichlet Laplacian. As a consequence, we obtain the following {\em lower} bounds for its counting function. For $\la\ge \la_1$, one has N(\la) > \dfrac{2}{n+2} \dfrac{1}{H_n} (\la-\la_1)^{n/2} \la_1^{-n/2}, and N(\la) > (\dfrac{n+2}{n+4})^{n/2} \dfrac{1}{H_n} (\la-(1+4/n) \la_1)^{n/2} \la_1^{-n/2}, where H_n=\dfrac{2 n}{j_{n/2-1,1}^2 J_{n/2}^2(j_{n/2-1,1})} is a constant which depends on $n$, the dimension of the underlying space, and Bessel functions and their zeros.

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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}.

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