Spectral Theory

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New submissions for Wed, 29 May 24

[1]  arXiv:2405.17715 [pdf, ps, other]

Title: Modified Jost solutions of Schrödinger operators with locally $H^{-1}$ potentials

Authors: Milivoje Lukić, Xingya Wang

Subjects: Spectral Theory (math.SP)

We study Jost solutions of Schr\"odinger operators with potentials which decay with respect to a local $H^{-1}$ Sobolev norm; in particular, we generalize to this setting the results of Christ--Kiselev for potentials between the integrable and square-integrable rates of decay, proving existence of solutions with WKB asymptotic behavior on a large set of positive energies. This applies to new classes of potentials which are not locally integrable, or have better decay properties with respect to the $H^{-1}$ norm due to rapid oscillations.

[2]  arXiv:2405.18079 [pdf, other]

Title: On the (growing) gap between Dirichlet and Neumann eigenvalues

Authors: Pedro Freitas

Comments: 15 pages, 2 figures

Subjects: Spectral Theory (math.SP)

We provide an answer to a question raised by Levine and Weinberger in their $1986$ paper concerning the difference between Dirichlet and Neumann eigenvalues of the Laplacian on bounded domains in $\R^{n}$. More precisely, we show that for a certain class of domains there exists a sequence $p(k)$ such that $\lambda_{k}\geq \mu_{k+ p(k)}$ for sufficiently large $k$. This sequence, which is given explicitly, grows with $k^{1-1/n}$ as $k$ goes to infinity, which we conjecture to be optimal, and may be chosen independently of the domain. We also prove the existence of a sequence, now not given explicitly and only of order $k^{1-3/n}$ but valid for bounded Lipschitz domains in $\R^{n} (n\geq4)$, for which a similar inequality holds for all $k$. From these results and the analysis of some particular examples we formulate a conjecture for general Euclidean domains.

[3]  arXiv:2405.18154 [pdf, ps, other]

Title: On the Laplace operator with a weak magnetic field in exterior domains

Authors: Ayman Kachmar, Vladimir Lotoreichik, Mikael Sundqvist

Comments: 24 pages

Subjects: Spectral Theory (math.SP); Mathematical Physics (math-ph); Analysis of PDEs (math.AP)

We study the magnetic Laplacian in a two-dimensional exterior domain with Neumann boundary condition and uniform magnetic field. For the exterior of the disk we establish accurate asymptotics of the low-lying eigenvalues in the weak magnetic field limit. For the exterior of a star-shaped domain, we obtain an asymptotic upper bound on the lowest eigenvalue in the weak field limit, involving the $4$-moment, and optimal for the case of the disk. Moreover, we prove that, for moderate magnetic fields, the exterior of the disk is a local maximizer for the lowest eigenvalue under a $p$-moment constraint.

Cross-lists for Wed, 29 May 24

[4]  arXiv:2405.18233 (cross-list from math.DG) [pdf, ps, other]

Title: First Eigenvalue of Jacobi operator and Rigidity Results for Constant Mean Curvature Hypersurfaces

Authors: Marcio Batista, Marcos P. Cavalcante, Luiz R. Melo

Comments: 17 pages. Comments welcome!

Subjects: Differential Geometry (math.DG); Analysis of PDEs (math.AP); Spectral Theory (math.SP)

In this paper, we obtain geometric upper bounds for the first eigenvalue $\lambda_1^J$ of the Jacobi operator for both closed and compact with boundary hypersurfaces having constant mean curvature (CMC). As an application, we derive new rigidity results for the area of CMC hypersurfaces under suitable conditions on $\lambda_1^J$ and the curvature of the ambient space. We also address the Jacobi-Steklov problem, proving geometric upper bounds for its first eigenvalue $\sigma_1^J$ and deriving rigidity results related to the length of the boundary. Additionally, we present some results in higher dimensions related to the Yamabe invariants.

Replacements for Wed, 29 May 24

[5]  arXiv:2301.03555 (replaced) [pdf, other]

Title: Energy Distribution for Dirichlet Eigenfunctions on Right Triangles

Authors: Hans Christianson, Daniel Pezzi

Subjects: Analysis of PDEs (math.AP); Spectral Theory (math.SP)

[6]  arXiv:2405.03046 (replaced) [pdf, ps, other]

Title: A note on the Huijsmans-de Pagter problem on finite dimensional ordered vector spaces

Authors: Catalin Badea, Jochen Glück

Comments: 8 pages ; to appear in Analysis Mathematica

Subjects: Functional Analysis (math.FA); Spectral Theory (math.SP)

[7]  arXiv:2405.17200 (replaced) [pdf, ps, other]

Title: A Mathematical Theory of Integer Quantum Hall Effect in Photonics

Authors: Jiayu Qiu, Hai Zhang

Subjects: Mathematical Physics (math-ph); Analysis of PDEs (math.AP); Spectral Theory (math.SP)

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