Luca Guido Molinari
Several sums of Neumann series with Bessel and trigonometric functions are evaluated, as finite sums of trigonometric functions. They arise from a generalization of the Neumann expansion of the eigenstates of the Laplacian in regular polygons.
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Integral representations and asymptotic expansions for the second type Neumann series of Bessel functions of the first kind
本文针对一类含参数的第一类贝塞尔函数第二类诺伊曼级数，推导出完全显式的积分表示，并给出带显式项的渐近展开，从而揭示贝塞尔函数导数级数的渐近行为。
Beyond Three Terms: Exponential Bounds in the Neumann Heat Trace of Polygons
本文通过建立角点、边和内部的热核局部性原理，用模型热核逼近多边形上的热核，给出了诺伊曼热迹渐近展开中指数小余项的显式估计，并讨论了该指数的锐度及与已知结果的关系，最后探讨了推广到Robin边界条件的问题。
We study the Laplace operator with Dirichlet or Neumann boundary condition on polygons in the Euclidean plane. We prove that almost every simply connected polygon with at least four vertices has simple spectrum. We also address the more general case of geodesic polygons in a constant curvature space form.
Given the Laplacian on a planar, convex domain with piecewise linear boundary subject to mixed Dirichlet-Neumann boundary conditions, we provide a sufficient condition for its lowest eigenvalue to dominate the lowest eigenvalue of the Laplacian with the complementary boundary conditions (i.e. with Dirichlet replaced by Neumann and vice versa). The application of this result to triangles gives an affirmative partial answer to a recent conjecture. Moreover, we prove a further observation of similar flavor for right triangles.
We study the existence of Neumann eigenfunctions which do not change sign on the boundary of some special domains. We show that eigenfunctions which are strictly positive on the boundary exist on regular polygons with at least 5 sides, while on equilateral triangles and cubes it is not even possible to find an eigenfunction which is nonnegative on the boundary. We use analytic methods combined with symmetry arguments to prove the result for polygons with six or more sides. The case for the regular pentagon is harder. We develop a validated numerical method to prove this case, which involves it…
Pólya's conjecture for higher-dimensional Neumann balls
该研究针对三维及更高维欧氏球上Laplacian的Neumann特征值，通过发展前期方法，结合变分估计、相位函数界、格点计数技术及计算机辅助论证，成功证明了Pólya猜想。
Bessel functions and Weyl's law for balls and spherical shells
本文研究贝塞尔函数交叉乘积及超球贝塞尔函数导数的零点性质，并应用于球体与球壳上Dirichlet/Neumann拉普拉斯算子的Weyl定律余项，获得所有维度上的新上界。
The paper is devoted to (combinations of) Bessel cross-products that arise from oblique derivative boundary value problems for the Laplacian in a circular annulus. We show that like their Neumann-Laplacian counterpart (and unlike the Dirichlet-Laplacian), they possess two kinds of zeros: those that can be derived by McMahon series and diverge to infinity in the limit, and exceptional ones that remain finite. For both cases we find asymptotic expressions for a fixed oblique angle and vanishing thickness of the annulus. We further present plots of numerically computed zeros and discuss their beh…
In this paper we consider eigenfunctions of the Laplacian on a planar domain with polygonal boundary with Dirichlet, Neumann, or mixed boundary conditions. The main result is a quantitative estimate on the $L^2$ mass of eigenfunctions near a point in terms of the distance to the nearest non-adjacent boundary face. In particular, eigenfunctions cannot concentrate completely at any one single point. The technique of proof is to use the commutator ideas from the recent work of the author \cite{Chr-tri,Chr-simp} on triangles and simplices.
As is known, the free heat-kernel on the integers (a modified Bessel function) is turned into the periodic free heat-kernel on the discrete circle by factoring, giving a pre-image sum. I generalise existing treatments by making the functions periodic up to a phase, thus introducing an extra parameter into the analysis. Identifying the classical paths form with the conventional eigenfunction expression, I find a combinatorial trace identity which allows various Bessel identities to be extracted, such as a generalisation of the Jacobi-Anger expansion.The free Dirichlet, Neumann and hybrid Dirich…
We give an elementary new proof of the hot spots conjecture for L-shaped domains. This result, in addition to a new eigenvalue inequality, allows us to locate the hot spots in Swiss cross translation surfaces. We then prove, in several cases, that first mixed Dirichlet-Neumann eigenfunctions of the Laplacian on L-shaped domains also have no interior critical points. As a combination of these results, we prove the hot spots conjecture for five classes of domains tiled by L-shaped domains, including a class of non-simply connected domains. An interesting feature of the proofs is that we make pos…
On the Pólya conjecture for the Neumann problem in planar convex domains
本文证明了平面凸域中Neumann问题谱计数函数满足Pólya猜想的一个弱化下界，系数为$(2\sqrt{3}j_0^2)^{-1}$，其中$j_0$为Bessel函数$J_0$的首个零点。
Corner contributions to Neumann jump determinants: three model calculations and a BFK conjecture
针对分段实解析切割曲线上诺伊曼跳跃算子的行列式，提出局部角因子猜想。三个模型计算支持该猜想，还将常曲率多边形狄利克雷行列式简化为封闭对偶的行列式。虽奇异异常公式适用于对偶，但非零曲率下显式求值需解决一致化问题，且讨论了与其他工作的联系。
We numerically compute the lowest Laplacian eigenvalues of several two-dimensional shapes with dihedral symmetry at arbitrary precision arithmetic. Our approach is based on the method of particular solutions with domain decomposition. We are particularly interested in asymptotic expansions of the eigenvalues $λ(n)$ of shapes with $n$ edges that are of the form $λ(n) \sim x\sum_{k=0}^{\infty} \frac{C_k(x)}{n^k}$ where $x$ is the limiting eigenvalue for $n\rightarrow \infty$. Expansions of this form have previously only been known for regular polygons with Dirichlet boundary condition and (quite…
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