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Senping Luo

Publications and source records attributed to Senping Luo.

15 recordsLinked to original sources
arXiv2026

On ratios of theta functions

Motivated by the average partition function of c free bosons $($Afhkami-Jeddi et al. \cite{Afhk2021}$)$ and the average of the genus 1 partition function over the Narain moduli space $($Maloney-Witten \cite{Witten2020}$)$, we investigate ratios of theta functions. In this paper, we completely classify the minimizers (or maximizers) for ratios of theta and Epstein zeta functions. We find that the hexagonal lattice plays a pivotal role there. These results have direct applications in conformal and Liouville field theory via partition functions. Additionally, they yield the minima of differences of theta and Epstein zeta functions, which have implications for the mathematics of crystallization and interacting particle theory (\cite{Bet2016,Bet2019AMP}).

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arXiv2024

A new type of minimizers in lattice energy and its application

Let $z\in \mathbb{H}:=\{z= x+ i y\in\mathbb{C}: y>0\}$ and $\mathcal{K}(\alpha;z):=\sum_{ (m,n)\in \mathbb{Z} ^2 }\frac{{\left| mz+n \right|}^2}{{{\Im}(z)}}e^{-\pi\alpha\frac{ \left|mz+n\right|^2}{\Im(z)}}.$ In this paper, we characterize the following minimization problem$:$ $\min_{ \mathbb{H} } \big(\mathcal{K}(\alpha;z)-b\mathcal{K}(2\alpha;z)\big).$ We prove that there exist hexagonal to skinny-rhombic minimizers, which is a novel finding in the literature.

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arXiv2024

Minimizing Lattice Energy and Hexagonal Crystallization

Consider the energy per particle on the lattice given by $\min_{ \Lambda }\sum_{ \mathbb{P}\in \Lambda} \left|\mathbb{P}\right|^4 e^{-\pi \alpha \left|\mathbb{P}\right|^2 }$, where $\alpha >0$ and $\Lambda$ is a two dimensional lattice. We prove that for $\alpha\geq\frac{3}{2}$, among two dimensional lattices with unit density, such energy minimum is attained at $e^{i\frac{\pi}{3}}$, corresponding to the hexagonal lattice. Our result partially answers some open questions proposed by B\'etermin.

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arXiv2023

On a variational model for the continuous mechanics exhibiting hexagonal to square Phase Transitions

Inspired by Conti and Zanzotto \cite{Conti2004A}, we reformulate a simple variational model for reconstructive phase transitions in crystals arising in continuum mechanics in the framework of Landau's theory of phase transition(with slight modification). We provide and prove that this class of modular invariant functions admit exactly hexagonal-square lattices minimizers without passing through rhombic lattices, being the first rigorous result in this regard. Our result gives an affirmative answer to an open problem by in \cite{Conti2004A}. In addition, our result has independent interest from number theory.

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arXiv2023

On lattice hexagonal crystallization for non-monotone potentials

Let $L =\sqrt{\frac{1}{\Im(z)}}\Big({\mathbb Z}\oplus z{\mathbb Z}\Big)$ where $z \in \mathbb{H}=\{z= x+ i y\;\hbox{or}\;(x,y)\in\mathbb{C}: y>0\}$ be the two dimensional lattices with unit density. Assuming that $\alpha\geq1$, we prove that \begin{equation}\aligned\nonumber \min_{L}\sum_{\mathbb{P}\in L, |L|=1}|\mathbb{P}|^2 e^{- \pi\alpha|\mathbb{P}|^2} \endaligned\end{equation} is achieved at hexagonal lattice. More generally we prove that for $\alpha \geq 1$ \begin{equation}\aligned\nonumber \min_{L}\sum_{\mathbb{P}\in L, |L|=1}(|\mathbb{P}|^2-\frac{b}{\alpha}) e^{- \pi\alpha|\mathbb{P}|^2} \endaligned\end{equation} is achieved at hexagonal lattice for $b\leq\frac{1}{2\pi}$ and does not exist for $b>\frac{1}{2\pi}$. As a consequence, we provide two classes of non-monotone potentials which lead to hexagonal crystallization among lattices. Our results partially answer some questions raised in \cite{Oreport, Bet2016, Bet2018, Bet2019AMP} and extend the main results in \cite{LW2022} on minima of difference of two theta functions.

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arXiv2022

On Minima of Difference of Epstein Zeta Functions and Exact Solutions to Lennard-Jones Lattice Energy

Let $\zeta(s,z)=\sum_{(m,n)\in\mathbb{Z}^2\backslash\{0\}}\frac{(\Im(z))^s}{|mz+n|^{2s}}$ be the Eisenstein series/Epstein Zeta function. Motivated by widely used Lennard-Jones potential \begin{equation}\aligned\nonumber \mathcal{V}(|\cdot|^2):=4\varepsilon\Big( (\frac{\sigma}{|\cdot|})^{12}-(\frac{\sigma}{|\cdot|})^{6} \Big), \endaligned\end{equation} in physics, in this paper, we consider the following lattice minimization problem \begin{equation}\aligned\nonumber \min_{z\in\mathbb{H}}\Big(\zeta(6,z)-b\zeta(3,z)\Big), \;\;b=\frac{1}{\sigma^6} \endaligned\end{equation} and completely classify the minimizers for all $b\in \R$. Our results resolve an open problem in Blanc-Lewin \cite{Bla2015}, and a conjecture by B\'etermin \cite{Bet2018}. Furthermore, our method of proofs works for general minimization problem \begin{equation}\aligned\nonumber \min_{z\in\mathbb{H}}\Big(\zeta(s_1,z)-b\zeta(s_2,z)\Big), \;\;s_1>s_2>1 \endaligned\end{equation} which corresponds to general Lennard-Jones potential \begin{equation}\aligned\nonumber \mathcal{V}(|\cdot|^2):=4\varepsilon\Big( (\frac{\sigma}{|\cdot|})^{2s_1}-(\frac{\sigma}{|\cdot|})^{2s_2} \Big),\;\;s_1>s_2>1. \endaligned\end{equation}

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arXiv2022

On minima of difference of theta functions and application to hexagonal crystallization

Let $z=x+iy \in \mathbb{H}:=\{z= x+ i y\in\mathbb{C}: y>0\}$ and $ \theta (\alpha;z)=\sum_{(m,n)\in\mathbb{Z}^2 } e^{-\alpha \frac{\pi }{y }|mz+n|^2}$ be the theta function associated with the lattice $L ={\mathbb Z}\oplus z{\mathbb Z}$. In this paper we consider the following minimization problem of difference of two theta functions \begin{equation}\aligned\nonumber \min_{ \mathbb{H} } \Big(\theta (\alpha; z)-\beta\theta (2\alpha; z)\Big) \endaligned\end{equation} where $\alpha \geq 1$ and $ \beta \in (-\infty, +\infty)$. We prove that there is a critical value $\beta_c=\sqrt2$ (independent of $\alpha$) such that if $\beta\leq\beta_c$, the minimizer is $\frac{1}{2}+i\frac{\sqrt3}{2}$ (up to translation and rotation) which corresponds to the hexagonal lattice, and if $\beta>\beta_c$, the minimizer does not exist. Our result partially answers some questions raised in \cite{Bet2016, Bet2018, Bet2020, Bet2019AMP} and gives a new proof in the crystallization of hexagonal lattice under Yukawa potential.

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arXiv2021

On universally optimal lattice phase transitions and energy minimizers of completely monotone potentials

We consider the minimizing problem for energy functionals with two types of competing particles and completely monotone potential on a lattice. We prove that the minima of sum of two completely monotone functions among lattices is located exactly on a special curve which is part of the boundary of the fundamental region. We also establish a universal result for square lattice being the optimal in certain interval, which is surprising. Our result establishes the hexagonal-rhombic-square-rectangular transition lattice shapes in many physical and biological system (such as Bose-Einstein condensates and two-component Ginzburg-Landau systems). It turns out, our results also apply to locating the minimizers of sum of two Eisenstein series, which is new in number theory.

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arXiv2020

On minima of sum of theta functions and Mueller-Ho Conjecture

Let $z=x+iy \in \mathbb{H}:=\{z= x+ i y\in\mathbb{C}: y>0\}$ and $ \theta (s;z)=\sum_{(m,n)\in\mathbb{Z}^2 } e^{-s \frac{\pi }{y }|mz+n|^2}$ be the theta function associated with the lattice $\Lambda ={\mathbb Z}\oplus z{\mathbb Z}$. In this paper we consider the following pair of minimization problems $$ \min_{ \mathbb{H} } \theta (2;\frac{z+1}{2})+\rho\theta (1;z),\;\;\rho\in[0,\infty),$$ $$ \min_{ \mathbb{H} } \theta (1; \frac{z+1}{2})+\rho\theta (2; z),\;\;\rho\in[0,\infty),$$ where the parameter $\rho\in[0,\infty)$ represents the competition of two intertwining lattices. We find that as $\rho$ varies the optimal lattices admit a novel pattern: they move from rectangular (the ratio of long and short side changes from $\sqrt3$ to 1), square, rhombus (the angle changes from $\pi/2$ to $\pi/3$) to hexagonal; furthermore, there exists a closed interval of $\rho$ such that the optimal lattices is always square lattice. This is in sharp contrast to optimal lattice shapes for single theta function ($\rho=\infty$ case), for which the hexagonal lattice prevails. As a consequence, we give a partial answer to optimal lattice arrangements of vortices in competing systems of Bose-Einstein condensates as conjectured (and numerically and experimentally verified) by Mueller-Ho \cite{Mue2002}.

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arXiv2019

Non-hexagonal lattices from a two species interacting system

A two species interacting system motivated by the density functional theory for triblock copolymers contains long range interaction that affects the two species differently. In a two species periodic assembly of discs, the two species appear alternately on a lattice. A minimal two species periodic assembly is one with the least energy per lattice cell area. There is a parameter $b$ in $[0,1]$ and the type of the lattice associated with a minimal assembly varies depending on $b$. There are several thresholds defined by a number $B=0.1867...$ If $b \in [0, B)$, a minimal assembly is associated with a rectangular lattice whose ratio of the longer side and the shorter side is in $[\sqrt{3}, 1)$; if $b \in [B, 1-B]$, a minimal assembly is associated with a square lattice; if $b \in (1-B, 1]$, a minimal assembly is associated with a rhombic lattice with an acute angle in $[\frac{\pi}{3}, \frac{\pi}{2})$. Only when $b=1$, this rhombic lattice is a hexagonal lattice. None of the other values of $b$ yields a hexagonal lattice, a sharp contrast to the situation for one species interacting systems, where hexagonal lattices are ubiquitously observed.

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arXiv2016

On finite Morse index solutions to the quadharmonic Lane-Emden equation

In this paper, we compute the Joseph-Lundgren exponent for the quadharmonic Lane-Emden equation, derive a monotonicity formula and classify the finite Morse index solution to the following quadharmonic Lane-Emden equation: \noindent \begin{equation}\nonumber \Delta^4 u=|u|^{p-1}u\;\;\;\;\hbox{in}\;\;\;\;\; \R^n. \end{equation} As a byproduct, we also get a monotonicity formula for the quadharmonic maps $ \Delta^4 u=0$.

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arXiv2016

Existence, nonexistence, symmetry and uniqueness of ground state for critical Schr\"odinger system involving Hardy term

We study the following elliptic system with critical exponent: \begin{displaymath} \begin{cases}-\Delta u_j-\frac{\lambda_j}{|x|^2}u_j=u_j^{2^*-1}+\sum\limits_{k\neq j}\beta_{jk}\alpha_{jk}u_j^{\alpha_{jk}-1}u_k^{\alpha_{kj}},\;\;x\in\R^N, u_j\in D^{1,2}(\R^N),\quad u_j>0 \;\; \hbox{in} \quad \R^N\setminus \{0\},\quad j=1,...,r.\end{cases}\end{displaymath} Here $N\geq 3, r\geq2, 2^*=\frac{2N}{N-2}, \lambda_j\in (0, \frac{(N-2)^2}{4})$ for all $ j=1,...,r $; $\beta_{jk}=\beta_{kj}$; \; $\alpha_{jk}>1, \alpha_{kj}>1,$ satisfying $\alpha_{jk}+\alpha_{kj}=2^* $ for all $k\neq j$. Note that the nonlinearities $u_j^{2^*-1}$ and the coupling terms all are critical in arbitrary dimension $N\geq3 $. The signs of the coupling constants $\bb_{ij}$'s are decisive for the existence of the ground state solutions. We show that the critical system with $r\geq 3$ has a positive least energy solution for all $\beta_{jk}>0$. However, there is no ground state solutions if all $\beta_{jk}$ are negative. We also prove that the positive solutions of the system are radially symmetric. Furthermore, we obtain the uniqueness theorem for the case $r\geq 3$ with $N=4$ and the existence theorem when $r=2$ with general coupling exponents.

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arXiv2016

On the triharmonic Lane-Emden equation

We derive a monotonicity formula and classify finite Morse index solutions (positive or sign-changing, radial or not) to the following triharmonic Lane-Emden equation: \begin{equation}\nonumber (-\Delta)^3 u=|u|^{p-1}u \hbox{ in } \mathbb{R}^n, \end{equation} where $p$ is below the Joseph-Lundgren exponent. As a byproduct we also obtain a new monotonicity formula for the triharmonic maps.

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arXiv2016

On the equation $p \frac{\Gamma(\frac{n}{2}-\frac{s}{p-1})\Gamma(s+\frac{s}{p-1})}{\Gamma(\frac{s}{p-1})\Gamma(\frac{n-2s}{2}-\frac{s}{p-1})} =\frac{\Gamma(\frac{n+2s}{4})^2}{\Gamma(\frac{n-2s}{4})^2}$

The note is aimed at giving a complete characterization of the following equation: $$\displaystyle p\frac{\Gamma(\frac{n}{2}-\frac{s}{p-1})\Gamma(s+\frac{s}{p-1})}{\Gamma(\frac{s}{p-1})\Gamma(\frac{n-2s}{2}-\frac{s}{p-1})} =\frac{\Gamma(\frac{n+2s}{4})^2}{\Gamma(\frac{n-2s}{4})^2}.$$ The method is based on some key transformation and the properties of the Gamma function. Applications to fractional nonlinear Lane-Emden equations will be given.

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