Related papers: On the spectral curve of the difference Lam\'e ope…
We consider $\Gamma=(X,E)$ a dual polar graph and we give a tight frame on each eigenspace of the Laplacian operator associated to $\Gamma$. We compute the constants associated to each tight frame and as an application we give a formula for…
The Eynard-Orantin recursion formula provides an effective tool for certain enumeration problems in geometry. The formula requires a spectral curve and the recursion kernel. We present a uniform construction of the spectral curve and the…
In this paper we construct the quantum spectral curve for the quantum dynamical elliptic gl(n) Gaudin model. We realize it considering a commutative family corresponding to the Felder's elliptic quantum group and taking the appropriate…
In this Thesis, we study the behavior of spectral curves of quantum graphs under certain families of vertex conditions, called the $\delta_s$ family, which we define in this work. We focus on studying two main quantities related to the…
There is a well known analogy between the Laughlin trial wave function for the fractional quantum Hall effect, and the Boltzmann factor for the two-dimensional one-component plasma. The latter requires analytic continuation beyond the…
Given a regular (connected) graph $\Gamma=(X,E)$ with adjacency matrix $A$, $d+1$ distinct eigenvalues, and diameter $D$, we give a characterization of when its distance matrix $A_D$ is a polynomial in $A$, in terms of the adjacency…
This survey is two-fold. We first report new progress on the spectral extremal results on the Tur\'{a}n type problems in graph theory. More precisely, we shall summarize the spectral Tur\'{a}n function in terms of the adjacency spectral…
In this paper we obtain several tight bounds on different types of alliance numbers of a graph: (global) defensive alliance number, global offensive alliance number and global dual alliance number. In particular, we investigate the…
We analyze the spectrum of the operator $\Delta^{-1} [\nabla \cdot (K\nabla u)]$, where $\Delta$ denotes the Laplacian and $K=K(x,y)$ is a symmetric tensor. Our main result shows that this spectrum can be derived from the spectral…
We establish a general link between integrable systems in algebraic geometry (expressed as Jacobian flows on spectral curves) and soliton equations (expressed as evolution equations on flat connections). Our main result is a natural…
Let $X=G/H$ be a reductive homogeneous space with $H$ noncompact, endowed with a $G$-invariant pseudo-Riemannian structure. Let $L$ be a reductive subgroup of $G$ acting properly on $X$ and $\Gamma$ a torsion-free discrete subgroup of $L$.…
We study measure perturbations of the Laplacian in $L^2(\R^2)$ supported by an infinite curve $\Gamma$ in the plane which is asymptotically straight in a suitable sense. We show that if $\Gamma$ is not a straight line, such a ``leaky…
We propose an alternative variational principle whose critical point is the algebraic plane curve associated to a matrix model (the spectral curve, i.e. the large $N$ limit of the resolvent). More generally, we consider a variational…
Let $\Gamma$ be a connected regular graph with an eigenvalue $\lambda$ and corresponding idempotent $E_{\lambda}$. Let ${\cal E}_{\lambda}=\langle J,E_{\lambda}\rangle^\circ$ be the algebra generated by $J$ and $E_\lambda$ with respect to…
Let $\Gamma < G := \operatorname{SO}(d+1, 1)$ for $d \geq 1$ be a Zariski dense, geometrically finite, discrete subgroup with critical exponent strictly greater than $d/2$. We show that $L^2(\Gamma\backslash G)$ admits a strong spectral…
The purpose of this note is to announce the results of our investigation into the role played by the continuous spectrum in the development of the Selberg trace formula vis-\`a-vis a pair $(G,\Gamma)$. For the sake of simplicity, we shall…
We introduce and study certain hyperbolic versions of automorphic Lie algebras related to the modular group. Let $\Gamma$ be a finite index subgroup of $\mathrm{SL}(2,\mathbb{Z})$ with an action on a complex simple Lie algebra $\mathfrak…
As a main result, we characterize prime spectra of abelian lattice ordered groups. Further we introduce some categories based on spectral spaces, lattices and Priestley spaces, and we relate these categories with each other and with the…
The Manin constant $c$ of an elliptic curve $E$ over $\mathbb{Q}$ is the nonzero integer that scales the differential $\omega_f$ determined by the normalized newform $f$ associated to $E$ into the pullback of a N\'{e}ron differential under…
We extend Sklyanin's method of separation of variables to quantum integrable models associated to elliptic curves. After reviewing the differential case, the elliptic Gaudin model studied by Enriquez, Feigin and Rubtsov, we consider the…