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Register a new userDiscrete Energy Asymptotics on a Riemannian circle
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We derive the complete asymptotic expansion in terms of powers of $N$ for the geodesic $f$-energy of $N$ equally spaced points on a rectifiable simple closed curve $Gamma$ in ${mathbb R}^p$, $pgeq2$, as $N to infty$. For $f$ decreasing and convex, such a point configuration minimizes the $f$-energy $sum_{j eq k}f(d(mathbf{x}_j, mathbf{x}_k))$, where $d$ is the geodesic distance (with respect to $Gamma$) between points on $Gamma$. Completely monotonic functions, analytic kernel functions, Laurent series, and weighted kernel functions $f$ are studied. % Of particular interest are the geodesic Riesz potential $1/d^s$ ($s eq 0$) and the geodesic logarithmic potential $log(1/d)$. By analytic continuation we deduce the expansion for all complex values of $s$.
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We prove a conjecture of Ambrus, Ball and Erd{e}lyi that equally spaced points maximize the minimum of discrete potentials on the unit circle whenever the potential is of the form sum_{k=1}^n f(d(z,z_k)), where $f:[0,pi]to [0,infty]$ is non-increasing and strictly convex and $d(z,w)$ denotes the geodesic distance between $z$ and $w$ on the circle.
We consider discrete spectra of bound states for non-relativistic motion in attractive potentials V_{sigma}(x) = -|V_{0}| |x|^{-sigma}, 0 < sigma leq 2. For these potentials the quasiclassical approximation for n -> infty predicts quantized energy levels e_{sigma}(n) of a bounded spectrum varying as e_{sigma}(n) ~ -n^{-2sigma/(2-sigma)}. We construct collective quantum states using the set of wavefunctions of the discrete spectrum taking into account this asymptotic behaviour. We give examples of states that are normalizable and satisfy the resolution of unity, using explicit positive functions. These are coherent states in the sense of Klauder and their completeness is achieved via exact solutions of Hausdorff moment problems, obtained by combining Laplace and Mellin transform methods. For sigma in the range 0<sigmaleq 2/3 we present exact implementations of such states for the parametrization sigma = 2(k-l)/(3k-l), with k and l positive integers satisfying k>l.
Let $Lambda$ be a lattice in ${bf R}^d$ with positive co-volume. Among $Lambda$-periodic $N$-point configurations, we consider the minimal renormalized Riesz $s$-energy $mathcal{E}_{s,Lambda}(N)$. While the dominant term in the asymptotic expansion of $mathcal{E}_{s,Lambda}(N)$ as $N$ goes to infinity in the long range case that $0<s<d$ (or $s=log$) can be obtained from classical potential theory, the next order term(s) require a different approach. Here we derive the form of the next order term or terms, namely for $s>0$ they are of the form $C_{s,d}|Lambda|^{-s/d}N^{1+s/d}$ and $-frac{2}{d}Nlog N+left(C_{log,d}-2zeta_{Lambda}(0)right)N$ where we show that the constant $C_{s,d}$ is independent of the lattice $Lambda$.
We survey known results and present estimates and conjectures for the next-order term in the asymptotics of the optimal logarithmic energy and Riesz $s$-energy of $N$ points on the unit sphere in $mathbb{R}^{d+1}$, $dgeq 1$. The conjectures are based on analytic continuation assumptions (with respect to $s$) for the coefficients in the asymptotic expansion (as $Nto infty$) of the optimal $s$-energy.
Given a 3-dimensional Riemannian manifold (M,g), we investigate the existence of positive solutions of the nonlinear Klein-Gordon-Maxwell system and nonlinear Schroedinger-Maxwell system with subcritical nonlinearity. We prove that the number of one peak solutions depends on the topological properties of the manifold M, by means of the Lusternik Schnirelmann category.
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