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Register a new userRepeated minimizers of $p$-frame energies
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For a collection of $N$ unit vectors $mathbf{X}={x_i}_{i=1}^N$, define the $p$-frame energy of $mathbf{X}$ as the quantity $sum_{i eq j} |langle x_i,x_j rangle|^p$. In this paper, we connect the problem of minimizing this value to another optimization problem, so giving new lower bounds for such energies. In particular, for $p<2$, we prove that this energy is at least $2(N-d) p^{-frac p 2} (2-p)^{frac {p-2} 2}$ which is sharp for $dleq Nleq 2d$ and $p=1$. We prove that for $1leq m<d$, a repeated orthonormal basis construction of $N=d+m$ vectors minimizes the energy over an interval, $pin[1,p_m]$, and demonstrate an analogous result for all $N$ in the case $d=2$. Finally, in connection, we give conjectures on these and other energies.
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We provide new answers about the placement of mass on spheres so as to minimize energies of pairwise interactions. We find optimal measures for the $p$-frame energies, i.e. energies with the kernel given by the absolute value of the inner product raised to a positive power $p$. Application of linear programming methods in the setting of projective spaces allows for describing the minimizing measures in full in several cases: we show optimality of tight designs and of the $600$-cell for several ranges of $p$ in different dimensions. Our methods apply to a much broader class of potential functions, those which are absolutely monotonic up to a particular order as functions of the cosine of the geodesic distance. In addition, a preliminary numerical study is presented which suggests optimality of several other highly symmetric configurations and weighted designs in low dimensions. In one case we improve the best known lower bounds on a minimal sized weighted design in $mathbb{CP}^4$. All these results point to the discreteness of minimizing measures for the $p$-frame energy with $p$ not an even integer.
The space of constructible functions form a dense subspace of the space of generalized valuations. In this note we prove a somewhat stronger property that the sequential closure, taken sufficiently many (in fact, infinitely many) times, of the former space is equal to the latter one. This stronger property is necessary for some applications in the theory of valuations on manifolds.
We consider the general physical situation of a quantum system $H_0$ interacting with a chain of exterior systems $bigotimes_N H$, one after the other, during a small interval of time $h$ and following some Hamiltonian $H$ on $H_0 otimes H$. We discuss the passage to the limit to continuous interactions ($h to 0$) in a setup which allows to compute the limit of this Hamiltonian evolution in a single state space: a continuous field of exterior systems $otimes_{R} H$. Surprisingly, the passage to the limit shows the necessity for 3 different time scales in $H$. The limit evolution equation is shown to spontaneously produce quantum noises terms: we obtain a quantum Langevin equation as limit of the Hamiltonian evolution. For the very first time, these quantum Langevin equations are obtained as the effective limit from repeated to continuous interactions and not only as a model. These results justify the usual quantum Langevin equations considered in continual quantum measurement or in quantum optics. We show that the three time scales correspond to the normal regime, the weak coupling limit and the low density limit. Our approach allows to consider these two physical limits altogether for the first time. Their combination produces an effective Hamiltonian on the small system, which had never been described before. We apply these results to give an Hamiltonian description of the von Neumann measurement. We also consider the approximation of continuous time quantum master equations by discrete time ones. In particular we show how any Lindblad generator is obtained as the limit of completely positive maps.
Let $mathcal F_0={f_i}_{iinmathbb{I}_{n_0}}$ be a finite sequence of vectors in $mathbb C^d$ and let $mathbf{a}=(a_i)_{iinmathbb{I}_k}$ be a finite sequence of positive numbers. We consider the completions of $cal F_0$ of the form $cal F=(cal F_0,cal G)$ obtained by appending a sequence $cal G={g_i}_{iinmathbb{I}_k}$ of vectors in $mathbb C^d$ such that $|g_i|^2=a_i$ for $iinmathbb{I}_k$, and endow the set of completions with the metric $d(cal F,tilde {mathcal F}) =max{ ,|g_i-tilde g_i|: iinmathbb{I}_k}$ where $tilde {cal F}=(cal F_0,,tilde {cal G})$. In this context we show that local minimizers on the set of completions of a convex potential $text{P}_varphi$, induced by a strictly convex function $varphi$, are also global minimizers. In case that $varphi(x)=x^2$ then $text{P}_varphi$ is the so-called frame potential introduced by Benedetto and Fickus, and our work generalizes several well known results for this potential. We show that there is an intimate connection between frame completion problems with prescribed norms and frame operator distance (FOD) problems. We use this connection and our results to settle in the affirmative a generalized version of Strawns conjecture on the FOD.
We consider a version of the fractional Sobolev inequality in domains and study whether the best constant in this inequality is attained. For the half-space and a large class of bounded domains we show that a minimizer exists, which is in contrast to the classical Sobolev inequalities in domains.
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