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Register a new userSome convexity and monotonicity results of trace functionals
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Haonan Zhang
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In this paper, we prove the convexity of trace functionals $$(A,B,C)mapsto text{Tr}|B^{p}AC^{q}|^{s},$$ for parameters $(p,q,s)$ that are best possible. We also obtain the monotonicity under unital completely positive trace preserving maps of trace functionals of this type. As applications, we extend some results in cite{HP12quasi,CFL16some} and resolve a conjecture in cite{RZ14}. Other conjectures in cite{RZ14} will also be discussed. We also show that some related trace functionals are not concave in general. Such concavity results were expected to hold in cite{Chehade20} to derive equality conditions of data processing inequalities for $alpha-z$ Renyi relative entropies.
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A version of Connes trace formula allows to associate a measure on the essential spectrum of a Schrodinger operator with bounded potential. In solid state physics there is another celebrated measure associated with such operators --- the density of states. In this paper we demonstrate that these two measures coincide. We show how this equality can be used to give explicit formulae for the density of states in some circumstances.
Consider a function $F(X,Y)$ of pairs of positive matrices with values in the positive matrices such that whenever $X$ and $Y$ commute $F(X,Y)= X^pY^q.$ Our first main result gives conditions on $F$ such that ${rm Tr}[ X log (F(Z,Y))] leq {rm Tr}[X(plog X + q log Y)]$ for all $X,Y,Z$ such that ${rm Tr} Z = {rm Tr} X$. (Note that $Z$ is absent from the right side of the inequality.) We give several examples of functions $F$ to which the theorem applies. Our theorem allows us to give simple proofs of the well known logarithmic inequalities of Hiai and Petz and several new generalizations of them which involve three variables $X,Y,Z$ instead of just $X,Y$ alone. The investigation of these logarithmic inequalities is closely connected with three quantum relative entropy functionals: The standard Umegaki quantum relative entropy $D(X||Y) = {rm Tr} [X(log X-log Y])$, and two others, the Donald relative entropy $D_D(X||Y)$, and the Belavkin-Stasewski relative entropy $D_{BS}(X||Y)$. They are known to satisfy $D_D(X||Y) leq D(X||Y)leq D_{BS}(X||Y)$. We prove that the Donald relative entropy provides the sharp upper bound, independent of $Z$, on ${rm Tr}[ X log (F(Z,Y))]$ in a number of cases in which $(Z,Y)$ is homogeneous of degree $1$ in $Z$ and $-1$ in $Y$. We also investigate the Legendre transforms in $X$ of $D_D(X||Y)$ and $D_{BS}(X||Y)$, and show how our results for these lead to new refinements of the Golden-Thompson inequality.
We establish quantitative bounds on the rate of approach to equilibrium for a system with infinitely many degrees of freedom evolving according to a one-dimensional focusing nonlinear Schrodinger equation with diffusive forcing. Equilibrium is described by a generalized grand canonical ensemble. Our analysis also applies to the easier case of defocusing nonlinearities
In a recent paper we studied an equation (called the simple equation) introduced by one of us in 1963 for an approximate correlation function associated to the ground state of an interacting Bose gas. Solving the equation yields a relation between the density $rho$ of the gas and the energy per particle. Our construction of solutions gave a well-defined function $rho(e)$ for the density as a function of the energy $e$. We had conjectured that $rho(e)$ is a strictly monotone increasing function, so that it can be inverted to yield the strictly monotone increasing function $e(rho)$. We had also conjectured that $rho e(rho)$ is convex as a function of $rho$. We prove both conjectures here for small densities, the context in which they have the most physical relevance, and the monotonicity also for large densities. Both conjectures are grounded in the underlying physics, and their proof provides further mathematical evidence for the validity of the assumptions underlying the derivation of the simple equation, at least for low or high densities, if not intermediate densities, although the equation gives surprisingly good predictions for all densities $rho$. Another problem left open in our previous paper was whether the simple equation could be used to compute accurate predictions of observables other than the energy. Here, we provide a recipe for computing predictions for any one- or two-particle observables for the ground state of the Bose gas. We focus on the condensate fraction and the momentum distribution, and show that they have the same low density asymptotic behavior as that predicted for the Bose gas. Along with the computation of the low density energy of the simple equation in our previous paper, this shows that the simple equation reproduces the known and conjectured properties of the Bose gas at low densities.
For a given Lipschitz domain $Omega$, it is a classical result that the trace space of $W^{1,p}(Omega)$ is $W^{1-1/p,p}(partialOmega)$, namely any $W^{1,p}(Omega)$ function has a well-defined $W^{1-1/p,p}(partialOmega)$ trace on its codimension-1 boundary $partialOmega$ and any $W^{1-1/p,p}(partialOmega)$ function on $partialOmega$ can be extended to a $W^{1,p}(Omega)$ function. Recently, Dyda and Kassmann (2019) characterize the trace space for nonlocal Dirichlet problems involving integrodifferential operators with infinite interaction ranges, where the boundary datum is provided on the whole complement of the given domain $mathbb{R}^dbackslashOmega$. In this work, we study function spaces for nonlocal Dirichlet problems with a finite range of nonlocal interactions, which naturally serves a bridging role between the classical local PDE problem and the nonlocal problem with infinite interaction ranges. For these nonlocal Dirichlet problems, the boundary conditions are normally imposed on a region with finite thickness volume which lies outside of the domain. We introduce a function space on the volumetric boundary region that serves as a trace space for these nonlocal problems and study the related extension results. Moreover, we discuss the consistency of the new nonlocal trace space with the classical $W^{1-1/p,p}(partialOmega)$ space as the size of nonlocal interaction tends to zero. In making this connection, we conduct an investigation on the relations between nonlocal interactions on a larger domain and the induced interactions on its subdomain. The various forms of trace, embedding and extension theorems may then be viewed as consequences in different scaling limits.
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