DISTRIBUTION OF INTEGER LATTICE POINTS IN A BALL CENTRED AT A DIOPHANTINE POINT

We study the variance of the fluctuations in the number of lattice points in a ball and in a thin spherical shell of large radius centred at a Diophantine point

Functions of self-adjoint operators in ideals of compact operators

For self-adjoint operators A,B, a bounded operator J, and a function f:R→C, we obtain bounds in quasi-normed ideals of compact operators for the difference f(A)J−Jf(B) in terms of the operator AJ−JB. The focus is on functions f that are smooth everywhere except for finitely many points. A typical example is the function f(t)=|t|γ with γ∈(0,1). The obtained results are applied to derive a two-term quasi-classical asymptotic formula for the trace trf(S) with S being a Wiener–Hopf operator with a discontinuous symbol

Eigenvalue estimates for the one-particle density matrix

It is shown that the eigenvalues \l_k, k=1, 2, \dots,\l k ​ ,k=1,2,…, of the one-particle density matrix satisfy the bound \l_k\le C k^{-8/3}\l k ​ ≤Ck −8/3 with a positive constant CC

Quasi-classical asymptotics for functions of Wiener-Hopf operators: smooth vs non-smooth symbols

We consider functions of Wiener–Hopf type operators on the Hilbert space L2(Rd). It has been known for a long time that the quasi-classical asymptotics for traces of resulting operators strongly depend on the smoothness of the symbol: for smooth symbols the expansion is power-like, whereas discontinuous symbols (e.g. indicator functions) produce an extra logarithmic factor. We investigate the transition regime by studying symbols depending on an extra parameter T≥0 in such a way that the symbol tends to a discontinuous one as T→0. The main result is two-parameter asymptotics (in the quasi-classical parameter and in T), describing a transition from the smooth case to the discontinuous one. The obtained asymptotic formulas are used to analyse the low-temperature scaling limit of the spatially bipartite entanglement entropy of thermal equilibrium states of non-interacting fermions

Multichannel scattering theory for Toeplitz operators with piecewise continuous symbols

Self-adjoint Toeplitz operators have purely absolutely continuous spectrum. For Toeplitz operators T with piecewise continuous symbols, we suggest a further spectral classification determined by propagation properties of the operator T , that is, by the behavior of exp ( − i T t ) f for t → ± ∞ . It turns out that the spectrum is naturally partitioned into three disjoint subsets: thick, thin and mixed spectra. On the thick spectrum, the propagation properties are modeled by the continuous part of the symbol, whereas on the thin spectrum, the model operator is determined by the jumps of the symbol. On the mixed spectrum, these two types of the asymptotic evolution of exp ( − i T t ) f coexist. This classification is justified in the framework of scattering theory. We prove the existence of wave operators that relate the model operators with the Toeplitz operator T . The ranges of these wave operators are pairwise orthogonal, and their orthogonal sum exhausts the whole space; i.e., the set of these wave operators is asymptotically complete

Sr, Nd, and Pb isotope evidence for a mantle origin of alkali chlorides and carbonates in the Udachnaya kimberlite, Siberia

The kimberlite rocks of the Udachnaya-East pipe (Siberia) are uniquely fresh and contain very high abundances of primary volatiles (Cl, CO2, S). Alkali elements and chlorine are extremely abundant in the reconstructed kimberlite melt compositions, and this enrichment is very important for our understanding of deep-mantle melting and melt transport. Here we present new isotopic data that confirm a mantle origin for these kimberlitic chlorides and carbonates, and constrain the kimberlite emplacement age as ca. 347 Ma. The initial Nd and Ph isotope ratios in a large salt aggregate, in a CI-S-enriched water leachate of the groundmass, and in the silicate fraction of the groundmass are very similar (epsilon(Nd) = +3 to +4, Pb-206/Pb-204 = 18.6, Pb-207/Pb-204 = 15.53), implying a comagmatic origin of the chlorides and carbonates and the silicates. Combined Sr, Nd, and Ph isotope data are used to rule out any significant contributions to the kimberlite chlorine budget from crustal sources, such as the Cambrian evaporite sequences of the Siberian platform. Our data support the interpretation that exsolved Na-K chloride and Na-K-Ca carbonate formed directly from original uncontaminated kimberlite magma. High Cl abundances in kimberlites suggest the presence of a Cl-rich reservoir in the deep sublithospheric mantl

Experimental and petrological studies of melt inclusions in phenocrysts from mantle-derived magmas: an overview of techniques, advantages and complications

Melt inclusions in phenocrysts are a potentially powerful tool in petrological research that can provide the only direct information available on the physical parameters ( P, T and melt composition) of crystallisation at various stages in the evolution of magmatic systems. However, melt inclusions also differ in principle from other parts of the magmatic system in that their composition, after trapping, may be controlled by the composition of the host phenocryst and therefore the direct application of our understanding of macro-scale magmatic processes to the interpretation of melt inclusion data can lead to erroneous conclusions. Our results indicate that the compositions of melt inclusions in early formed phenocrysts (olivine, pyroxene, plagioclase and spinel), often of most interest in petrological studies, can be affected by processes such as volatile dissociation, oxidation and/or partial re-equilibration with their host, both during natural cooling and homogenisation experiments. In particular, melt inclusions in all minerals are prone to hydrogen diffusion into or out of the inclusions after trapping and prior to eruption, and during homogenisation experiments. If not taken into account, this can significantly affect the crystallisation temperatures derived from the homogenisation experiments. Melt inclusions in highmagnesian olivine phenocrysts commonly have lower Fe contents compared to the initially trapped composition due to reequilibration with the host at lower temperatures. This often leads to the appearance of sulphide globules and in some cases high-magnesian clinopyroxene daughter crystals, and may cause an increase in the oxidation state of the inclusions. Homogenised melt inclusions in plagioclase phenocrysts in MORB usually have lower Ti and Fe, and higher Si contents compared to the melt composition at the moment of trapping. However, homogenisation experiments can provide reliable estimates of trapping temperature and the MgO, Al2O3, CaO, Na2O, and K2O contents of the host magma at the moment of trapping. Some of these processes can be identified by observing the behaviour of melt inclusions during homogenisation experiments using low-inertia visually controlled heating stages, and their effects can be minimised by using appropriate experimental conditions as determined by kinetic experiments, ideally completed for each phenocryst type in every sample. We also discuss general aspects of melt inclusion studies aimed at recovering H2O content of primary mantle-derived magmas and demonstrate that, in cases of low-pressure crystallisation, it is important to identify the first liquidus (most magnesian) olivine that crystallised from these magmas

Trace formulas for Wiener-Hopf operators with applications to entropies of free fermionic equilibrium states

We consider non-smooth functions of (truncated) Wiener–Hopf type operators on the Hilbert space L2(Rd) . Our main results are uniform estimates for trace norms ( d≥1 ) and quasiclassical asymptotic formulas for traces of the resulting operators ( d=1 ). Here, we follow Harold Widom's seminal ideas, who proved such formulas for smooth functions decades ago. The extension to non-smooth functions and the uniformity of the estimates in various (physical) parameters rest on recent advances by one of the authors (AVS). We use our results to obtain the large-scale behaviour of the local entropy and the spatially bipartite entanglement entropy (EE) of thermal equilibrium states of non-interacting fermions in position space Rd ( d≥1 ) at positive temperature, T>0 . In particular, our definition of the thermal EE leads to estimates that are simultaneously sharp for small T and large scaling parameter α>0 provided that the product Tα remains bounded from below. Here α is the reciprocal quasiclassical parameter. For d=1 we obtain for the thermal EE an asymptotic formula which is consistent with the large-scale behaviour of the ground-state EE (at T=0 ), previously established by the authors for d≥1

Wiener–Hopf Operators in Higher Dimensions: The Widom Conjecture for Piece-Wise Smooth Domains

We prove a two-term quasi-classical trace asymptotic formula for the functions of multi-dimensional Wiener–Hopf operators with discontinuous symbols. The discontinuities occur on surfaces which are assumed to be piece-wise smooth. Such a two-term formula was conjectured by H. Widom (On a Class of Integral Operators with Discontinuous Symbol, Toeplitz centennial (Tel Aviv, 1981), pp. 477–500. Operator Theory: Advances and Applications, vol. 4. Birkhäuser, Basel, 1982), and proved by A. V. Sobolev for smooth surfaces in 2009 (Mem. AMS 222(1043), 2013)

On Szeg Formulas for Truncated Wiener-Hopf Operators

We consider functions of multi-dimensional versions of truncated Wiener–Hopf operators with smooth symbols, and study the scaling asymptotics of their traces. The obtained results extend the asymptotic formulas obtained by H. Widom in the 1980’s to non-smooth functions, and non-smooth truncation domains. The obtained asymptotic formulas are used to analyse the scaling limit of the spatially bipartite entanglement entropy of thermal equilibrium states of noninteracting fermions at positive temperature