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A class of purely-nonperturbative (PNP) composite operators is defined in Quantum Chromodynamics, which is perturbative scheme and scale independent and are useful to describe the internal structure of a strong interacting system. The operator product expansion in terms of the new operators cleanly separates the perturbative and nonperturbative physics without introducing any factorization scale. A number of PNP observables of the nucleon is briefly discussed including the PNP parton distributions. In particular, the fraction of the nucleon momentum carried by the purely-nonperturbative gluons is found to be around 16%.

We have investigated skewed parton distributions in coordinate space. We found that their evolution can be described in a simple manner in terms of non-local, conformal operators introduced by Balitsky and Braun. The resulting formula is given by a Neumann series expansion. Its structure resembles, for all values of the asymmetry parameter, the well-known solution of the ERBL equation in the momentum space. Performing Fourier transformation we have reproduced known results for evolution of momentum-space distributions.

We discuss skewed parton distributions in the coordinate space. Solution of the corresponding LO evolution equation is constructed in terms of eigenfunctions of the evolution kernel and its relation to the conformal symmetry is explained.

A group theoretical procedure for parametrizing non-forward matrix elements of non-local QCD operators is introduced. The related (two-variable) distribution amplitudes are given as sum over power corrections of the double distributions. For totally symmetric (local) operators the infinite twist decomposition is given completely and for operators with non-trivial symmetry type up to twist 3. The power corrections for most of the phenomenological relevant double distributions and the vector meson wave functions are determined. They may be obtained, by harmonic extension, from the corresponding operators or distribution amplitudes on the light-cone.

In this paper, we generalise Hardy's uncertainty principle to vector-valued functions, and hence to operators. The principle for operators can be formulated loosely by saying that the kernel of an operator cannot be localised near the diagonal if the spectrum is also localised.

Short-range correlations in 4He are investigated using many-body wave functions obtained in the no-core shell model. The similarity renormalization group (SRG) is used to evolve the Argonne V8' interaction and the density operators. The effects of short-range correlations are reflected in the two-body densities in coordinate space as a function of the distance between two nucleons, or alternatively in in momentum space as function of the relative momentum between two nucleons. The SRG transformation is performed in two-body approximation. The importance of missing three-body and higher-body contributions is investigated by comparing results obtained for different flow parameters and by comparing to exact results with the bare Argonne V8' interaction obtained in the correlated Gaussian approach.

In this note we compare two recent results about the distribution of eigenvalues for semi-classical pseudodifferential operators in two dimensions. For classes of analytic operators A. Melin and the author obtained a complex Bohr-Sommerfeld rule, showing that the eigenvalues are situated on a distorted lattice. On the other hand, with M. Hager we showed in any dimension that Weyl asymptotics holds with probability close to 1 for small random perturbations of the operator. In both cases the eigenvalues distribute to leading order according two smooth densities and we show here that the two densities are in general different.

In these survey lectures, we investigate the geometric and analytic properties of transverse Dirac operators. In particular, we define a transverse Dirac operator associated to a distribution that is essentially self-adjoint (Prokhorenkov-R result). We describe the Habib-R Theorem showing that the invariance of the spectrum of a basic Dirac operator on a Riemannian foliation. The Bruening-Kamber-R theorems give Atiyah-Singer type formulas for the equivariant index of transversally elliptic operators on G-manifolds and the index of basic Dirac operators on Riemannian foliations. These notes contain exercises at the end of each subsection and are meant to be accessible to graduate students.

We study the problem of efficient compression of a stochastic source of probability distributions. It can be viewed as a generalization of Shannon's source coding problem. It has relation to the theory of common randomness, as well as to channel coding and rate--distortion theory: in the first two subjects ``inverses'' to established coding theorems can be derived, yielding a new approach to proving converse theorems, in the third we find a new proof of Shannon's rate--distortion theorem. After reviewing the known lower bound for the optimal compression rate, we present a number of approaches to achieve it by code constructions. Our main results are: a better understanding of the known lower bounds on the compression rate by means of a strong version of this statement, a review of a construction achieving the lower bound by using common randomness which we complement by showing the optimal use of the latter within a class of protocols. Then we review another approach, not dependent on common randomness, to minimizing the compression rate, providing some insight into its combinatorial structure, and suggesting an algorithm to optimize it. The second part of the paper is concerned with the generalization of the problem to quantum information theory: the compression of mixed quantum states. Here, after reviewing the known lower bound we contribute a strong version of it, and discuss the relation of the problem to other issues in quantum information theory.

This work is devoted to explore fundamental aspects of the spectral properties of few-body general operators. We first consider the following question: when we know the probability distributions of a set of observables, what can we way on the probability distribution of the summation of them? In considering arbitrary operators, we could not obtain a useful information over third order moment, while under the assumption of the few-body operators, we can rigorously prove a much stronger bound on the moment generating function for arbitrary quantum states. Second, by the use of this bound, we generalize the Chernoff inequality (or the Hoeffding inequality), which characterizes the asymptotic decay of the probability distribution for the product states by the Gaussian decay. In the present form, the Chernoff inequality can be applied to a summation of independent local observables (e.g., single-site operators). We extend the range of application of the Chernoff inequality to the generic few-body observables.

This work is devoted to explore fundamental aspects of the spectral properties of few-body general operators. We first consider the following question: when we know the probability distributions of a set of observables, what can we way on the probability distribution of the summation of them? In considering arbitrary operators, we could not obtain a useful information over third order moment, while under the assumption of the few-body operators, we can rigorously prove a much stronger bound on the moment generating function for arbitrary quantum states. Second, by the use of this bound, we generalize the Chernoff inequality (or the Hoeffding inequality), which characterizes the asymptotic decay of the probability distribution for the product states by the Gaussian decay. In the present form, the Chernoff inequality can be applied to a summation of independent local observables (e.g., single-site operators). We extend the range of application of the Chernoff inequality to the generic few-body observables.

We deal with the Sturm--Liouville operator $Ly=l(y)=-\dfrac{d^2y}{dx^2}+q(x)y,$ with Dirichlet--Neumann boundary conditions $ y(0)=y'(\pi)=0 $ in the space $L_2[0,\pi]$. We assume that the potential $q$ is complex-valued and has the form $q(x)=u'(x)$, where $u\in L_2[0,\pi]$. Here the derivative is treated in the distributional sense. Our aim is to obtain the detailed asymptotic formulae for eigenvalues and eigen- and associated functions of the operator $L$.

Analogues of Brown measure and Haagerup-Schultz projections for tuples of commuting operators in a von Neumann algebra equipped with a faithful tracial state are constructed. These are called joint spectral distribution measures and decomposing projections, respectively, and they satisfy properties similar to their single operator precursors (i.e., Brown measure and Haagerup-Schultz projections). It is shown that the support of the joint spectral distribution measure is contained in the Taylor joint spectrum of the tuple, and also in the ostensibly smaller left Harte spectrum. A simultaneous upper triangularization result for finite commuting tuples is proved and natural results are shown to hold for the Arens multivariate holomorphic functional calculus of such a commuting tuple.