On the solutions of generalized discrete Poisson equation

The set of common numerical and analytical problems is introduced in the form of the generalized multidimensional discrete Poisson equation. It is shown that its solutions with square-summable discrete derivatives are unique up to a constant. The proof uses the Fourier transform as the main tool. The necessary condition for the existence of the solution is provided.Comment: 8 pages, LaTe

On the approximation of real powers of sparse, infinite, bounded and Hermitian matrices

We describe a way to approximate the matrix elements of a real power $\alpha$ of a positive (for $\alpha \ge 0$) or non-negative (for $\alpha \in \mathbb{R}$), infinite, bounded, sparse and Hermitian matrix $W$. The approximation uses only a finite part of the matrix $W$.Comment: 9 pages; LaTeX; revised version (minor corrections); to appear in "Linear Algebra and its Applications

A simple renormalization group approximation of the groundstate properties of interacting bosonic systems

We present a new, simple renormalization group method of investigating groundstate properties of interacting bosonic systems. Our method reduces the number of particles in a system, which makes numerical calculations possible for large systems. It is conceptually simple and easy to implement, and allows to investigate the properties unavailable through mean field approximations, such as one- and two-particle reduced density matrices of the groundstate. As an example, we model a weakly interacting 1D Bose gas in a harmonic trap. Compared to the mean-field Gross-Pitaevskii approximation, our method provides a more accurate description of the groundstate one-particle density matrix. We have also obtained the Hall-Post lower bounds for the groundstate energy of the gas. All results have been obtained by the straightforward numerical diagonalization of the Hamiltonian matrix.Comment: RevTex 4, 12 figures (revised and updated); to appear in Physical Review

Universality of affine formulation in General Relativity theory

Affine variational principle for General Relativity, proposed in 1978 by one of us (J.K.), is a good remedy for the non-universal properties of the standard, metric formulation, arising when the matter Lagrangian depends upon the metric derivatives. Affine version of the theory cures the standard drawback of the metric version, where the leading (second order) term of the field equations depends upon matter fields and its causal structure violates the light cone structure of the metric. Choosing the affine connection (and not the metric one) as the gravitational configuration, simplifies considerably the canonical structure of the theory and is more suitable for purposes of its quantization along the lines of Ashtekar and Lewandowski (see http://www.arxiv.org/gr-qc/0404018). We show how the affine formulation provides a simple method to handle boundary integrals in general relativity theory.Comment: 38 pages, no figures, LaTeX+BibTex, corrected (restructured contents, one example removed, no additional results, typos fixed) versio