Partially fundamentally reducible operators in Krein spaces

A self-adjoint operator $A$ in a Krein space $\bigl({\mathcal K},[\,\cdot\,,\cdot\,]\bigr)$ is called partially fundamentally reducible if there exist a fundamental decomposition ${\mathcal K} = {\mathcal K}_+ [\dot{+}] {\mathcal K}_-$ (which does not reduce $A$) and densely defined symmetric operators $S_+$ and $S_-$ in the Hilbert spaces $\bigl({\mathcal K}_+,[\,\cdot\,,\cdot\,]\bigr)$ and $\bigl({\mathcal K}_-,-[\,\cdot\,,\cdot\,]\bigr)$, respectively, such that each $S_+$ and $S_-$ has defect numbers $(1,1)$ and the operator $A$ is a self-adjoint extension of $S =S_+ \oplus (-S_-)$ in the Krein space $\bigl({\mathcal K},[\,\cdot\,,\cdot\,]\bigr)$. The operator $A$ is interpreted as a coupling of operators $S_+$ and $-S_-$ relative to some boundary triples $\bigl({\mathbb C},\Gamma_0^+,\Gamma_1^+\bigr)$ and $\bigl({\mathbb C},\Gamma_0^-,\Gamma_1^-\bigr)$. Sufficient conditions for a nonnegative partially fundamentally reducible operator $A$ to be similar to a self-adjoint operator in a Hilbert space are given in terms of the Weyl functions $m_+$ and $m_-$ of $S_+$ and $S_-$ relative to the boundary triples $\bigl({\mathbb C},\Gamma_0^+,\Gamma_1^+\bigr)$ and $\bigl({\mathbb C},\Gamma_0^-,\Gamma_1^-\bigr)$. Moreover, it is shown that under some asymptotic assumptions on $m_+$ and $m_-$ all positive self-adjoint extensions of the operator $S$ are similar to self-adjoint operators in a Hilbert space.Comment: 45 pages, results presented at the 21st IWOTA 2010 held in Berlin, German

Coupling of symmetric operators and the third Green identity

The principal aim of this paper is to derive an abstract form of the third Green identity associated with a proper extension $T$ of a symmetric operator $S$ in a Hilbert space $\mathfrak H$, employing the technique of quasi boundary triples for $T$. The general results are illustrated with couplings of Schr\"{o}dinger operators on Lipschitz domains on smooth, boundaryless Riemannian manifolds.Comment: 26 page

On linear fractional transformations associated with generalized J-inner matrix functions

In this paper we study generalized J-inner matrix valued functions which appear as resolvent matrices in various indefinite interpolation problems. Reproducing kernel indefinite inner product spaces associated with a generalized J-inner matrix valued function W are studied and intensively used in the description of the range of the linear fractional transformation associated with W and applied to the Schur class. For a subclass of generalized J-inner matrix valued function W the notion of associated pair is introduced and factorization formulas for W are found.Comment: 41 page

Multi-pomeron exchange model for pppp and ppˉp\bar{p} collisions at ultra-high energy

A new variant of the effective pomeron exchange model is proposed for the description of the correlation, observed in $pp$ and $p\bar{p}$ collisions at center-of-mass energy from SPS to LHC, between mean transverse momentum and charged particles multiplicity. The model is based on the Regge-Gribov approach. Smooth logarithmic growth with the collision energy was established for the parameter k, the mean rapidity density of charged particles produced by a single string. It was obtained in the model by the fitting of the available experimental data on charged particles rapidity density in $pp$ and $p\bar{p}$ collisions. The main effect of the model, a gradual onset of string collectivity with the growth of collision energy, is accounted by a free parameter {\beta} that is responsible in an effective way for the string fusion phenomenon. Another free parameter, t, is used to define string tension. We extract parameters {\beta} and t from the available experimental results on -multiplicity correlation at nucleon collision energy $\sqrt{s}$ from 17 GeV to 7 TeV. Smooth dependence of both {\beta} and t on energy allows to make predictions for the correlation behavior at the collision energy of 14 TeV. The indications to the string interaction effects in high multiplicity events in $pp$ collisions at the LHC energies are also discussed.Comment: 7 pages, 7 figures, to appear in proc. QFTHEP'201