Multivariate orthogonal polynomials and integrable systems

Multivariate orthogonal polynomials in D real dimensions are considered from the perspective of the Cholesky factorization of a moment matrix. The approach allows for the construction of corresponding multivariate orthogonal polynomials, associated second kind functions, Jacobi type matrices and associated three term relations and also Christoffel-Darboux formulae. The multivariate orthogonal polynomials, their second kind functions and the corresponding Christoffel-Darboux kernels are shown to be quasi-determinants as well as Schur complements of bordered truncations of the moment matrix; quasi-tau functions are introduced. It is proven that the second kind functions are multivariate Cauchy transforms of the multivariate orthogonal polynomials. Discrete and continuous deformations of the measure lead to Toda type integrable hierarchy, being the corresponding flows described through Lax and Zakharov-Shabat equations; bilinear equations are found. Varying size matrix nonlinear partial difference and differential equations of the 2D Toda lattice type are shown to be solved by matrix coefficients of the multivariate orthogonal polynomials. The discrete flows, which are shown to be connected with a Gauss-Borel factorization of the Jacobi type matrices and its quasi-determinants, lead to expressions for the multivariate orthogonal polynomials and their second kind functions in terms of shifted quasi-tau matrices, which generalize to the multidimensional realm, those that relate the Baker and adjoint Baker functions to ratios of Miwa shifted tau-functions in the 1D scenario. In this context, the multivariate extension of the elementary Darboux transformation is given in terms of quasi-determinants of matrices built up by the evaluation, at a poised set of nodes lying in an appropriate hyperplane in R^D, of the multivariate orthogonal polynomials. The multivariate Christoffel formula for the iteration of m elementary Darboux transformations is given as a quasi-determinant. It is shown, using congruences in the space of semi-infinite matrices, that the discrete and continuous flows are intimately connected and determine nonlinear partial difference-differential equations that involve only one site in the integrable lattice behaving as a Kadomstev-Petviashvili type system. Finally, a brief discussion of measures with a particular linear isometry invariance and some of its consequences for the corresponding multivariate polynomials is given. In particular, it is shown that the Toda times that preserve the invariance condition lay in a secant variety of the Veronese variety of the fixed point set of the linear isometry.

The ionized gas in the CALIFA early-type galaxies I. Mapping two representative cases: NGC 6762 and NGC 5966

As part of the ongoing CALIFA survey, we have conducted a thorough bidimensional analysis of the ionized gas in two E/S0 galaxies, NGC 6762 and NGC 5966, aiming to shed light on the nature of their warm ionized ISM. Specifically, we present optical (3745–7300 Å) integral field spectroscopy obtained with the PMAS/PPAK integral field spectrophotometer. Its wide field-of-view (1′ × 1′) covers the entire optical extent of each galaxy down to faint continuum surface brightnesses. To recover the nebular lines, we modeled and subtracted the underlying stellar continuum from the observed spectra using the STARLIGHT spectral synthesis code. The pure emission-line spectra were used to investigate the gas properties and determine the possible sources of ionization. We show the advantages of IFU data in interpreting the complex nature of the ionized gas in NGC 6762 and NGC 5966. In NGC 6762, the ionized gas and stellar emission display similar morphologies, while the emission line morphology is elongated in NGC 5966, spanning ~6 kpc, and is oriented roughly orthogonal to the major axis of the stellar continuum ellipsoid. Whereas gas and stars are kinematically aligned in NGC 6762, the gas is kinematically decoupled from the stars in NGC 5966. A decoupled rotating disk or an “ionization cone” are two possible interpretations of the elongated ionized gas structure in NGC 5966. The latter would be the first “ionization cone” of such a dimension detected within a weak emission-line galaxy. Both galaxies have weak emission-lines relative to the continuum[EW(Hα) ≲ 3 Å] and have very low excitation, log([OIII]λ5007/Hβ) ≲ 0.5. Based on optical diagnostic ratios ([OIII]λ5007/Hβ, [NII]λ6584/Hα, [SII]λ6717, 6731/Hα, [OI]λ6300/Hα), both objects contain a LINER nucleus and an extended LINER-like gas emission. The emission line ratios do not vary significantly with radius or aperture, which indicates that the nebular properties are spatially homogeneous. The gas emission in NGC 6762 can be best explained by photoionization by pAGB stars without the need of invoking any other excitation mechanism. In the case of NGC 5966, the presence of a nuclear ionizing source seems to be required to shape the elongated gas emission feature in the “ionization cone” scenario, although ionization by pAGB stars cannot be ruled out. Further study of this object is needed to clarify the nature of its elongated gas structure.