Time-dependent calculation of ionization in Potassium at mid-infrared wavelengths

We study the dynamics of the Potassium atom in the mid-infrared, high intensity, short laser pulse regime. We ascertain numerical convergence by comparing the results obtained by the direct expansion of the time-dependent Schroedinger equation onto B-Splines, to those obtained by the eigenbasis expansion method. We present ionization curves in the 12-, 13-, and 14-photon ionization range for Potassium. The ionization curve of a scaled system, namely Hydrogen starting from the 2s, is compared to the 12-photon results. In the 13-photon regime, a dynamic resonance is found and analyzed in some detail. The results for all wavelengths and intensities, including Hydrogen, display a clear plateau in the peak-heights of the low energy part of the Above Threshold Ionization (ATI) spectrum, which scales with the ponderomotive energy Up, and extends to 2.8 +- 0.5 Up.Comment: 15 two-column pages with 15 figures, 3 tables. Accepted for publication in Phys. Rev A. Improved figures, language and punctuation, and made minor corrections. We also added a comparison to the ADK theor

On numerical solution of Fredholm and Hammerstein integral equations via Nyström method and Gaussian quadrature rules for splines

The first author acknowledges partial financial support by the IMAG-Maria de Maeztu grant CEX2020-001105-M/AEI/10.13039/501100011033. The second author has been partially supported by Spanish State Research Agency (Spanish Min-istry of Science, Innovation and Universities) : BCAM Severo Ochoa excellence accreditation SEV-2017-0718 and by Ramon y Cajal with reference RYC-2017-22649. The fourth author is member of GNCS-INdAM.Nyström method is a standard numerical technique to solve Fredholm integral equations of the second kind where the integration of the kernel is approximated using a quadrature formula. Traditionally, the quadrature rule used is the classical polynomial Gauss quadrature. Motivated by the observation that a given function can be better approximated by a spline function of a lower degree than a single polynomial piece of a higher degree, in this work, we investigate the use of Gaussian rules for splines in the Nyström method. We show that, for continuous kernels, the approximate solution of linear Fredholm integral equations computed using spline Gaussian quadrature rules converges to the exact solution for m →∞, m being the number of quadrature points. Our numerical results also show that, when fixing the same number of quadrature points, the approximation is more accurate using spline Gaussian rules than using the classical polynomial Gauss rules. We also investigate the non-linear case, considering Hammerstein integral equations, and present some numerical tests.IMAG-Maria de Maeztu grant CEX2020-001105-M/AEI/10.13039/501100011033Spanish State Research Agency (Spanish Min-istry of Science, Innovation and Universities) SEV-2017-0718Spanish Government RYC-2017-2264

On numerical solution of Fredholm and Hammerstein integral equations via Nystr\"{o}m method and Gaussian quadrature rules for splines

Nystr\"{o}m method is a standard numerical technique to solve Fredholm integral equations of the second kind where the integration of the kernel is approximated using a quadrature formula. Traditionally, the quadrature rule used is the classical polynomial Gauss quadrature. Motivated by the observation that a given function can be better approximated by a spline function of a lower degree than a single polynomial piece of a higher degree, in this work, we investigate the use of Gaussian rules for splines in the Nystr\"{o}m method. We show that, for continuous kernels, the approximate solution of linear Fredholm integral equations computed using spline Gaussian quadrature rules converges to the exact solution for $m \rightarrow \infty$, $m$ being the number of quadrature points. Our numerical results also show that, when fixing the same number of quadrature points, the approximation is more accurate using spline Gaussian rules than using the classical polynomial Gauss rules. We also investigate the non-linear case, considering Hammerstein integral equations, and present some numerical tests.RYC-2017-2264

Multivariate Splines and Algebraic Geometry

Multivariate splines are effective tools in numerical analysis and approximation theory. Despite an extensive literature on the subject, there remain open questions in finding their dimension, constructing local bases, and determining their approximation power. Much of what is currently known was developed by numerical analysts, using classical methods, in particular the so-called Bernstein-B´ezier techniques. Due to their many interesting structural properties, splines have become of keen interest to researchers in commutative and homological algebra and algebraic geometry. Unfortunately, these communities have not collaborated much. The purpose of the half-size workshop is to intensify the interaction between the different groups by bringing them together. This could lead to essential breakthroughs on several of the above problems