Analytical solutions to some generalized and polynomial eigenvalue problems

It is well-known that the finite difference discretization of the Laplacian eigenvalue problem −Δu = λu leads to a matrix eigenvalue problem (EVP) Ax =λx where the matrix A is Toeplitz-plus-Hankel. Analytical solutions to tridiagonal matrices with various boundary conditions are given in a recent work of Strang and MacNamara. We generalize the results and develop analytical solutions to certain generalized matrix eigenvalue problems (GEVPs) Ax = λBx which arise from the finite element method (FEM) and isogeometric analysis (IGA). The FEM matrices are corner-overlapped block-diagonal while the IGA matrices are almost Toeplitz-plus-Hankel. In fact, IGA with a correction that results in Toeplitz-plus-Hankel matrices gives a better numerical method. In this paper, we focus on finding the analytical eigenpairs to the GEVPs while developing better numerical methods is our motivation. Analytical solutions are also obtained for some polynomial eigenvalue problems (PEVPs). Lastly, we generalize the eigenvector-eigenvalue identity (rediscovered and coined recently for EVPs) for GEVPs and derive some trigonometric identities.

An Iterative Ab Initio Non-Hermitian Floquet Method for Photoionization Resonances

We present an efficient ab initio non-Hermitian Floquet method for computing the photoionization resonances of an electronic system interacting with linearly polarized<br>monochromatic laser light. Unlike the direct "brute force" diagonalization method, which has been used for huge Floquet matrix eigenvalue problems, the new method follows a simple iterative process. The computational advantages of<br>the iterative method are very remarkable as it avoids computation, storage, and diagonalization of the huge Floquet matrix. The new method can also be used in<br>conjunction with the ab initio computational techniques that were originally developed for the field-free bound state calculations. The method is best illustrated<br>with the photoionization resonance of the hydrogen atom.<br>

An Iterative Ab Initio Non-Hermitian Floquet Method for Photoionization Resonances

We present an efficient ab initio non-Hermitian Floquet method for computing the photoionization resonances of an electronic system interacting with linearly polarized<br>monochromatic laser light. Unlike the direct "brute force" diagonalization method, which has been used for huge Floquet matrix eigenvalue problems, the new method follows a simple iterative process. The computational advantages of<br>the iterative method are very remarkable as it avoids computation, storage, and diagonalization of the huge Floquet matrix. The new method can also be used in<br>conjunction with the ab initio computational techniques that were originally developed for the field-free bound state calculations. The method is best illustrated<br>with the photoionization resonance of the hydrogen atom.<br>

Kernel-Based Approximation of the Koopman Generator and Schrödinger Operator

Many dimensionality and model reduction techniques rely on estimating dominant eigenfunctions of associated dynamical operators from data. Important examples include the Koopman operator and its generator, but also the Schrödinger operator. We propose a kernel-based method for the approximation of differential operators in reproducing kernel Hilbert spaces and show how eigenfunctions can be estimated by solving auxiliary matrix eigenvalue problems. The resulting algorithms are applied to molecular dynamics and quantum chemistry examples. Furthermore, we exploit that, under certain conditions, the Schrödinger operator can be transformed into a Kolmogorov backward operator corresponding to a drift-diffusion process and vice versa. This allows us to apply methods developed for the analysis of high-dimensional stochastic differential equations to quantum mechanical systems.