arXiv subjects

Arieh Iserles

Publications and source records attributed to Arieh Iserles.

34 records · Page 2Linked to original sources

arXiv2022

Solving the linear semiclassical Schrödinger equation on the real line

The numerical solution of a linear Schrödinger equation in the semiclassical regime is very well understood in a torus $\mathbb{T}^d$. A raft of modern computational methods are precise and affordable, while conserving energy and resolving high oscillations very well. This, however, is far from the case with regard to its solution in $\mathbb{R}^d$, a setting more suitable for many applications. In this paper we extend the theory of splitting methods to this end. The main idea is to derive the solution using a spectral method from a combination of solutions of the free Schrödinger equation and of linear scalar ordinary differential equations, in a symmetric Zassenhaus splitting method. This necessitates detailed analysis of certain orthonormal spectral bases on the real line and their evolution under the free Schrödinger operator.

math.NA↗

arXiv2021

The kissing polynomials and their Hankel determinants

In this paper we investigate algebraic, differential and asymptotic properties of polynomials $p_n(x)$ that are orthogonal with respect to the complex oscillatory weight $w(x)=e^{iωx}$ on the interval $[-1,1]$, where $ω>0$. We also investigate related quantities such as Hankel determinants and recurrence coefficients. We prove existence of the polynomials $p_{2n}(x)$ for all values of $ω>0$, as well as degeneracy of $p_{2n+1}(x)$ at certain values of $ω$ (called kissing points). We obtain detailed asymptotic information as $ω\to\infty$, using recent theory of multivariate highly oscillatory integrals, and we complete the analysis with the study of complex zeros of Hankel determinants, using the large $ω$ asymptotics obtained before.

math.CA↗

arXiv2020

A Differential Analogue of Favard's Theorem

Favard's theorem characterizes bases of functions $\{p_n\}_{n\in\mathbb{Z}_+}$ for which $x p_n(x)$ is a linear combination of $p_{n-1}(x)$, $p_n(x)$, and $p_{n+1}(x)$ for all $n \geq 0$ with $p_{0}\equiv1$ (and $p_{-1}\equiv 0$ by convention). In this paper we explore the differential analogue of this theorem, that is, bases of functions $\{φ_n\}_{n\in\mathbb{Z}_+}$ for which $φ_n'(x)$ is a linear combination of $φ_{n-1}(x)$, $φ_n(x)$, and $φ_{n+1}(x)$ for all $n \geq 0$ with $φ_{0}(x)$ given (and $φ_{-1}\equiv 0$ by convention). We answer questions about orthogonality and completeness of such functions, provide characterisation results, and also, of course, give plenty of examples and list challenges for further research. Motivation for this work originated in the numerical solution of differential equations, in particular spectral methods which give rise to highly structured matrices and stable-by-design methods for partial differential equations of evolution. However, we believe this theory to be of interest in its own right, due to the interesting links between orthogonal polynomials, Fourier analysis and Paley--Wiener spaces, and the resulting identities between different families of special functions.

math.CA↗

arXiv2019

A family of orthogonal rational functions and other orthogonal systems with a skew-Hermitian differentiation matrix

In this paper we explore orthogonal systems in $\mathrm{L}_2(\mathbb{R})$ which give rise to a skew-Hermitian, tridiagonal differentiation matrix. Surprisingly, allowing the differentiation matrix to be complex leads to a particular family of rational orthogonal functions with favourable properties: they form an orthonormal basis for $\mathrm{L}_2(\mathbb{R})$, have a simple explicit formulae as rational functions, can be manipulated easily and the expansion coefficients are equal to classical Fourier coefficients of a modified function, hence can be calculated rapidly. We show that this family of functions is essentially the only orthonormal basis possessing a differentiation matrix of the above form and whose coefficients are equal to classical Fourier coefficients of a modified function though a monotone, differentiable change of variables. Examples of other orthogonal bases with skew-Hermitian, tridiagonal differentiation matrices are discussed as well.

math.NA↗

arXiv2019

Fast Computation of Orthogonal Systems with a Skew-symmetric Differentiation Matrix

Orthogonal systems in $\mathrm{L}_2(\mathbb{R})$, once implemented in spectral methods, enjoy a number of important advantages if their differentiation matrix is skew-symmetric and highly structured. Such systems, where the differentiation matrix is skew-symmetric, tridiagonal and irreducible, have been recently fully characterised. In this paper we go a step further, imposing the extra requirement of fast computation: specifically, that the first $N$ coefficients {of the expansion} can be computed to high accuracy in $\mathcal{O}(N\log_2N)$ operations. We consider two settings, one approximating a function $f$ directly in $(-\infty,\infty)$ and the other approximating $[f(x)+f(-x)]/2$ and $[f(x)-f(-x)]/2$ separately in $[0,\infty)$. In each setting we prove that there is a single family, parametrised by $α,β> -1$, of orthogonal systems with a skew-symmetric, tridiagonal, irreducible differentiation matrix and whose coefficients can be computed as Jacobi polynomial coefficients of a modified function. The four special cases where $α, β= \pm 1/2$ are of particular interest, since coefficients can be computed using fast sine and cosine transforms. Banded, Toeplitz-plus-Hankel multiplication operators are also possible for representing variable coefficients in a spectral method. In Fourier space these orthogonal systems are related to an apparently new generalisation of the Carlitz polynomials.

math.NA↗

arXiv2018

Magnus-Zassenhaus methods for the semiclassical Schrödinger equation with oscillatory time-dependent potentials

Schrödinger equations with time-dependent potentials are of central importance in quantum physics and theoretical chemistry, where they aid in the simulation and design of systems and processes at atomic scales. Numerical approximation of these equations is particularly difficult in the semiclassical regime because of the highly oscillatory nature of solution. Highly oscillatory potentials such as lasers compound these difficulties even further. Altogether, these effects render a large number of standard numerical methods less effective in this setting. In this paper we will develop a class of high-order exponential splitting schemes that are able to overcome these challenges by combining the advantages of integral-preserving simplified-commutator Magnus expansions with those of symmetric Zassenhaus splittings. This allows us to use large time steps in our schemes even in the presence of highly oscillatory potentials and solutions.

math.NA↗

arXiv2018

Compact schemes for laser-matter interaction in Schrödinger equation

Numerical solutions for laser-matter interaction in Schrödinger equation has many applications in theoretical chemistry, quantum physics and condensed matter physics. In this paper we introduce a methodology which allows, with a small cost, to extend any fourth-order scheme for Schrödinger equation with time-indepedent potential to a fourth-order method for Schrödinger equation with laser potential. These fourth-order methods improve upon many leading schemes of order six due to their low costs and small error constants.

math.NA↗

arXiv2018

Magnus-Lanczos methods with simplified commutators for the Schrödinger equation with a time-dependent potential

The computation of the Schrödinger equation featuring time-dependent potentials is of great importance in quantum control of atomic and molecular processes. These applications often involve highly oscillatory potentials and require inexpensive but accurate solutions over large spatio-temporal windows. In this work we develop Magnus expansions where commutators have been simplified. Consequently, the exponentiation of these Magnus expansions via Lanczos iterations is significantly cheaper than that for traditional Magnus expansions. At the same time, and unlike most competing methods, we simplify integrals instead of discretising them via quadrature at the outset -- this gives us the flexibility to handle a variety of potentials, being particularly effective in the case of highly oscillatory potentials, where this strategy allows us to consider significantly larger time steps.

math.NA↗

arXiv2017

Magnus expansions and pseudospectra of Master Equations

New directions in research on master equations are showcased by example. Magnus expansions, time-varying rates, and pseudospectra are highlighted. Exact eigenvalues are found and contrasted with the large errors produced by standard numerical methods in some cases. Isomerisation provides a running example and an illustrative application to chemical kinetics. We also give a brief example of the totally asymmetric exclusion process.

math.NA↗

arXiv2016

Efficient methods for time-dependence in semiclassical Schrödinger equations

We build an efficient and unitary (hence stable) method for the solution of the semi-classical Schrödinger equation subject with explicitly time-dependent potentials. The method is based on a combination of the Zassenhaus decomposition (Bader, Iserles, Kropielnicka & Singh 2014) with the Magnus expansion of the time-dependent Hamiltonian. We conclude with numerical experiments.

math.NA↗

arXiv2015

Asymptotic solvers for ordinary differential equations with multiple frequencies

We construct asymptotic expansions for ordinary differential equations with highly oscillatory forcing terms, focussing on the case of multiple, non-commensurate frequencies. We derive an asymptotic expansion in inverse powers of the oscillatory parameter and use its truncation as an exceedingly effective means to discretize the differential equation in question. Numerical examples illustrate the effectiveness of the method.

math.NA↗

arXiv2006

A Class of Integrable Geodesic Flows on the Symplectic Group and the Symmetric Matrices

This paper shows that the left-invariant geodesic flow on the symplectic group relative to the Frobenius metric is an integrable system that is not contained in the Mishchenko-Fomenko class of rigid body metrics. This system may be expressed as a flow on symmetric matrices and is bi-Hamiltonian. This analysis is extended to cover flows on symmetric matrices when an isomorphism with the symplectic Lie algebra does not hold. The two Poisson structures associated with this system, including an analysis of its Casimirs, are completely analyzed. Since the system integrals are not generated by its Casimirs it is shown that the nature of integrability is fundamentally different from that exhibited in the Mischenko-Fomenko setting.

math-ph↗

arXiv1999

Methods for the approximation of the matrix exponential in a Lie-algebraic setting

Discretization methods for ordinary differential equations based on the use of matrix exponentials have been known for decades. This set of ideas has come off age and acquired greater urgency recently, within the context of geometric integration and discretization methods on manifolds based on the use of Lie-group actions. In the present paper we study the approximation of the matrix exponential in a particular context: given a Lie group $G$ and its Lie algebra $g$, we seek approximants $F(tB)$ of $\exp(tB)$ such that $F(tB)\in G$ if $B\in g$. Having fixed a basis of the Lie algebra, we write $F(tB)$ as a composition of exponentials of the basis elements pre-multiplied by suitable scalar functions.

math.NA↗

arXiv1994

Biorthogonal polynomials and zero-mapping transformations

The authors have presented in \cite{IN2} a technique to generate transformations $\cal T$ of the set ${\Bbb P}_n$ of $n$th degree polynomials to itself such that if $p\in{\Bbb P}_n$ has all its zeros in $(c,d)$ then ${\cal T}\{p\}$ has all its zeros in $(a,b)$, where $(a,b)$ and $(c,d)$ are given real intervals. The technique rests upon the derivation of an explicit form of biorthogonal polynomials whose Borel measure is strictly sign consistent and such that the ratio of consecutive generalized moments is a rational $[1/1]$ function of the parameter. Specific instances of strictly sign consistent measures that have been debated in \cite{IN2} include $x^μ\Dψ(x)$, $μ^x\Dψ(x)$ and $x^{\log_qμ}\Dψ(x)$, $q\in(0,1)$. In this paper we identify all measures $ψ$ such that their consecutive generalized moments have a rational $[1/1]$ quotient, thereby characterizing all possible zero-mapping transformations of this kind.

math.CA↗

arXiv1994

Explicit representations of biorthogonal polynomials

Given a parametrised weight function $ω(x,μ)$ such that the quotients of its consecutive moments are Möbius maps, it is possible to express the underlying biorthogonal polynomials in a closed form \cite{IN2}. In the present paper we address ourselves to two related issues. Firstly, we demonstrate that, subject to additional assumptions, every such $ω$ obeys (in $x$) a linear differential equation whose solution is a generalized hypergeometric function. Secondly, using a generalization of standard divided differences, we present a new explicit representation of the underlying orthogonal polynomials.

math.CA↗

arXiv1993

From Schrödinger spectra to orthogonal polynomials, via a functional equation

The main difference between certain spectral problems for linear Schrödinger operators, e.g. the almost Mathieu equation, and three-term recurrence relations for orthogonal polynomials is that in the former the index ranges across $\ZZ$ and in the latter only across $\Zp$. We present a technique that, by a mixture of Dirichlet and Taylor expansions, translates the almost Mathieu equation and its generalizations to three term recurrence relations. This opens up the possibility of exploiting the full power of the theory of orthogonal polynomials in the analysis of Schrödinger spectra. Aforementioned three-term recurrence relations share the property that their coefficients are almost periodic. We generalize a method of proof, due originally to Jeff Geronimo and Walter van Assche, to investigate essential support of the Borel measure of associated orthogonal polynomials, thereby deriving information on the underlying absolutely continuous spectra of Schrödinger operators.

math.CA↗