Arieh Iserles

Arieh Iserles, Karolina Kropielnicka, Pranav Singh

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.

Arieh Iserles, Karolina Kropielnicka, Pranav Singh

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.

Arieh Iserles, Karolina Kropielnicka, Pranav Singh

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.

Arieh Iserles, Shev MacNamara

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.

Philipp Bader, Arieh Iserles, Karolina Kropielnicka et al.

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.

Arieh Iserles, Marcus Webb

The starting point of this paper is that a spectral method is essentially a combination of an orthonormal basis of the underlying Hilbert space with Galerkin conditions. The choice of an orthonormal basis depends on a number of desirable features which we explore in the context of spectral methods for time-dependent partial differential equations in a single space dimension. A central role in ensuring many of the above features is played by the differentiation matrix of the underlying orthonormal system. In particular, it is beneficial if this matrix is skew-symmetric and tridiagonal. While orthonormal systems with this feature have been characterised in A. Iserles & M. Webb, ``Orthogonal systems with a skew-symmetric differentiation matrix'', Found. Comput. Maths, 19 (2019), 1191--1221, employing Fourier transforms, in this paper we provide an alternative characterisation using the differential Lanczos algorithm, which can be implemented constructively. It is valid for inner products that obey an `integration-by-parts condition', inclusive of $L_2$ and Sobolev norms on the real line. Motivated by quest for integration methods that conserve Hamiltonian energy, we conclude the paper replacing inner products by more general sesquilinear forms and presenting preliminary results. Here the Fourier transform characterisation generalises to spectral theory of Schrödinger operators and the differential Lanczos algorithm generalises to the differential Arnoldi algorithm.

Marissa Condon, Alfredo Deano, Jing Gao et al.

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.