## Approximation of wave packets on the whole real line

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In this thesis we compare six different orthonormal systems in L₂(R) which can be used in the construction of a spectral method for solving the semi-classically scaled time dependent linear Schrödinger equation (TDSE) on the real line. All bases have banded skew-symmetric (or skew-Hermitian) differentiation matrices, which greatly simplifies their implementation in a spectral method, while ensuring that the numerical solution is unitary. This is essential in order to respect the Born interpretation in quantum mechanics and, as a byproduct, ensures numerical stability with respect to the L₂(R) norm. However, this does not guarantee that the basis is a suitable candidate for a spectral method to solve TDSE. Thus, the overarching question that each chapter answers is: how well does the chosen basis function approximate wave packets?

Chapter 1 provides the necessary background to understand the motivation and results presented in the subsequent chapters. In Chapters 2, 3 and 4 we derive asymptotic estimates of the expansion coefficients for a wave packet in stretched Fourier, Hermite and Malmquist--Takenaka bases respectively. These estimates are extremely accurate in the high frequency regime. The results within these chapters allow us to compare the speed of convergence of the expansion coefficients to determine which orthonormal system would be more suitable in a spectral method to solve TDSE. Our criteria consider how the number of coefficients grows with respect to parameters n and ω for a given accuracy ε, and the computational efficiency of evaluating the expansion coefficients. We assume that the parameter n is large as it denotes the number of basis functions used to approximate a function. Moreover, we are interested in approximating highly oscillatory wave packets so ω, which denotes the frequency of the wave packet, is also a large parameter. This work appears in [Iserles et al., 2022]. A similar analysis is applied to the basis functions explored in the remaining three chapters. We provide original estimates to the approximation properties of three basis functions based on numerical results.

A breakdown of the results in each chapter is as follows: in Chapter 2, we look at the stretched Fourier basis, the results in this chapter were developed in collaboration with Arieh Iserles and are included in the thesis because stretched Fourier is an intuitive basis to use in solving TDSE. We find that the basis can be optimised to exhibit exponential decay (see Theorem 2.0.1). However the method is limited in practice as it requires more knowledge of the end solution than is typically available. This limitation highlights why Hermite and Malmquist--Takenaka basis are more suitable. We consider Hermite basis functions in Chapter 3, which is joint work with Arieh Iserles. The proven result in Theorem 3.0.1 shows that the expansion coefficients for wave packets exhibit asymptotic exponential decay. In addition, for a given accuracy ε, we find that the number of coefficients which are greater than ε grow quadratically with ω. In Chapter 4 we show that for Malmquist--Takenaka basis functions, the number of coefficients which are greater than a given tolerance ε, grow linearly with ω. Furthermore, the expansion coefficients can be approximated using the fast Fourier transform. This shows that the Malmquist--Takenaka basis is superior, in a practical sense, to the more commonly used Hermite functions. The result proved in Theorem 4.4.1 highlights a phenomenon which is somewhat counterintuitive; in the presence of oscillations, the coefficients decay exponentially fast. In fact, the higher the oscillations, the bigger the region of exponential decay. The work in Chapter 4 is joint with Marcus Webb.

Chapters 5 and 6 look at Freud polynomials in two different L₂(R) function forms called T-Freud functions and attenuated Freud functions respectively. Freud polynomials do not have an explicit form, which makes them more difficult to study. However, they have the desirable property that their recurrence coefficients grow slower than both Hermite and Malmquist--Takenaka functions which is useful to have when evolving our solutions to TDSE in time. We provide a practical way of generating Freud polynomials and a framework to evaluate their approximating properties numerically, which has not been done before. We show that T-Freud functions do not approximate wave packets well, but attenuated Freud functions show potential. In Theorem 6.0.2 we prove that the family of attenuated Freud functions have a banded, skew-symmetric, irreducible differentiation matrix. For a specific case of attenuated Freud functions, we explicitly find the differentiation matrix in Theorem 6.0.1. In addition, attenuated Freud functions display interesting approximating properties; we observe exponential convergence up to some transition point, subsequently the coefficients exhibit algebraic decay.

In Chapter 7, we consider generalised Hermite polynomials in T-generalised Hermite function form. Similarly to attenuated Freud functions, this basis also exhibits two different convergence rates for increasing n. Generalised Hermite functions do not display any computational or approximating advantages when compared to the classical Hermite functions. However, they raise many interesting questions regarding the approximating properties of orthonormal systems on the whole real line.

Karen Min Luong