WKB Methods for Finite Difference Schrodinger Equations

Abstract

In this thesis, we develop WKB techniques for the finite difference Schrodinger equation, following the construction of the WKB approach for the standard differential Schrodinger equation. In particular, we will develop an all-order WKB algorithm to get arbitrary hbar-corrections and construct a general quantum momentum, underlining the various properties of its coefficients and the quantities that will be used when constructing the quantization condition. In doing so, we discover the existence of additional periodic factors that need to be considered in order to obtain the most general solution to the problem at hand. We will then proceed to study the simplest non trivial example, the linear potential case and the Bessel functions, that provide a solution to the linear problem. After studying the resurgence properties of the Bessel functions from an analytical and numerical point of view, we will then proceed to use those results in order to build general connection formulae, allowing us to connect the local solutions defined on two sides of a turning point into a smooth solution on the whole real line. With those connection formulae, we will analyse a selection of problems, constructing the discrete spectrum of various finite difference Schrodinger problems and comparing our results with existing literature.

Figures (41)

• Figure 1: Turning points for the example potential $V(x)=8(x^2-1)^2$. On the top, we have the a plot of the potential, shaped as a double well. The points marked in purple are turning points for the condition $E=V(x^*)$. The points marked in brown are turning points for the condition $E=V(x^*)+2$. On the bottom, we have real (blue) and imaginary (orange) parts of $p(x)$ in the same potential. Between two points of the same colour, only the real part of $p(x)$ varies, while between points of different colour only the imaginary part changes. We have included both $\pm$ determinations of \ref{['eq:impulse_def']}.
• Figure 2: Integration path for the integrals of $P_n$, with $x_0$ the turning point and the wavy line being the branch cut. The dashed line is the part of the path that runs on the second sheet after crossing the branch cut. We stress that $\gamma_{0,x}$ is an open path, as its endpoints are different.
• Figure 3: Two different choices for the base point, $x_0$ or $x_1$, both turning points.
• Figure 4: The path of integration $\gamma_{0,1}$, running around the two singularities $x_0$ and $x_1$. Contrary to $\gamma_{0,x}$ and $\gamma_{1,x}$, this is a closed path as it crosses the square root branch cuts twice.
• Figure 5: The path of integration $\gamma_{0,1}$ where the alternative choice of branch cuts has been made. This contour is also closed, as it never crosses the branch cuts.