Background

From Mathematical Physics II you known that the one-dimensional harmonic oscillator has real, positive eigenvalues and the corresponding normalised eigenfunctions form an orthonormal basis of $L^2(\mathbb R)$. This project is an introduction to non-Hermitian spectral theory. We replace the real potential $x^2$ by the imaginary potential ${\rm i}x^2$, so the complex oscillator acts on a function $\psi$ as $$H\psi=-\frac{{\rm d}^2\psi}{{\rm d} x^2}+{\rm i}x^2\psi.$$ This results in a rotation of the eigenvalues into the complex plane. For $\varepsilon>0$ we define the so-called $\varepsilon$-pseudospectrum by $$\sigma_{\varepsilon}(H):=\left\{z\in\mathbb C:\,\exists\,\psi\text{ such that }\|(H-z)\psi\|<\varepsilon \|\psi\|\right\}.$$ If $\psi$ is an eigenfunction to an eigenvalue $\lambda$ of $H$, then $\|(H-z)\psi\|=|\lambda-z|\|\psi\|$, hence $\sigma_{\varepsilon}(H)$ contains all open $\varepsilon$-disks around eigenvalues. So one might think that this set is located near the eigenvalues.

As discovered by E.B. Davies (1999), the set $\sigma_{\varepsilon}(H)$ is actually much larger than just the union of these $\varepsilon$-disks. In fact, it contains points that are arbitrarily far away from the eigenvalues. On the left you see the eigenvalues (blue dots) and the boundaries of $\sigma_{\varepsilon}(H)$ for $\varepsilon=10^{-1}, 10^{-2},\dots,10^{-8}$. This is a typical feature of non-Hermitian linear operators and its discovery resulted in a rapidly growing literature on pseudospectral theory, which was invented to explore just such possibilities.

Description of the project

You will read Davies' celebrated paper, see reference [1] below. A very good introduction to pseudospectral theory is given in [2], relevant here are Sections 4,5,10,11. Then, individually, you will focus on different aspects of the problem. This can be purely theoretic, by studying the pseudospectra of the complex oscillator truncated to a finite interval $(-a,a)$ (subject to some boundary conditions); or you can focus on numerical aspects - can you reproduce the above picture?

Prerequisites

Mathematical Physics II. If you are interested in numerical implementations, you should have some prior knowledge in any programming language/software.

References

[1] E.B. Davies. Pseudo-spectra, the harmonic oscillator and complex resonances. Proceedings of the Royal Society London A (1999).

[2] L.N. Trefethen and M. Embree. Spectra and pseudospectra: the behavior of nonnormal matrices and operators. Princeton University Press (2005).