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Pseudospectra of Non-Hermitian Hamiltonians

Research article:

Overview:

This was the first project of my PhD. The work is concerned with the Schrödinger equation with complex-valued potentials. In general, the eigenvalues (or the spectrum) of such operators could be any complex number, however, some potentials exhibit special symmetries (called PT-symmetry), which constrain their eigenvalues to lie on the real axis. An example is the imaginary cubic oscillator, defined as $$ \mathsf{H = -\frac{d^2}{dx^2} + ix^3} \quad \textsf{ on }\quad \mathsf{L^2(\mathbb R)} \tag{1} $$ Indeed, this operator can be shown to be PT-symmetric and its eigenvalues are purely real, as the following numerical plot illustrates.

Operators like $(1)$ are of interest in the theory of Non-Hermitian Quantum Mechanics1. This example shows that, given only the spectrum of an operator, it is impossible to decide whether its potential is real-valued or not (i.e. whether the operator is selfadjoint or not). This motivates the introduction of a finer indicator, called the pseudospectrum. The pseudospectrum is the collection of level sets of the function $\mathsf{R(z) := \|(H-z)^{-1}\|}$. It can be shown that the eigenvalues of $\mathsf H$ are precisely the poles of $\mathsf{R(z)}$ and that for real-valued potentials $\mathsf{R(z)}$ is determined completely by the distance of $\mathsf z$ to the set of eigenvalues of $\mathsf H$ (accordingly, the level sets of $\mathsf{R(z)}$ are circles around the eigenvalues). For non-real potential, however, the pseudospectrum can deviate very strongly from the set of eigenvalues. Indeed, the next plot shows a contour plot of $\mathsf{R(z)}$ for the imaginary cubic oscillator $(1)$.
As the figure shows, the function $\mathsf{R(z)}$ may assume very large values far away from the eigenvalues. This behaviour would be impossible for a hamiltonian with a real-valued potential.
Other examples from the literature show that the pseudospectrum of non-hermitian operators can indeed look arbitrarily chaotic. Our own contribution in the paper mentioned above was to prove that for a class of examples such as the imaginary cubic oscillator the situation is not all bad: Even though the pseudospectrum is not simply given by the distance to the spectrum, we prove certain bounds on $\mathsf{R(z)}$, which imply that on any half plane which extends to the left in the complex plane, the function is determined, to some extent, by the eigenvalues in that half plane. Even though suggested by the numerical figure above, this result is highly nontrivial and not guaranteed except for a very limited class of operators.

  1. [Bender, Carl M. Making sense of non-Hermitian Hamiltonians. Reports on Progress in Physics 70.6 (2007): 947.] ↩︎