Abstract

Spectral Pathologies of Non-Self-Adjoint Operators E.B. Davies It is by now well known that quite small ($30\times 30$) non-self-adjoint matrices may have very peculiar spectral properties: small perturbations of the matrix may lead to very large perturbations of the eigenvalues, and the norm of the inverse matrix may be very large even though the matrix does not have any small eigenvalues. These questions may be investigated systematically using the notion of pseudospectrum. Together with A Aslanyan, I have studied similar questions for Sturm-Liouville operators with complex potentials, and report the conclusions. Consider the \Schrodinger operator Hf(x)=-\frac{\rmd ^{2}f}{\rmd x^{2}} +V(x) f(x) acting in $L^{2}(\R)$. The complex harmonic oscillator corresponds to the choice $V(x)=cx^{2}$ where $\Re c>0$ and $\Im c>0$. The spectrum of this operator consists of the numbers $\lam_{n}=\sqrt{c}(2n+1)$ where $n=0,1,\ldots$ and the eigenfunctions are Hermite functions. Nevertheless the operator has several strong pathologies. The eigenfunctions span the Hilbert space but do not form a basis, let alone an unconditional basis. The norms of the spectral projections, which are all of rank $1$, are not uniformly bounded: indeed there cannot be a polynomial bound on the rate of growth of the norms as $n$ increases. Numerical evidence strongly suggests exponential increase of the norms. These facts are correlated with very strong instability of the eigenvalues under small perturbations of the potential for all $n\geq 30$. Moreover the $L^{2}$ norms of the resolvent operators $(H-z)^{-1}$ increase faster than polynomially in $|z|$ as $z$ moves to infinity on certain rays through the origin, even though the distance of $z$ from the spectrum increases indefinitely. We have investigated complex resonances of \Schrodinger operators associated with potentials such as V(x)=x^{2}\rme^{-kx^{2}} for various values of $k>0$. Such self-adjoint operators have spectrum equal to $[0,\infty)$ and no eigenvalues. However, they have some complex resonances, defined and computed using dilation analyticity of the potential. For certain values of $k$ we find that most of the resonances are very unstable under small perturbations of the potential. Moreover some of the corresponding eigenfunctions do not have a simple form. While highly oscillatory, their absolute values may have two different local maxima. We have been able to approximate the eigenfunctions globally by a linear combination of two different JWKB solutions with a fair degree of accuracy. The method can be applied to a variety of similar operators, and depends on code which takes into account the rapid oscillations of the functions involved.

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