School of Mathematical Sciences

External seminar: Frank Rösler (Durham University)

Wednesday 21st October 2015 (16:00-17:00)

Yves Van Gennip


[Algebra and Analysis seminar]

A Bound on the Pseudospectrum of the Harmonic Oscillator with Imaginary Cubic Potential

We are concerned with the non-normal operator

H = −d²/dx² + ix³ + cx² + bix

on dom(H) = {φ L²(R ) | Hφ L²(R)}, where c > 0, b 0 are constants. It is known that the spectrum of H is entirely real and discrete. The ε-pseudospectrum of the operator, however, contains an unbounded set for any ε > 0 and thus does not approximate the spectrum in a global sense.

By exploiting the fact that the semigroup exp(−tH) is eventually compact we show a complementary result, namely that the pseudospectrum is contained in the unification of small discs around the eigenvalues and a right half-plane which moves towards +∞ as ε decreases.

Furthermore, semiclassical methods can be employed to show that for c < 0 no inclusion of the above type is possible. In fact, H does not even generate a bounded semigroup in this case.

Joint work with Patrick Dondl and Patrick Dorey.

School of Mathematical Sciences

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