- Guaranteed resonance enclosures and exclosures for atoms and molecules
S. Bögli, B. M. Brown, M. Marletta, C. Tretter, and M. Wagenhofer, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng Sci. 470 (2014), 17 pp. (doi:10.1098/rspa.2014.0488)
Abstract: In this paper, we confirm, with absolute certainty, a conjecture on a certain oscillatory behaviour of higher auto-ionizing resonances of atoms and molecules beyond a threshold. These results not only definitely settle a more than 30 year old controversy in Rittby et al. (1981 Phys. Rev. A 24, 1636–1639 (doi:10.1103/PhysRevA.24.1636)) and Korsch et al. (1982 Phys. Rev. A 26, 1802–1803 (doi:10.1103/PhysRevA.26.1802)), but also provide new and reliable information on the threshold. Our interval-arithmetic-based method allows one, for the first time, to enclose and to exclude resonances with guaranteed certainty. The efficiency of our approach is demonstrated by the fact that we are able to show that the approximations in Rittby et al. (1981 Phys. Rev. A 24, 1636–1639 (doi:10.1103/PhysRevA.24.1636)) do lie near true resonances, whereas the approximations of higher resonances in Korsch et al. (1982 Phys. Rev. A 26, 1802–1803 (doi:10.1103/PhysRevA.26.1802)) do not, and further that there exist two new pairs of resonances as suggested in Abramov et al. (2001 J. Phys. A 34, 57–72 (doi:10.1088/0305-4470/34/1/304)).
- Remarks on the convergence of pseudospectra
S. Bögli and P. Siegl, Integral Equations Operator Theory 80 (2014), pp. 303-321 (doi:10.1007/s00020-014-2178-1, arXiv:1408.3431 [math.SP])
Abstract: We establish the convergence of pseudospectra in Hausdorff distance for closed operators acting in different Hilbert spaces and converging in the generalised norm resolvent sense. As an assumption, we exclude the case that the limiting operator has constant resolvent norm on an open set. We extend the class of operators for which it is known that the latter cannot happen by showing that if the resolvent norm is constant on an open set, then this constant is the global minimum. We present a number of examples exhibiting various resolvent norm behaviours and illustrating the applicability of this characterisation compared to known results.
- Schrödinger operator with non-zero accumulation points of complex eigenvalues
S. Bögli, Comm. Math. Phys. 352 (2017), pp. 629-639 (doi:10.1007/s00220-016-2806-5, arXiv:1605.09356 [math.SP])
Abstract: We study Schrödinger operators H=−Δ+V in L^2(Ω) where Ω is R^d or the half-space R_+^d, subject to (real) Robin boundary conditions in the latter case. For p>d we construct a non-real potential V∈L^p(Ω)∩L^∞(Ω) that decays at infinity so that H has infinitely many non-real eigenvalues accumulating at every point of the essential spectrum σ_ess(H)=[0,∞). This demonstrates that the Lieb-Thirring inequalities for selfadjoint Schrödinger operators are no longer true in the non-selfadjoint case.
- Approximations of spectra of Schrödinger operators with complex potentials on R^d
S. Bögli, P. Siegl, and C. Tretter, Comm. Part. Diff. Eq. 42, pp. 1001-1041 (doi:10.1080/03605302.2017.1330342, arXiv:1512.01826 [math.SP])
Abstract: We study spectral approximations of Schrödinger operators T=−Δ+Q with complex potentials on Ω=R^d, or exterior domains Ω⊂R^d, by domain truncation. Our weak assumptions cover wide classes of potentials Q for which T has discrete spectrum, of approximating domains Ω_n, and of boundary conditions on ∂Ω_n such as mixed Dirichlet/Robin type. In particular, Re Q need not be bounded from below and Q may be singular. We prove generalized norm resolvent convergence and spectral exactness, i.e. approximation of all eigenvalues of T by those of the truncated operators T_n without spectral pollution. Moreover, we estimate the eigenvalue convergence rate and prove convergence of pseudospectra. Numerical computations for several examples, such as complex harmonic and cubic oscillators for d=1,2,3, illustrate our results.
- Convergence of sequences of linear operators and their spectra
S. Bögli, Integral Equations Operator Theory 88 (2017), pp. 559-599 (doi:10.1007/s00020-017-2389-3, arXiv:1604.07732 [math.SP])
Abstract: We establish spectral convergence results of approximations of unbounded non-selfadjoint linear operators with compact resolvents by operators that converge in generalized strong resolvent sense. The aim is to establish general assumptions that ensure spectral exactness, i.e. that every true eigenvalue is approximated and no spurious eigenvalues occur. A main ingredient is the discrete compactness of the sequence of resolvents of the approximating operators. We establish sufficient conditions and perturbation results for strong convergence and for discrete compactness of the resolvents.
- Local convergence of spectra and pseudospectra
S. Bögli, Journal of Spectral Theory 8 (2018), pp. 1051–1098
(doi:10.4171/JST/222, arXiv:1605.01041 [math.SP])
Abstract: We prove local convergence results for the spectra and pseudospectra of sequences of linear operators acting in different Hilbert spaces and converging in generalised strong resolvent sense to an operator with possibly non-empty essential spectrum. We establish local spectral exactness outside the limiting essential spectrum, local ε-pseudospectral exactness outside the limiting essential ε-near spectrum, and discuss properties of these two notions including perturbation results.
(Eigenvalues (blue dots) and pseudospectra (different shades of grey) of a compactly perturbed Toeplitz operator (symbol curve in red), approximation by truncating infinite matrix to finite sections)