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. A24, 1636–1639 (doi:10.1103/PhysRevA.24.1636)) and Korsch et al. (1982 Phys. Rev. A26, 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. A24, 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. A26, 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. A34, 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.
The essential numerical range for unbounded linear operators S. Bögli, M. Marletta, C. Tretter, Journal of Functional Analysis 108509 (2020)
(doi:10.1016/j.jfa.2020.108509, arXiv:1907.09599 [math.SP]) Abstract: We introduce the concept of essential numerical range W_e(T) for unbounded Hilbert space operators T and study its fundamental properties including possible equivalent characterizations and perturbation results. Many of the properties known for the bounded case do not carry over to the unbounded case, and new interesting phenomena arise which we illustrate by some striking examples. A key feature of the essential numerical range W_e(T) is that it captures spectral pollution in a unified and minimal way when approximating T by projection methods or domain truncation methods for PDEs.
Essential numerical ranges for linear operator pencils S. Bögli, M. Marletta, IMA Journal of Numerical Analysis (2020)
(doi:10.1093/imanum/drz049, arXiv:1909.01301 [math.SP]) Abstract: We introduce concepts of essential numerical range for the linear operator pencil λ↦A−λB. In contrast to the operator essential numerical range, the pencil essential numerical ranges are, in general, neither convex nor even connected. The new concepts allow us to describe the set of spectral pollution when approximating the operator pencil by projection and truncation methods. Moreover, by transforming the operator eigenvalue problem Tx=λx into the pencil problem BTx=λBx for suitable choices of B, we can obtain non-convex spectral enclosures for T and, in the study of truncation and projection methods, confine spectral pollution to smaller sets than with hitherto known concepts. We apply the results to various block operator matrices. In particular, Theorem 4.12 presents substantial improvements over previously known results for Dirac operators while Theorem 4.5 excludes spectral pollution for a class of non-selfadjoint Schrödinger operators which it has not been possible to treat with existing methods.
On the eigenvalues of the Robin Laplacian with a complex parameter S. Bögli, J. B. Kennedy, R. Lang, submitted for publication
(arXiv:1908.06041 [math.SP]) Abstract: We study the spectrum of the Robin Laplacian with a complex Robin parameter α on a bounded Lipschitz domain Ω. We start by establishing a number of properties of the corresponding operator, such as generation properties, local analytic dependence of the eigenvalues and eigenspaces on α∈C, and basis properties of the eigenfunctions. Our focus, however, is on bounds and asymptotics for the eigenvalues as functions of α: we start by providing estimates on the numerical range of the associated operator, which lead to new eigenvalue bounds even in the case α∈R. For the asymptotics of the eigenvalues as α→∞ in C, in place of the min-max characterisation of the eigenvalues and Dirichlet-Neumann bracketing techniques commonly used in the real case, we exploit the duality between the eigenvalues of the Robin Laplacian and the eigenvalues of the Dirichlet-to-Neumann map. We use this to show that every Robin eigenvalue either diverges to ∞ in C or converges to a point in the spectrum of the Dirichlet Laplacian, and also to give a comprehensive treatment of the special cases where Ω is an interval, a hyperrectangle or a ball. This leads to the conjecture that on a general smooth domain in dimension d≥2 all eigenvalues converge to the Dirichlet spectrum if Reα remains bounded from below as α→∞, while if Reα→−∞, then there is a family of divergent eigenvalue curves, each of which behaves asymptotically like −α^2.
(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)