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Hermitian matrices of three parameters: perturbing coalescing eigenvalues and a numerical method

Authors: Luca Dieci and Alessandro Pugliese
Journal: Math. Comp. 84 (2015), 2763-2790
MSC (2010): Primary 15A18, 15A23, 65P30
Published electronically: May 27, 2015
MathSciNet review: 3378847
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Abstract: In this work we consider Hermitian matrix-valued functions of 3 (real) parameters, and are interested in generic coalescing points of eigenvalues, conical intersections. Unlike our previous works [L. Dieci, A. Papini and A. Pugliese, Approximating coalescing points for eigenvalues of Hermitian matrices of three parameters, SIAM J. Matrix Anal. Appl., 2013] and [L. Dieci and A. Pugliese, Hermitian matrices depending on three parameters: Coalescing eigenvalues, Linear Algebra Appl., 2012], where we worked directly with the Hermitian problem and monitored variation of the geometric phase to detect conical intersections inside a sphere-like region, here we consider the following construction: (i) Associate to the given problem a real symmetric problem, twice the size, all of whose eigenvalues are now (at least) double, (ii) perturb this enlarged problem so that, generically, each pair of consecutive eigenvalues coalesce along curves, and only there, (iii) analyze the structure of these curves, and show that there is a small curve, nearly planar, enclosing the original conical intersection point. We will rigorously justify all of the above steps. Furthermore, we propose and implement an algorithm following the above approach, and illustrate its performance in locating conical intersections.

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Additional Information

Luca Dieci
Affiliation: School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332

Alessandro Pugliese
Affiliation: Dipartimento di Matematica, Università degli Studi di Bari Aldo Moro, Via Orabona 4, Bari, 70125 Italy

Keywords: Hermitian matrix-valued functions, conical intersection, perturbed eigenproblem, diabolical points
Received by editor(s): May 16, 2013
Received by editor(s) in revised form: April 1, 2014
Published electronically: May 27, 2015
Additional Notes: This work was supported in part under INDAM GNCS. Support from the School of Mathematics of the Georgia Institute of Technology is also gratefully acknowledged.
Article copyright: © Copyright 2015 American Mathematical Society