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

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 [7, 4], 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. Notation. Below, Ω ⊂ R3 indicates an open region of R3 diffeomorphic to the open unit ball; ξ = (α, β, γ) ∈ Ω will indicate a general point in Ω. The metric is the Euclidean metric. We write A ∈ Ck(Ω,Cn×n), k ≥ 1, to indicate a smooth complex matrix-valued function defined on Ω and further A ∈ Cω if the dependence on parameter(s) is analytic. We write A∗ for the conjugate transpose of a matrix A, and have A = A∗ for a Hermitian matrix. The word Hermitian will imply complex valued entries. Similarly, the word symmetric will be restricted to matrices with real entries. With σ(A) we indicate the set of eigenvalues (repeated by multiplicity) of A.