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arXiv · 2002.05015

Eigenvalues of non-hermitian matrices: a dynamical and an iterative approach. Application to a truncated Swanson model

Fabio Bagarello·Francesco Gargano
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Abstract

We propose two different strategies to find eigenvalues and eigenvectors of a given, not necessarily Hermitian, matrix $A$. Our methods apply also to the case of complex eigenvalues, making the strategies interesting for applications to physics, and to pseudo-hermitian quantum mechanics in particular. We first consider a {\em dynamical} approach, based on a pair of ordinary differential equations defined in terms of the matrix $A$ and of its adjoint $A^\dagger$. Then we consider an extension of the so-called power method, for which we prove a fixed point theorem for $A\neq A^\dagger$ useful in the determination of the eigenvalues of $A$ and $A^\dagger$. The two strategies are applied to some explicit problems. In particular, we compute the eigenvalues and the eigenvectors of the matrix arising from a recently proposed quantum mechanical system, the {\em truncated Swanson model}, and we check some asymptotic features of the Hessenberg matrix.

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Fabio Bagarello, Francesco Gargano. 2020-02-12. Eigenvalues of non-hermitian matrices: a dynamical and an iterative approach. Application to a truncated Swanson model. https://doi.org/10.1002/mma.6317

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