Abstract
We present an intuitive and scalable algorithm for the diagonalization of complex symmetric matrices, which arise from the projection of pseudo-Hermitian and complex scaled Hamiltonians onto a suitable basis set of “trial” states. The algorithm diagonalizes complex and symmetric (non-Hermitian) matrices and is easily implemented in modern computer languages. It is based on generalized Householder transformations and relies on iterative similarity transformations T → T′ = Q T T Q, where Q is a complex and orthogonal, but not unitary, matrix, i.e. Q T = Q −1 but Q + ≠ Q −1. We present numerical reference data to support the scalability of the algorithm. We construct the generalized Householder transformations from the notion that the conserved scalar product of eigenstates Ψ n and Ψ m of a pseudo-Hermitian quantum mechanical Hamiltonian can be reformulated in terms of the generalized indefinite inner product ∫ dx Ψ n (x, t) Ψ m (x, t), where the integrand is locally defined, and complex conjugation is avoided. A few example calculations are described which illustrate the physical origin of the ideas used in the construction of the algorithm.
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Noble, J.H., Lubasch, M. & Jentschura, U.D. Generalized Householder transformations for the complex symmetric eigenvalue problem. Eur. Phys. J. Plus 128, 93 (2013). https://doi.org/10.1140/epjp/i2013-13093-1
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DOI: https://doi.org/10.1140/epjp/i2013-13093-1