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Properties of Definite Bethe–Salpeter Eigenvalue Problems

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Book cover Eigenvalue Problems: Algorithms, Software and Applications in Petascale Computing (EPASA 2015)

Part of the book series: Lecture Notes in Computational Science and Engineering ((LNCSE,volume 117))

Abstract

The Bethe–Salpeter eigenvalue problem is solved in condense matter physics to estimate the absorption spectrum of solids. It is a structured eigenvalue problem. Its special structure appears in other approaches for studying electron excitation in molecules or solids also. When the Bethe–Salpeter Hamiltonian matrix is definite, the corresponding eigenvalue problem can be reduced to a symmetric eigenvalue problem. However, its special structure leads to a number of interesting spectral properties. We describe these properties that are crucial for developing efficient and reliable numerical algorithms for solving this class of problems.

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Notes

  1. 1.

    A vector \(v\) is called \(C\)-positive, \(C\)-negative, \(C\)-neutral, respectively, if \(v^{{\ast}}C_{n}v> 0\), \(v^{{\ast}}C_{n}v <0\), \(v^{{\ast}}C_{n}v = 0\). An eigenvalue of \(\varOmega -\lambda C_{n}\) is called \(C\)-positive (\(C\)-negative) if its associated eigenvector is \(C\)-positive (\(C\)-negative).

  2. 2.

    In the case when \(\phi (X,Y )^{{\ast}}\varOmega H\phi (X,Y )\) is singular, we assign half of the zero eigenvalues with the positive sign in the notation \(\lambda _{i}^{+}\big(\phi (X,Y )^{{\ast}}\varOmega H\phi (X,Y )\big)\).

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Acknowledgements

The authors thank Ren-Cang Li for helpful discussions. Support for this work was provided through Scientific Discovery through Advanced Computing (SciDAC) program funded by U.S. Department of Energy, Office of Science, Advanced Scientific Computing Research.

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Authors and Affiliations

  1. Lawrence Berkeley National Laboratory, Berkeley, CA, 94720, USA

    Meiyue Shao & Chao Yang

Authors
  1. Meiyue Shao
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  2. Chao Yang
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Correspondence to Chao Yang .

Editor information

Editors and Affiliations

  1. Department of Computer Science, University of Tsukuba, Tsukuba, Ibaraki, Japan

    Tetsuya Sakurai

  2. Department of Electrical Engineering and Computer Science, Nagoya University, Nagoya, Aichi, Japan

    Shao-Liang Zhang

  3. RIKEN Advanced Institute for Computational Science, Kobe, Hyogo, Japan

    Toshiyuki Imamura

  4. Department of Communication Engineering and Informatics, University of Electro-Communications, Graduate School of Informatics and Engineering, Tokyo, Japan

    Yusaku Yamamoto

  5. Department of Particle Physics, University of Tsukuba, Tsukuba, Ibaraki, Japan

    Yoshinobu Kuramashi

  6. Department of Applied Mathematics and Physics, Tottori University, Tottori, Japan

    Takeo Hoshi

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Shao, M., Yang, C. (2017). Properties of Definite Bethe–Salpeter Eigenvalue Problems. In: Sakurai, T., Zhang, SL., Imamura, T., Yamamoto, Y., Kuramashi, Y., Hoshi, T. (eds) Eigenvalue Problems: Algorithms, Software and Applications in Petascale Computing. EPASA 2015. Lecture Notes in Computational Science and Engineering, vol 117. Springer, Cham. https://doi.org/10.1007/978-3-319-62426-6_7

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