| Recently,the Bethe-Salpeter equation,named after Hans Bethe and Edwin Salpeter,is widely used in many academic disciplines such as quantum physics and quantum chemistry.In multiphysics,the two-particle collective excitation is usually described by a two-particle Green’s function whose excited state energy level corresponds to the pole of this function.At present,studies have shown that the two-particle Green’s function satisfies the Bethe-Salpeter equation,and its pole can be obtained by solving the eigenvalues of a related Hamiltonian.Using an appropriate discrete method,the operator can be expressed as a Bethe-Salpeter eigenvalue problem with a special structure where A,B∈Cn×n with AH=A,Bτ=B.The special structure of the matrix(?)makes its eigenvalues have(λ,λ,-λ,-λ)important properties of pairing.In order to maintain these special structure,in this paper we have proposed two structure-preserving algorithms,ΓQR algorithm and Γ-Lanczos algorithm,which are used to solve the small-sized and medium-sized or large-sized sparse Bethe-Salpeter eigenvalue problems.Using the matrix Π ≡(?),define the Π±-matrix,Π-Hermitian,Π-tridiagonal,Π-diagonal matrix structure,and Γ-orthogonal re-lationship and ΓQR decomposition of Π±-matrix.In the paper,above(?)is a class of Π--Hermitian matrix.Based on the classic QR method,we propose theΓQR method for the dense small-sized or medium-sized Bethe-Salpeter eigenvalue problem,which maintains the Π-Hermitian structure of the matrix during the it-erative process.Using the techniques of implicit double shift,we give the detailed algorithm and theoretical proof,the numerical experiments show the effectiveness of the algorithm.In addition,for the large-scale sparse Bethe-Salpeter eigenvalue problem,we construct the Γ-Lanczos decomposition of the Π-Hermitian matrix using the properties such as Γ-orthogonal,combined with the technique of dis-placement inversion.Based on the ΓQR method,a Γ-Lanczos method for theΠ-Hermitian structure is proposed.This method can effectively find a few required eigenvalues and corresponding eigenvectors of(?).The convergence theorem and error analysis guarantee the effectiveness of the methodThe structure of this paper is as follows:The first chapter introduces the background of Bethe-Salpeter eigenvalue problem,and briefly describes two com-mon methods for solving eigenvalues.It clarifies the current research status and the main direction of this topic.The second chapter gives some related defini-tions and theories,as well as the ΓQR algorithm for solving the small-sized and medium-sized Bethe-Salpeter eigenvalue problem.The third chapter proposes the Γ-Lanczos algorithm for the structure-preserving of large-scale Bethe-Salpeter eigenvalue problems.In Chapter 4 we show some experimental results of numerical experiments. |
