We obtain an exact analytical solution of the Klein Gordon equation for the equal vector and scalar Rosen Morse and Eckart potentials as well as the parity-time （PT） symmetric version of the these potentials by usin...We obtain an exact analytical solution of the Klein Gordon equation for the equal vector and scalar Rosen Morse and Eckart potentials as well as the parity-time （PT） symmetric version of the these potentials by using the asymptotic iteration method. Although these PT symmetric potentials are non-Hermitian, the corresponding eigenvalues are real as a result of the PT symmetry.展开更多
The present paper deals with the eigenvalues of complex nonlocal Sturm-Liouville boundary value problems.The bounds of the real and imaginary parts of eigenvalues for the nonlocal Sturm-Liouville differential equation...The present paper deals with the eigenvalues of complex nonlocal Sturm-Liouville boundary value problems.The bounds of the real and imaginary parts of eigenvalues for the nonlocal Sturm-Liouville differential equation involving complex nonlocal potential terms associated with nonlocal boundary conditions are obtained in terms of the integrable conditions of coefficients and the real part of the eigenvalues.展开更多
The authors present their analysis of the differential equation d X(t)/ d t = AX(t)-X T (t)BX(t)X(t) , where A is an unsymmetrical real matrix, B is a positive definite symmetric real matrix, X ∈...The authors present their analysis of the differential equation d X(t)/ d t = AX(t)-X T (t)BX(t)X(t) , where A is an unsymmetrical real matrix, B is a positive definite symmetric real matrix, X ∈R n; showing that the equation characterizes a class of continuous type full feedback artificial neural network; We give the analytic expression of the solution; discuss its asymptotic behavior; and finally present the result showing that, in almost all cases, one and only one of following cases is true. 1. For any initial value X 0∈R n, the solution approximates asymptotically to zero vector. In this case, the real part of each eigenvalue of A is non positive. 2. For any initial value X 0 outside a proper subspace of R n, the solution approximates asymptotically to a nontrivial constant vector (X 0) . In this case, the eigenvalue of A with maximal real part is the positive number λ=‖(X 0)‖ 2 B and (X 0) is the corresponding eigenvector. 3. For any initial value X 0 outside a proper subspace of R n, the solution approximates asymptotically to a non constant periodic function (X 0,t) . Then the eigenvalues of A with maximal real part is a pair of conjugate complex numbers which can be computed.展开更多
1 IntroductionLet R<sup>n×n</sup> be the set of all n×n real matrices.R<sup>n</sup>=R<sup>n×1</sup>.C<sup>n×n</sup>denotes the set of all n×n co...1 IntroductionLet R<sup>n×n</sup> be the set of all n×n real matrices.R<sup>n</sup>=R<sup>n×1</sup>.C<sup>n×n</sup>denotes the set of all n×n complex matrices.We are interested in solving the following inverse eigenvalue prob-lems:Problem A （Additive inverse eigenvalue problem） Given an n×n real matrix A=(a<sub>ij</sub>),and n distinct real numbers λ<sub>1</sub>,λ<sub>2</sub>,…,λ<sub>n</sub>,find a real n×n diagonal matrix展开更多
文摘We obtain an exact analytical solution of the Klein Gordon equation for the equal vector and scalar Rosen Morse and Eckart potentials as well as the parity-time （PT） symmetric version of the these potentials by using the asymptotic iteration method. Although these PT symmetric potentials are non-Hermitian, the corresponding eigenvalues are real as a result of the PT symmetry.
基金Supported by the National Nature Science Foundation of China(12101356,12101357,12071254,11771253)the National Science Foundation of Shandong Province(ZR2021QA065,ZR2020QA009,ZR2021MA047)the China Postdoctoral Science Foundation(2019M662313)。
文摘The present paper deals with the eigenvalues of complex nonlocal Sturm-Liouville boundary value problems.The bounds of the real and imaginary parts of eigenvalues for the nonlocal Sturm-Liouville differential equation involving complex nonlocal potential terms associated with nonlocal boundary conditions are obtained in terms of the integrable conditions of coefficients and the real part of the eigenvalues.
文摘The authors present their analysis of the differential equation d X(t)/ d t = AX(t)-X T (t)BX(t)X(t) , where A is an unsymmetrical real matrix, B is a positive definite symmetric real matrix, X ∈R n; showing that the equation characterizes a class of continuous type full feedback artificial neural network; We give the analytic expression of the solution; discuss its asymptotic behavior; and finally present the result showing that, in almost all cases, one and only one of following cases is true. 1. For any initial value X 0∈R n, the solution approximates asymptotically to zero vector. In this case, the real part of each eigenvalue of A is non positive. 2. For any initial value X 0 outside a proper subspace of R n, the solution approximates asymptotically to a nontrivial constant vector (X 0) . In this case, the eigenvalue of A with maximal real part is the positive number λ=‖(X 0)‖ 2 B and (X 0) is the corresponding eigenvector. 3. For any initial value X 0 outside a proper subspace of R n, the solution approximates asymptotically to a non constant periodic function (X 0,t) . Then the eigenvalues of A with maximal real part is a pair of conjugate complex numbers which can be computed.
文摘1 IntroductionLet R<sup>n×n</sup> be the set of all n×n real matrices.R<sup>n</sup>=R<sup>n×1</sup>.C<sup>n×n</sup>denotes the set of all n×n complex matrices.We are interested in solving the following inverse eigenvalue prob-lems:Problem A （Additive inverse eigenvalue problem） Given an n×n real matrix A=(a<sub>ij</sub>),and n distinct real numbers λ<sub>1</sub>,λ<sub>2</sub>,…,λ<sub>n</sub>,find a real n×n diagonal matrix
