Hermitian equations couplings

The equations (17) are a set of N coupled equations for the N components of ll. Since the Hamiltonian is required to be Hermitian, the matrices must also be Hermitian, such that a = ad, f3 = ft. 

The effect of time correlations in the medium can be displayed working with a factorized coupling hamiltonian, which we choose here as Hps = AB, with A = a hermitian operator dependent only on the p-variables, and similarly for B dependent only on s-variables. This expression can be easily generalized to include several factorized terms. It leads to a residual Hamiltonian Hp = (A — ((A)))(B — ((B))) = AA(t).AB(t). Turning to the IP, the equation of motion for the RDOp is... 

It is clear that the necessary and sufficient number of equations in the coupled set [Eq. (78)] is equal to the number of unique cluster coefficients, provided that a solution exists. Since the coupled-cluster equations are non-Hermitian and nonlinear, the existence of solutions and the reality of eigenvalues corresponding to solutions are not guaranteed. However, Zivkovic and Monkhorst3536 have recently shown, using analytic continuations of solutions to the Cl problem, that for physically reasonable cases both the existence of solutions and the reality of eigenvalues are assured. 

From this equation, noting that Vve = Veep because H is Hermitian, and dropping the subscript (p on V (for just one bound state), we obtain, by the usual device of premultiplication by the complementary ket, a pair of coupled equations ... 

In Section 2.3 below we are going to show to what extend the anticommutation relations determine the properties of the Dirac matrices. Here we just note that these relations do not define the Dirac matrices uniquely. If (/ , 0, ) is a set of Hermitian matrices satisfying (5), then 0 = S0S and a). = SakS with some unitary matrix S is another set of Hermitian matrices obeying the same relations. Any specific set is said to define a representation of Dirac matrices. With respect to a given representation, the Dirac equation is a system of coupled linear partial differential equations. It is of first order in space and time derivatives. 

There is a separate solution corresponding to each possible incoming channel, and the solution is characterized at long range by the S-matrix with elements Sji. The S-matrix is an A open x A open complex symmetric matrix, where A open is the number of open channels. It is unitary, that is, SS = I, where indicates the Hermitian conjugate and / is a unit matrix. If the physical problem is factorized into separate sets of coupled equations for different symmetries (such as total angular momentum or parity), there is a separate S-matrix for each symmetry. All properties that correspond to completed collisions, such as elastic and inelastic integral and differential cross-sections, can be written in terms of S-matrices. 

Hermicity of the Hamiltonian H implies that the coupled equations conserve probability (unitarity) even with the approximation of using a finite basis set. However, further approximations in solving the coupled equations may result in the loss of unitarity. Such loss may occur since the coupling matrix js not necessarily Hermitian symmetric. Note that H is not equal to HL, in general. A well-known method to ensure unitarity is to force the coupling matrix to be Hermitian symmetric by... 

Although the matrix (H ) is not Hermitian symmetric, unitarity is retained when the coupled equations (20) are solved exactly. This has been done numerically by Lin and collaborators using wave functions with translational factors and realistic interaction potentials. Similarly, Pfeiffer and Garcia applied nonHermitian symmetric matrix elements under conditions preserving unitarity. They have been able to solve the coupled equations analytically on the basis of hydrogenic orbitals. However, in the calculation the overlap of the wave functions and the distortion in the diabatic potential have been neglected. 

We treat the coupling between the ICD electron and the dication by means of the Lippmann-Schwinger equation extended to non Hermitian Hamiltonians ... 

The remaining problem is how to solve the reaction-diffusion equations effectively. It is noted that the coupled equations, eqn (12.49) and eqn (12.50), can be transformed into Hermitian forms. To do so, one may use the substitution ... 

The use of the orbital concept in the Hartree-Fock (HE) and Kohn-Sham (KS) methods leads to similar variational equations a coupled set of eigenvalue equations with a hermitian operator (See for example [7, 25]). This system of integro-differential equations is transformed into a matrix problem when we use a basis set. In both methods, one has to solve a generalized eigenvalue equation ... 

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