Outgoing boundary condition

Associated with the pole of the S-matrix is a Seigert state, I-Ves, which has purely outgoing boundary conditions and satisfies (with some caveats) the equation, // I res = z les,H being the system Hamiltonian.44 If a square integrable approximation to I res is constructed, then its time evolution, k . (/,), wiH exhibit pure exponential decay after a transient induction period. Of course any L2 state will show quadratic, and hence non-exponential, decay at short times since... 

As in Eqs (18.19) we impose outgoing boundary conditions on the out state, by assigning a small damping rate rj to this channel. The corresponding terms containing r/ that appear in (18.58d,e) insure that a steady state is achieved if we also impose a constant CT,n,in on the dynamics (equivalent to the driving boundary condition f i (Z) = exp(—iEiat/K)Cin(O ) imposed on Eqs (18.19)). 

Thus A + ) has the outgoing boundary condition while A ) has the incoming boundary condition and they are called scattered waves. From Eqs. (63-67), the asymptotic boundary conditions for (A ) are... 

As long as the Hamiltonian matrix is defined in a minimal size (interaction plus edge) region the e — 0 limit has nothing to do with the outgoing boundary conditions for the Green s function i.e., the relation... 

V (z) describes a decreasing in time quasi-stationary state. Contrary to the Lippmann-Schwinger equation, which requires scattering boundary conditions, V (z) does require outgoing boundary conditions commensurate with the Gammow-Siegert method. It is inherent in the complex technique and defined in a nonambiguous manner as a continued wavefunction in the second Riemann sheet. 

The fulfillmenf of Eq. (53) generates a complex energy if fhe inward matrix incorporates outgoing boundary conditions (with a real coordinate) or bound state boundary conditions (with a complex rotated coordinate). 

The reality of Z(E) in Eq. (21) is fhe manifesfafion of the time-symmetric, Hermitian structure of the theory. On the other hand, it was shown [101] that k(E) becomes imaginary if the outgoing boundary conditions are imposed on the asymptotic form of T, (E), whereby fhe problem becomes non-Hermitian and time-asymmetric, and described by two adjoint CESEs, one of which corresponds to the physically relevant solution of fhe decaying sfafe. 

Abstract The theory of resonant states provides an exact analytical approach for the description of open quantum systems. Resonant states are defined by imposing purely outgoing boundary conditions to the solutions to the Schrodinger... 

Notice that the time reversed state U-n r) = u r) satisfies the conjugate of Eq. (5) with complex eigenvalue = k and the outgoing boundary-condition with wavenumber... 

We have shown that demanding purely outgoing boundary conditions on the solutions to the Schrbdinger equation for an arbitrary potential of finite range, i.e. V(r) = 0,r > a defines the resonant states of the system and leads... 

In equation (3), is the stationary value of the T-matrix and Tt ° is the trial value which appears in the trial wave functions, and corresponding to incoming and outgoing boundary conditions in channels F and To respectively. The working equations for and all other variational... 

Up to this point we have not discussed how complex conjugation is treated in the matrix elements appearing in equation (9). The trial functions in equation (3) are and with incoming and outgoing boundary conditions respectively. The only functions in the basis with a complex radi part are... 

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