Siegert boundary condition

Consider the one dimensional TISE in Eq. (24), where we allow x to vary between -L and L, i.e., in the interaction region only. In order to solve this equation, we must supplement it with some boundary conditions. Although motivated from different quantum phenomena, Siegert [30] was the first to introduce the idea of solving the TISE with outgoing BCs, also known as Siegert boundary conditions or radiation boundary conditions. In one dimension, these outgoing BCs read... 

In general, the eigenvalues and eigenstates are complex. It is common to divide the spectrum of the Hamiltonian with Siegert boundary conditions into four parts ... 

Electron Propagator with Siegert Boundary Condition... 

Electron Propagator with Siegert Boundary condition /24/ 125.47 0.02... 

An interesting aspect of this case is that Siegert boundary conditions with a real coordinate are difficult to formulate in spherical coordinates (the diagonal elements of the potential matrix go to zero and the off-diagonal ones to infinity). Complex rotation obviates these difficulties. The initial boundary condition for inward propagation can be taken in all cases to be (r2) 0. 

Siegert boundary conditions also imply that the open-channel wave numbers are complex, since Eq. (41) is now used with a complex energy. Thus k can be written as... 

It is possible to implement explicitly Siegert boundary conditions in the determination of the multichannel wavefunction. An alternative approach is to transform the reaction coordinate R into a complex one R = p exp i )... 

Associated with the pole of the S-matrix is a Siegert state, s, which has purely outgoing boundary conditions and satisfies (with some caveats) the equation, HTy, = 2 Fir. H being the system Hamiltonian [73]. If a square integrate approximation to T res is constructed, then its time evolution, (f), will exhibit pure exponential decay after a transient induction period. Of course, any L2 state will show quadratic, and hence nonexponential, decay at short times since... 

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. 

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