Lippmann-Schwinger equations

Kouri D J, Huang Y, Zhu W and Hoffman D K 1994 Variational principles for the time-independent wave-packet-Schrddinger and wave-packet-Lippmann-Schwinger equations J. Chem. Phys. 100... 

The basic equations to be used are the Lippmann-Schwinger equations for the alloy wave function... 

General Time-to-Energy Transform of Wavepackets. Time-Independent Wavepacket-Schroedinger and Wavepacket-Lippmann-Schwinger Equations. 

Energy-Separable Polynomial Representation of the Time-Independent Full Green Operator with Application to Time-Independent Wavepacket Forms of Schrodinger and Lippmann-Schwinger Equations. 

Variational Principles for the Time-Independent Wave-Packet-Schrodinger and Wave-Packet-Lippmann-Schwinger Equations. 

The collision integrals of the kinetic equations (3.68) and (3.69) are essentially given by the scattering matrix ( T ). For this reason, the determination of such quantities from the Lippmann-Schwinger equations... 

When the PWCs are orthogonal among themselves (an assumption which is in fact not necessary and that thus far was not made) and to the localized channel, as is the case for the present treatment of the helium atom, the closecoupling ansatz [Eq. (52)] is equivalent to the Lippmann-Schwinger equation with the principal-value Green function [65]... 

From the Lippmann-Schwinger equation and the expansion of Go in some specified cell r/x, the coefficients in these expansions must be related by = cl + ft N jyf. Consistency conditions are derived by considering the alternative... 

In any cell r = r/(. with surface a = a/x, any solution ijr of the Lippmann-Schwinger equation dehnes auxiliary funchons xm = f fr Govf and = Jm3 r Govifr, which are equal by construction. After integration by parts,... 

The notation Wtl defines a Wronskian integral over ct/2. By the surface matching theorem, xm = X> the interior component of 4> Since 4r is a solution of the Lippmann-Schwinger equation, this implies xout = Xm when evaluated in the interior of Tjj. This is a particular statement of the tail-cancellation condition. To show this in detail, after integration by parts... 

In a true scattering problem, an incident wave is specified, and scattered wave components of ifr are varied. In MST or KKR theory, the fixed term x in the full Lippmann-Schwinger equation, f = x + / GqVms required to vanish, x is a solution of the Helmholtz equation. In each local atomic cell r of a space-filling cellular model, any variation of i// in the orbital Hilbert space induces an infinitesimal variation of the KR functional of the form 8 A = fr Govi/s) + he. This... 

These properties of the model Green function imply that the Lippmann-Schwinger equation [228],... 

Specializing the present derivation to the principal value Green function, the unsymmetrical expression tan = — 2(wo Av f) is exact for an exact solution of the Lippmann-Schwinger equation, but it is not stationary with respect to infinitesimal variations about such a solution. Since w0 = / + G Avf for such a solution, this can be substituted into the unsymmetrical formula to give an alternative, symmetrical expression tan r] = —2(/1 At> + AvG At> /), which is also not stationary. However, these expressions can be combined to define the Schwinger functional... 

This implies that [tan ] is stationary if and only if /, in the range of Av, satisfies the Lippmann-Schwinger equation. 

This multichannel matrix Green function determines a multichannel Lippmann-Schwinger equation... 

The multichannel Lippmann-Schwinger equation implies that w0 = u + GAvu. Substituting this into the second expression for K gives an alternative formula,... 

This expression is stationary for variations of u if and only if u is an exact solution of the multichannel Lippmann-Schwinger equation. The proof follows exactly as in the single-channel case, if the order of matrix products is maintained in the derivation. If the multichannel continuum solution is expanded as... 

We now use the Lippmann-Schwinger equation to explore the long-time behavior fif the continuum part of the wave packet P (0 that we have created with the laser Jilise [Eq. (2,28)]. We can use either the outgoing or incoming states as the basis set f exfianding P (r). In what follows we shall expose the different boundary condi-hs and see which type of solution is best suited for which purpose. Substituting (2 52) in Eq. (2.28), we obtain that... 

The adiabatic switching is introduced via the slowly varying e c term that guarantees that the interaction vanishes as t —> ice. It is the exact time-dependent analog of the procedure used in the iE derivation of the Lippmann-Schwinger equation [Eq. (2.52)] in the energy domain. 

The momentum-space integral equation is the Lippmann—Schwinger equation. It is an equation for the T matrix. We multiply (4.I0I) on the left by V, take the matrix element for the eigenstate (k of the final momentum, and introduce the spectral representation of K. 

The partial-wave Lippmann—Schwinger equation is (4.121). We retain the convention that k is on shell, that is... 

We form the T-matrix element (6.72) in the integral equation (6.70) and expand in the complete set of channel states jk ) to obtain the Lippmann—Schwinger equation for the T-matrix element. 

Some of the early calculations of electron—atom scattering assumed that the potential was small compared with the total energy so that it is a good approximation to iterate the Lippmann—Schwinger equation (6.73). Using the notation (6.65,6.71) the resultant series... 

Subsequent chapters will be concerned with non-perturbative solutions of the Lippmann—Schwinger equation. 

It would be convenient for solving the Lippmann—Schwinger equation (6.73) if we could make the potential matrix elements as small as possible. For example, we could hope to find a transformed equation whose iteration would converge much more quickly. This is achieved by a judicious choice of a local, central potential U, which is called the distorting potential since the problem is reformulated in terms of the distorted-wave eigenstates of U rather than the plane waves of (6.73). An important particular case of U is the Coulomb potential Vc in the case where the target is charged. The Hamiltonian (6.2) is repartitioned as follows... 

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