Purely bound wave

Wigner s study of the correlation effects for high-electron densities is closely connected to the standard methods described in Section III.E., and the main errors come from the restricted form of the wave function (Eq. III.7) and the fact that this function does not represent a pure spin state. Hence, Wigner obtains only an upper bound for the correlation energy in this case. 

M. Rosina, (a) Direct variational calculation of the two-body density matrix (b) On the unique representation of the two-body density matrices corresponding to the AGP wave function (c) The characterization of the exposed points of a convex set bounded by matrix nonnegativity conditions (d) Hermitian operator method for calculations within the particle-hole space in Reduced Density Operators with Applications to Physical and Chemical Systems—II (R. M. Erdahl, ed.), Queen s Papers in Pure and Applied Mathematics No. 40, Queen s University, Kingston, Ontario, 1974, (a) p. 40, (b) p. 50, (c) p. 57, (d) p. 126. 

If k is purely imaginary and positive, then these states correspond to bound states with asymptotic behavior L) bound state solutions are the only solutions with positive imaginary values of the wave vector [31]. 

Of these, the pure electron-electron Coulomb interaction (4.14a) appears to be the obvious choice and, indeed, has been widely used [12,14,16]. The electron-electron contact interaction (4.14b), which only acts if both electrons are at the position of the ion (in effect, a three-body contact interaction), has also been frequently employed [15], Both interactions have been compared in various regards in [17,18,40]. More recently, the Coulomb interaction (4.14c), which is only effective if the second (bound) electron is located at the position of the ion, and the electron-electron contact interaction (4.14d), which is not restricted to the position of the ion, have also been studied [27]. The interactions (4.14b) and (4.14c) are effective three-body interactions, which attempt to take into account that the effective electron-electron interaction will depend on the positions of the electrons relative to the ion. An alternative interpretation, which formally leads to the same results, is to consider a two-body interaction Vi2 in (4.17) and a wave function (rlip ) in (4.18) that is extremely strongly localized at the position of the ion for details, see [27]. 

In Fig. 4.8 the effect of the initial-state wave functions is explored, for the case where the crucial electron-electron interaction is the two-body Coulomb interaction (4.14a) and for the case where this interaction is the two-body contact interaction (4.14d), which is not restricted to the position of the ion. In both cases, the form factor includes the function (4.23), which favors momenta such that pi + p2 is large. This is clearly visible for the contact interaction (4.14d) and less so for the Coulomb interaction (4.14a) whose form factor also includes the factor (4.19), which favors pi = 0 (or p2 = 0)- We conclude that (i) the effect of the specific bound state of the second electron is marginal and (ii) that a pure two-body interaction, be it of Coulomb type as in (4.14a) or contact type as in (4.14d), yields a rather poor description of the data. A three-body effective interaction, which only acts if the second electron is positioned at the ion, provides superior results, notably the three-body contact interaction (4.14b), cf. the left-hand panel (d). This points to the significance of the interaction of the electrons with the ion, which so far has not been incorporated into the S-matrix theory beyond the very approximate description via effective three-body interactions such as (4.14b) or (4.14c). 

In this section we shall discuss in some detail the formalism needed to apply the so(4, 2) algebraic methods to problems whose unperturbed Hamiltonian is hydrogenic. First a scaling transformation is applied to obtain a new Hamiltonian whose unperturbed part is just the so(2, 1) generator T3, which has a purely discrete spectrum. Next we use the scaled hydrogenic eigenfunctions of T3 as a basis for the expansion of the exact wave function. This discrete basis is complete with respect to the expansion of bound-state wave functions whereas the usual bound-state eigenfunctions do not form a complete set continuum functions must also be included to ensure completeness (cf. Section VI,A)-... 

From, the purely niatlHunatica, point of vit w oiu might b( . tempted to assume a similar expr( Ssion but with th( ])lus sign in the exponent from the physica.1 point of view, however, siudi an a,ssumption is not permissil)lc, as in that eas( the wave function would increase beyond all bounds as x inen a-sed.)... 

The wavenumber interval on which new diffusion instability modes exist is bounded from below and not bounded from above. Consequently, these perturbations are mainly shortwave. At the same time, the formation of regular nonlinear wave structures in falling film of a pure liquid is associated with the fact that only perturbations on the bounded interval 0 < a < are unstable. 

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