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As discussed in Secs. II A and IV B the in-plane magnetic field effectively enhances the confinement potential of a parabolic well. Therefore any small deviation of the bare confinement potential from a perfect parabolic one is relatively less important when a strong in-plane magnetic field is applied. Also if needed, nonparabolic effects can be included in our analysis simply by calculating the matrix element in the corresponding noninteracting basis (see Appendix A), which has to be carried out numerically for nonparabolic confinement potentials.
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There are dual purposes for our development of the screening theory in this work. The first is simply to develop an improved static theory of screening in the quantum Hall situation which transcends the simple theory of Ref. 35, and includes ladder diagrams in the polarizability function by going beyond the RPA. This is the primary motivation for our developing the screening theory including the ladder-exchange diagrams in the polarizability. We believe this to be a qualitative improvement on the ordinary Hartree-Fock screening, but it is difficult to precisely quantify the improvement. The second (and equally important) reason for our screening theory is to obtain the collective mode dispersion in an "improved" approximation where screened static exchange diagrams have been included in calculation. The screened static exchange approximation developed by us in this work in the quantum Hall regime is similar to the widely used Hubbard approximation [J. Hubbard, Proc. R. Soc. London, Ser. A 243, 336 (1957)] in the electron-gas problem.
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