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\def\prb#1#2#3{, Phys.\ Rev.\ B {\bf #1}, #2 (#3)}
\def\pra#1#2#3{, Phys.\ Rev.\ A {\bf #1}, #2 (#3)}
\def\jcp#1#2#3{, J.\ Chem.\ Phys.\ {\bf #1}, #2 (#3)}
\def\rmp#1#2#3{, Rev.\ Mod.\ Phys.\ {\bf #1}, #2 (#3)}
\def\ssc#1#2#3{, Solid State Commun.~{\bf #1}, #2 (#3)}
%\def\mp#1#2#3{, Mol.~Phys.~{\bf #1}, #2 (#3)}
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\def\cpl#1#2#3{, Chem.~Phys.~Lett.~{\bf #1}, #2 (#3)}
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\def\iniv{\int d^3x\,}
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\def\fto{\frac1{\sqrt{2\pi}}\int_{-\infty}^\infty d\om\,}
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\def\ekzokt{e^{i(kz-\om(k)t)}}
\def\ekdod{e^{i[\k\cdot(\x-\xp)-\om(t-t')]}}
\def\ekxxpa{e^{ik\xxpa}}
\def\qko{q(\k,\om)_}
\def\ekx{e^{i\k\cdot\x}}
\def\emkx{e^{-i\k\cdot\x}}
\def\ekz{e^{ikz}}
\def\emkz{e^{-ikz}}
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\def\ekrp{e^{ikr'}}
\def\ekR{e^{ikR}}
\def\ekRp{e^{ikR'}}
\def\emknxp{e^{-ik(\nn\cdot\xp)}}
\def\emkxp{e^{-i\k\cdot\xp}}
\def\emkzot{e^{-i(kz+\om t)}}
\def\ekrot{e^{i(kr-\om t)}}
\def\uvk{u_\k}
\def\uk{u_k}
\def\lep{\left(}
\def\rip{\right)}
\def\Phx{\Ph(\x)}
\def\Phxt{\Ph(\x,t)}
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\def\Phxp{\Ph(\xp)}
\def\phx{\ph(\x)}
\def\psx{\ps(\x)}
\def\fxt{f(\x,t)}
\def\Fxt{F(\x,t)}
\def\Fx{\F(\x)}
\def\Fxo{F(\x,\om)}
\def\Fko{F(\k,\om)}
\def\fxtp{f(\xp,t')}
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\def\psxp{\ps(\xp)}
\def\Ak{A(k)}
\def\uzt{U(z,t)}
\def\Ex{\E(\x)}
\def\Ext{\E(\x,t)}
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\def\Cx{\C(\x)}
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\def\Px{\P(\x)}
\def\Pxt{\P(\x,t)}
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\def\Bxt{\B(\x,t)}
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\def\Bko{\B(\k,\om)}
\def\Hx{\H(\x)}
\def\Hxt{\H(\x,t)}
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\def\Jx{\J(\x)}
\def\Jxt{\J(\x,t)}
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\def\Jxo{\J(\x,\om)}
\def\Jxop{\J(\x,\om')}
\def\Jxp{\J(\xp)}
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\def\N{{\bf N}}
\def\eq#1{Eq.~(#1)}
\def\rhx{\rh(\x)}
\def\rhr{\rh(\vr)}
\def\rhxt{\rh(\x,t)}
\def\rhko{\rh(\k,\om)}
\def\rhxp{\rh(\xp)}
\def\rhxo{\rh(\x,\om)}
\def\rhxop{\rh(\x,\om')}
\def\leb{\left[}
\def\rib{\right]}
\def\six{\si(\x)}
\def\sixp{\si(\xp)}
\def\gxxp{G(\x,\xp)}
\def\gxt{G(\x,t;\xp,t')}
\def\gko{g(\k,\om)}
\def\dexxp{\de(\x-\xp)}
\def\dettp{\de(t-t')}
\def\dekkp{\de(\k-\k')}
\def\deoop{\de(\om-\om')}
\def\fxxp{F(\x,\xp)}
\def\lapp{\nabla'^2}
\def\xxpi{\frac{1}{|\x-\xp|}}
\def\xxpa{|\x-\xp|}
\def\xxp{(\x-\xp)}
\def\dvl{d{\bf l}}
\def\gradp{\nabla'}
\def\nnp{{\bf n}'}
\def\hsp#1{\hspace*{#1 in}}
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\def\p{{\bf p}}
\def\r{{\bf r}}
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\def\rhat{\hat{{\bf r}}}
\def\eqs#1#2{Eqs.~(#1) and (#2)}
\def\eqss#1#2#3{Eqs.~(#1),~(#2), and~(#3)}
\def\sde#1#2{\frac{d^2#1}{d#2^2}}
\def\csde#1#2#3{\frac{d^2#1}{d#2 d#3}}
\def\y{{\bf y}}
\def\yp{{\bf y'}}
\def\unet{U_n(\et)}
\def\umet{U_m(\et)}
\def\unets{U_n^*(\et)}
\def\umets{U_m^*(\et)}
\def\inabet{\int_a^bd\et}
\def\Ps{\Psi}
\def\sth{\sin\th}
\def\cth{\cos\th}
\def\sph{\sin\ph}
\def\cph{\cos\ph}
\def\inoou{\int_{-1}^1du}
\def\lec{\left\{}
\def\ric{\right\}}
\def\ylm{Y_{l,m}(\th,\ph)}
\def\ylms{Y_{l,m}^*(\th,\ph)}
\def\ylmsp{Y_{l,m}^*(\th',\ph')}
\def\ylmp{Y_{l'm',}(\th,\ph)}
\def\ylmps{Y_{l'm'}^*(\th,\ph)}
\def\inom{\int d\Om}
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\def\Ga{\Gamma}
\def\De{\Delta}
\def\Th{\Theta}
\def\jnx{J_\nu(x)}
\def\nnx{N_\nu(x)}
\def\jnxy{J_\nu(x_{\nu n}y)}
\def\jxy{J_\nu(xy)}
\def\jxpy{J_\nu(x'y)}
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\def\jnxpy{J_\nu(x_{\nu n'}y)}
\def\xnnp{x_{\nu n'}}
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\def\edm{{\bf p}}
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\def\vereq#1#2{\lower3pt\vbox{\baselineskip1.5pt \lineskip1.5pt
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\def\alt{\mathrel{\mathpalette\vereq<}}
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\def\SE{{\Sc E}}
\def\SL{{\Sc L}}
\def\j{{\bf j}}
\def\jd{{\dot{\bf j}}}
\def\La{\Lambda}
\def\hb{\hbar}
\def\hbo{\hb \omega }
\def\hod{\hb \omega _D}
\def\kpr{k ^{\prime }}
\def\thk{\th _k}
\def\thkp{\th _{k^{\prime }}}
