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 @@ -381,7 +381,8 @@ \section{Bogoliubov transformation} 381 381  \|_{H^1}^{4n}, \tag{iv} 382 382  \end{align} 383 383  where $C$ is a constant that depends only on $a_0$ and $\rho$ from Lemma 384 - \ref{l:w}. Consequently, 384 + \ref{l:w}. Consequently, there is a constant $C_1$ that depends only on $\| \varphi \|_{H^1}$, and 385 + $a_0$ and $\rho$ from Lemma \ref{l:w} such that 385 386   386 387  \begin{alignedat}{2} 387 388  \| p \|_{L^2} & \le C_1, \qquad & \| \nabla_1 p \|_{L^2} & \le C_1, \\ @@ -394,18 +395,14 @@ \section{Bogoliubov transformation} 394 395  \| s \|_{L^2} \le C_1, \qquad \| \nabla_1 s \|_{L^2} \apprle \| \varphi 395 396  \|_{H^1}^2 \sqrt{N}, \tag{vi} 396 397   397 - where $C_1$ is a constant that depends only on $\| \varphi \|_{H^1}$, and 398 - $a_0$ and $\rho$ from Lemma \ref{l:w}. 399 - 400 -INSERTED \marginpar{please integrate in a way you like}: ------------------------------------------------------\\ 401 -We also make use of the following estimates (with $C$ independent of $N$ and $t$): \marginpar{I have a proof for this, but we should discuss first who edits the proof in the file.} 398 + and 402 399  \bd 403 -\sup_{x} \norm{s_x}_2^2 \leq C \norm{\ph}_{H^2}^2, \quad \sup_{x} \norm{p_x}_2^2 \leq C \norm{\ph}_{H^2}^2, 400 +\sup_{x} \norm{s_x}_2^2 \leq C \norm{\varphi}_{H^2}^2, \quad \sup_{x} \norm{p_x}_2^2 \leq C \norm{\varphi}_{H^2}^2, 404 401  \ed 405 402  \bd 406 -\lvert r(x,y)\rvert \leq C \lvert\ph(x)\rvert \lvert\ph(y)\rvert, \quad \lvert k(x,y)\rvert \leq N \lvert \ph(x)\rvert \lvert \ph(y)\rvert. 403 +\lvert r(x,y)\rvert \leq C \lvert\varphi(x)\rvert \lvert\varphi(y)\rvert, \quad \lvert k(x,y)\rvert \leq N \lvert \varphi(x)\rvert \lvert \varphi(y)\rvert. 407 404  \ed 408 ------------------------------------------------------- 405 +We emphasize that $C$ and $C_1$ do not depend on $N$. 409 406  \end{prp} 410 407   411 408   @@ -546,11 +543,15 @@ \section{Bogoliubov transformation} 546 543   547 544   548 545  (vi) Recall that $s = k + r$. Observe that part (vi) follows from (i), (ii) 549 - and (v). 546 + and (v). \marginpar{my part of the proof has to be added. N} 550 547  \end{proof} 551 548   552 549   553 550  \section{The generator} 551 +Define 552 +\bd 553 +U_N(t) := T^\ast_t W^\ast_t e^{-it \Hcal_N} W_0 T_0. 554 +\ed 554 555  The generator $\Lcal_N(t)$ is defined by 555 556  \bd 556 557  \Lcal_N(t) U_N(t) = i \partial_t U_N(t). @@ -570,8 +571,6 @@ \section{The generator} 570 571  & + N b(N,t), 571 572  \end{align*} 572 573  with a phase (which, like all phases, will be dropped from now on without any further comment) 573 -\marginpar{It's clear, but perhaps we should mention that these functions 574 -converge as $N \to \infty$. Is $b(N,t) \to 0$ as $N \to \infty$?} 575 574  \bd 576 575  b(N,t) = \norm{\nabla \ph}_{L^2}^2 - \Im \scal{\ph}{\phdot} + \frac{1}{2}\int \di x \di y\, NV_N(x-y) \lvert \ph(x)\rvert^2 \lvert \ph(y) \rvert^2. 577 576  \ed @@ -655,7 +654,7 @@ \section{The generator} 655 654   656 655  \section{Estimates for the terms of the generator} 657 656  \label{ch:generatorestimates} 658 -In this chapter we denote the expectation value $\scal{\psi}{A\psi}$ by $\ev{A}$. In this chapter, $C$ is a constant which can change from line to line in inequalities, but which is always independent of $N$ and $t$. 657 +In this chapter we denote the expectation value $\scal{\psi}{A\psi}$ by $\ev{A}$. In this chapter, $C$ is a constant which can change from line to line in inequalities, but which is always independent of $N$ and $t$. The constant $C$ can depend on $\norm{\ph}_{H^1}$. 659 658   660 659  All bounds proven in this chapter hold as operator inequalities, independent of the choice of $\psi$. In particular, the structure of the time evolution $\psi_N(t) = U_N(t)\psi_0$ is not used in this chapter. 661 660   @@ -1110,33 +1109,7 @@ \section{Estimates for the terms of the generator} 1110 1109  \leq & C\norm{\ph}_{H^2}^2 \frac{1}{\sqrt{N}}\ev{\Ncal+1}.  1111 1110  \end{align*} 1112 1111   1113 -\iffalse 1114 -\begin{lem} 1115 - \bd 1116 - \norm{k} \leq C, \quad \norm{\nabla k} \leq \sqrt{N} 1117 - \ed 1118 - \bd 1119 - \lvert p(x,y) \rvert \leq C \lvert \ph(x) \rvert\cdot \lvert \ph(y) \rvert, \quad \norm{p} \leq C, \quad \norm{\nabla p} \leq C 1120 - \ed 1121 -\bd 1122 -\mbox{maybe } \sup_x \norm{s_x}_2^2 \leq C \mbox{ (can be used ?)} 1123 -\ed 1124 -\end{lem} 1125 - 1126 -\begin{lem} 1127 -\bd 1128 -\Ncal \leq C T^*_t (\Ncal + 1) T_t, \quad \Ncal^2 \leq C T^*_t (\Ncal + 1)^2 T_t, \quad T^*_t (\Ncal + 1) T_t \leq \Ncal 1129 -\ed 1130 -\end{lem} 1131 - 1132 -\begin{lem} 1133 -\label{lem:nsquaredbound} 1134 -For $\ev{\cdot} = \scal{U_N(t)\Omega}{\cdot U_N(t)\Omega}$ or maybe easier $\ev{\cdot} = \scal{U_N(t)T^*\Omega}{\cdot U_N(t)T^*\Omega}$, we have \marginpar{$\Ncal\Kcal$ not needed anymore?} 1135 - \bd 1136 - \frac{1}{N}\ev{\Ncal^2} \leq C \ev{\Ncal}, \quad \frac{1}{N}\ev{\Ncal \Kcal} \leq \ev{\Kcal} 1137 - \ed 1138 -\end{lem} 1139 -\fi 1112 +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 1140 1113   1141 1114  We still have to give a bound for the $\ev{(\partial_t T^*_t)T_t}$ part of the generator. We start with a number of simple lemmata. 1142 1115  \begin{lem} @@ -1147,64 +1120,29 @@ \section{Estimates for the terms of the generator} 1147 1120  \ed 1148 1121  and 1149 1122  \bd 1150 -\lvert \scal{\psi}{\frac{1}{2} \int \di x\di y \left( f_1(x,y) a^\ast_x a_y + f_2(x,y) a_x a^\ast_y \right) \psi} \rvert \leq \ev{\Ncal} \frac{\norm{f_1}+\norm{f_2}}{2} + \frac{1}{2} \left\lvert \int \di x\, f_2(x,x) \right\rvert \scal{\psi}{\psi}. 1123 +\lvert \scal{\psi}{\frac{1}{2} \int \di x\di y \left( f_1(x,y) a^\ast_x a_y + f_2(x,y) a_x a^\ast_y \right) \psi} \rvert \leq \ev{\Ncal} \frac{\norm{f_1}+\norm{f_2}}{2} + \frac{1}{2} \left\lvert \int f_2(x,x) \di x \right\rvert \scal{\psi}{\psi}. 1151 1124  \ed 1152 1125  \end{lem} 1153 -\begin{proof} Proof of the first estimate: 1126 +\begin{proof} Proof of the first estimate, using Hoelder's inequality: 1154 1127  \begin{align*} 1155 1128  & \lvert \scal{\psi}{\frac{1}{2} \int \di x\di y \left( f_1(x,y) a^\ast_x a^\ast_y + f_2(x,y) a_x a_y \right) \psi} \rvert \\ 1156 -\leq & \frac{1}{2} \sum_{i=1}^2 \lvert \scal{\psi}{\int \di x\di y\, b_i(x,y) a^\ast_x a^\ast_y \psi }\rvert \\ 1157 -\leq & \frac{1}{2} \sum_{i=1}^2 \int \di y \norm{a_y \psi} \norm{a^\ast(b_i(\cdot,y))\psi} \\ 1158 -\leq & \frac{1}{2} \sum_{i=1}^2 \left( \int \di y_1\, \norm{a_{y_1}\psi}^2 \int \di y_2\, \norm{b_i(\cdot,y_2)}^2 \norm{(\Ncal+1)^{1/2} \psi}^2 \right)^{1/2} \\ 1159 -\leq & \ev{\Ncal+1} \frac{\norm{f_1}+\norm{f_2}}{2}. 1129 +& \leq \frac{1}{2} \sum_{i=1}^2 \lvert \scal{\psi}{\int \di x\di y\, f_i(x,y) a^\ast_x a^\ast_y \psi }\rvert \\ 1130 +& \leq \frac{1}{2} \sum_{i=1}^2 \int \di y \norm{a_y \psi} \norm{a^\ast(f_i(\cdot,y))\psi} \\ 1131 +& \leq \frac{1}{2} \sum_{i=1}^2 \left( \int \di y_1\, \norm{a_{y_1}\psi}^2 \int \di y_2\, \norm{f_i(\cdot,y_2)}^2 \norm{(\Ncal+1)^{1/2} \psi}^2 \right)^{1/2} \\ 1132 +& \leq \ev{\Ncal+1} \frac{\norm{f_1}+\norm{f_2}}{2}. 1160 1133  \end{align*} 1161 -Proof of the second estimate: 1134 +Proof of the second estimate, using the CCR and Hoelder's inequality: 1162 1135  \begin{align*} 1163 1136  & \lvert \scal{\psi}{\frac{1}{2} \int \di x\di y \left( f_1(x,y) a^\ast_x a_y + f_2(x,y) a_x a^\ast_y \right) \psi} \rvert\\ 1164 -\leq & \frac{1}{2} \lvert \scal{\psi}{\int \di x\di y\, f_1(x,y) \psi} \rvert + \frac{1}{2}\lvert \scal{\psi}{\int \di x \di y\, f_2(x,y) \psi}\rvert + \frac{1}{2} \lvert \scal{\psi}{\int \di x \di y\, f_2(x,y) \delta(x-y) \psi}\rvert \\ 1165 -\leq & \frac{1}{2} \int \di x\, \norm{a_x \psi} \norm{a(f_1(x,\cdot))\psi} + \frac{1}{2} \int \di y\, \norm{a_y\psi} \norm{a(f_2(\cdot,y))\psi} + \frac{1}{2} \lvert \int \di x f_2(x,x) \rvert \scal{\psi}{\psi} \\ 1166 -\leq & \frac{1}{2} \left( \int \di x_1 \norm{a_{x_1}\psi}^2 \int \di x_2 \norm{f_1(x_2,\cdot)}^2 \norm{\Ncal^{1/2}\psi}^2 \right)^{1/2}\\ & \quad + \frac{1}{2} \left( \int \di x_1 \norm{a_{x_1}\psi}^2 \int \di x_2 \norm{f_2(\cdot,x_2)}^2 \norm{\Ncal^{1/2}\psi}^2 \right)^{1/2} + \frac{1}{2}\lvert \int \di x\, f_2(x,x) \rvert \scal{\psi}{\psi} \\ 1167 -\leq & \frac{1}{2}\ev{\Ncal} \left(\left(\int \di x \norm{f_1(x,\cdot)}^2 \right) + \left(\int \di x \norm{f_2(\cdot,x)}^2 \right) \right) + \frac{1}{2}\lvert \int \di x\, f_2(x,x) \rvert \scal{\psi}{\psi}. \qedhere 1137 +& \leq \frac{1}{2} \lvert \scal{\psi}{\int \di x\di y\, f_1(x,y) a^\ast_x a_y \psi} \rvert + \frac{1}{2}\lvert \scal{\psi}{\int \di x \di y\, f_2(x,y) a^\ast_y a_x \psi}\rvert \\ 1138 +&\quad + \frac{1}{2} \lvert \scal{\psi}{\int \di x \di y\, f_2(x,y) \delta(x-y) \psi}\rvert \\ 1139 +& \leq \frac{1}{2} \int \di x\, \norm{a_x \psi} \norm{a(f_1(x,\cdot))\psi} + \frac{1}{2} \int \di y\, \norm{a_y\psi} \norm{a(f_2(\cdot,y))\psi} + \frac{1}{2} \lvert \int f_2(x,x) \di x \rvert \scal{\psi}{\psi} \\ 1140 +& \leq \frac{1}{2} \left( \int \di x_1 \norm{a_{x_1}\psi}^2 \int \di x_2 \norm{f_1(x_2,\cdot)}^2 \norm{\Ncal^{1/2}\psi}^2 \right)^{1/2}\\ 1141 +& \quad + \frac{1}{2} \left( \int \di x_1 \norm{a_{x_1}\psi}^2 \int \di x_2 \norm{f_2(\cdot,x_2)}^2 \norm{\Ncal^{1/2}\psi}^2 \right)^{1/2} + \frac{1}{2}\lvert \int f_2(x,x) \di x\rvert \scal{\psi}{\psi} \\ 1142 +& \leq \frac{1}{2}\ev{\Ncal} \left(\left(\int \di x \norm{f_1(x,\cdot)}^2 \right)^{1/2} + \left(\int \di x \norm{f_2(\cdot,x)}^2 \right)^{1/2} \right) + \frac{1}{2}\lvert \int f_2(x,x) \di x\rvert \scal{\psi}{\psi}. \qedhere 1168 1143  \end{align*} 1169 1144  \end{proof} 1170 1145   1171 -\begin{lem}[Regularity of $\phdot$] 1172 -\label{lm:phdotregularity} 1173 -Let $\ph$ be a solution of the modified Hartree equation. Then 1174 - \bd 1175 -\norm{\phdot}_{L^2} \leq \norm{\ph}_{H^2} + 8\pi a_0 \norm{\ph}_{H^2}^2. 1176 -\ed 1177 -For $k(x,y) = -N w_N(x-y) \ph(x) \ph(y)$, we have the bound 1178 -\bd 1179 -\norm{\dot k} \leq 4 \max\{R,a_0\} \norm{\ph}_{H^1} \left( \norm{\ph}_{H^2} + 8\pi a_0 \norm{\ph}_{H^2}^2\right) \leq C \norm{\ph}_{H^2}^2. 1180 -\ed 1181 -\end{lem} 1182 -\begin{proof} The modified Hartree equation reads 1183 -\bd 1184 -i \phdot = -\Delta \ph + \left(N F_N V_N \ast \lvert \ph \rvert^2 \right) \ph. 1185 -\ed 1186 -It follows that 1187 -\bd 1188 -\norm{\phdot}_{L^2} \leq \norm{\ph}_{H^2} + \norm{\left(N f_N V_N \ast \lvert \ph\rvert^2 \right)\ph}_{L^2}. 1189 -\ed 1190 -Now we calculate 1191 -\begin{align*} 1192 -& \norm{\left(N f_N V_N \ast \lvert \ph\rvert^2 \right)\ph}_{L^2} \\ 1193 -= & \int \di x\, \lvert \ph(x)\rvert^2 \left\lvert \int \di y\, N f_NV_N(x-y) \lvert \ph(y)\rvert^2 \right\rvert^2 \\ 1194 -\leq & \int \di x\, \lvert \ph(x)\rvert^2 \left\lvert \norm{\ph}_\infty^2 \int \di y\, N f_N V_N(x-y) \right\rvert^2 \\ 1195 -\leq & \norm{\ph}_{L^2}^2 \norm{\ph}_\infty^4 (8\pi a_0)^2. 1196 -\end{align*} 1197 -We now prove the bound for $\norm{\dot k}$: 1198 -\begin{align*} 1199 -\norm{\dot k}^2 & = \int \di x\di y\, \lvert \frac{\di}{\di t} \left( -N w_N(x-y) \ph(x)\ph(y) \right) \rvert^2 \\ 1200 -& \leq 2 \int \di x\di y\, \lvert N w_N(x-y) \phdot(x) \ph(y) \rvert^2 + 2 \int \di x \di y\, \lvert N w_N(x-y) \ph(x) \phdot(y) \rvert^2 \\ 1201 -& \leq 4 \left( \max\{R,a_0\} \right)^2 \int \di x\di y\, \lvert \phdot(x)\rvert^2 \frac{\lvert\ph(y)\rvert^2}{\lvert x-y\rvert^2} \\ 1202 -& \leq 16 \left( \max\{R,a_0\} \right)^2 \int \di x\di y\, \lvert \phdot(x)\rvert^2 \lvert \nabla_y \ph(y)\rvert^2 \\ 1203 -& \leq \left( 4 \max\{R,a_0\} \right)^2 \norm{\phdot}_{L^2}^2 \norm{\ph}_{H^1}^2. 1204 -\end{align*} 1205 -Now use the estimate from the first part of the lemma. 1206 -\end{proof} 1207 - 1208 1146  \begin{lem} 1209 1147  \label{lm:highercommutators} 1210 1148  For each $n \in \Nbb$ and each $i \in \{1,2\}$, there exists $f_{n,i} \in L^2(\Rbb^3 \times \Rbb^3)$ such that @@ -1221,17 +1159,17 @@ \section{Estimates for the terms of the generator} 1221 1159  and the following bounds hold: $\norm{f_{n,i}} \leq 2 \norm{\dot k} (2\norm{k})^n$ and $\int \lvert f_{n,i}(x,x)\rvert \di x \leq 2 \norm{\dot k} (2\norm{k})^n$ for all $n \in \Nbb$ and $i \in \{1,2\}$.  1222 1160  \end{itemize} 1223 1161  \end{lem} 1224 -\begin{proof} The proof is by induction in $n$. In the inductive step, we have to treat the cases of even $n$ and odd $n$ separately (though the calculations are similar).\\ 1162 +\begin{proof} The proof is by induction in $n$. In the inductive step, we have to treat the cases of even $n$ and odd $n$ separately (but the calculations are similar).\\ 1225 1163  \underline{basis:}\\ 1226 1164  We take the case of $n=0$ as the basis. For 1227 1165  \bd 1228 1166  \ad^0_B(\dot B) = \dot B = \frac{1}{2}\int \di x\di y\left( \dot k(x,y) a^\ast_x a^\ast_y - \cc{\dot k(x,y)} a_x a_y \right) 1229 1167  \ed 1230 1168  the estimate stated in the even case is clearly fulfilled.\\ 1231 -\underline{inductive step, case of even $n$:}\\ 1169 +\underline{inductive step, starting from even $n$:}\\ 1232 1170  We calculate (by using the CCR and appropriately renaming integration variables) 1233 1171  \begin{align*} 1234 -\ad^{n+1}_B(\dot B) & = [B,\ad^n_B(\dot B)] \\ 1172 +& \ad^{n+1}_B(\dot B) = [B,\ad^n_B(\dot B)] \\ 1235 1173  & = \left[\frac{1}{2} \int \di x\di y\left( k(x,y)a^\ast_x a^\ast_y - \cc{k(x,y)}a_x a_y \right), \frac{1}{2}\int \di x\di y\left( f_{n,1}(x,y) a^\ast_x a^\ast_y + f_{n,2}(x,y) a_x a_y \right)\right] \\ 1236 1174  & = \frac{1}{2} \int \di x\di z \left(f_{n+1,1}(x,z) a^\ast_x a_z + f_{n+1,2}(x,z) a_x a^\ast_z \right), 1237 1175  \end{align*} @@ -1240,7 +1178,7 @@ \section{Estimates for the terms of the generator} 1240 1178  f_{n+1,1}(x,z) & = -\frac{1}{2} \int \di y \left( k(x,y) \left( f_{n,2}(z,y) + f_{n,2}(y,z) \right) + \cc{k(y,z)}\left( f_{n,2}(x,y) + f_{n,2}(y,x)\right) \right)\\ 1241 1179  f_{n+1,2}(x,z) & = -\frac{1}{2} \int \di y \left( k(y,z) \left( f_{n,1}(x,y) + f_{n,1}(y,x) \right) + \cc{k(x,y)}\left( f_{n,1}(z,y) + f_{n,1}(y,z)\right) \right),  1242 1180  \end{align*} 1243 -so, as $n+1$ is odd, we have the correct expression for $\ad^{n+1}_B(\dot B)$. We still have to check validity of the estimates. 1181 +so, as $n+1$ is odd, we have the correct expression for $\ad^{n+1}_B(\dot B)$. We still have to check for validity of the estimates. 1244 1182  Clearly 1245 1183  \be{normnorm} 1246 1184  \begin{split} @@ -1248,25 +1186,26 @@ \section{Estimates for the terms of the generator} 1248 1186  & \quad + \norm{\int \di y\, \cc{k(y,z)}f_{n,2}(x,y)}_{L^2} + \norm{\int \di y\, \cc{k(y,z)} f_{n,2}(y,x)}_{L^2} \bigg), 1249 1187  \end{split} 1250 1188  \ee 1251 -where by abuse of notation, $x$ and $z$ are actually the variables which are integrated over in calculating the $L^2(\Rbb^3\times \Rbb^3)$-norm (see next equation). 1189 +where by abuse of notation, $x$ and $z$ are actually the variables which are integrated over in calculating the $L^2(\Rbb^3\times \Rbb^3)$-norm (cf.\ next equation). 1252 1190  Using Hoelder's inequality we get 1253 1191  \begin{align*} 1254 -\norm{\int \di y k(x,y) f_{n,2}(z,y)}^2 &= \int \di x\di z \left\lvert \int \di y k(x,y) f_{n,2}(z,y) \right\rvert^2 \\ 1255 -& \leq \int \di x \di z \int \di y_1 \lvert k(x,y_1) \rvert^2 \int \di y_2 \lvert f_{n,2}(z,y_2) \rvert^2 = \norm{k}^2 \norm{f_{n,2}}^2.  1192 +\norm{\int \di y\,k(x,y) f_{n,2}(z,y)}^2 &= \int \di x\di z \left\lvert \int \di y\,k(x,y) f_{n,2}(z,y) \right\rvert^2 \\ 1193 +& \leq \int \di x \di z \int \di y_1\,\lvert k(x,y_1) \rvert^2 \int \di y_2\,\lvert f_{n,2}(z,y_2) \rvert^2 = \norm{k}^2 \norm{f_{n,2}}^2.  1256 1194  \end{align*} 1257 1195  The other three summands in \eqr{normnorm} obey the same bound, so by making use of the inductive hypothesis, we obtain 1258 1196  \bd 1259 1197  \norm{f_{n+1,1}}_{L^2} \leq 2\norm{\dot k} (2\norm{k})^{n+1}, 1260 1198  \ed 1261 -which was to be proven. The calculation for $\norm{f_{n+1,2}}_{L^2}$ works the same way. We still have to proof the second extimate of the lemma, so we calculate using Hoelder's inequality that 1199 +which was to be proven. The calculation for $\norm{f_{n+1,2}}_{L^2}$ works the same way. We still have to proof the second estimate of the lemma, so we calculate using Hoelder's inequality that 1262 1200  \bd 1263 -\int \lvert f_{n+1,1}(x,x) \rvert \di x \leq 2\norm{k} \norm{f_{n,2}} \leq 2\norm{\dot k} (2\norm{k})^2 1201 +\int \lvert f_{n+1,1}(x,x) \rvert \di x \leq 2\norm{k} \norm{f_{n,2}} \leq 2\norm{\dot k} (2\norm{k})^{n+1} 1264 1202  \ed 1265 -and in the same way $\int \lvert f_{n+1,2}(x,x) \rvert \di x$. Herewith the inductive step startin from even $n$ is proven.\\ 1266 -\underline{inductive step, case of odd $n$:}\\ 1203 +and in the same way $\int \lvert f_{n+1,2}(x,x) \rvert \di x$.\\ 1204 +\underline{inductive step, starting from odd $n$:}\\ 1267 1205  We calculate (by using the CCR and appropriately renaming integration variables) 1268 1206  \begin{align*} 1269 -\ad^{n+1}_B(\dot B) & = \left[ \frac{1}{2}\int \di x \di y \left( k(x,y)a^\ast_x a^\ast_y - \cc{k(x,y)} a_x a_y \right) , \frac{1}{2}\int \di x\di y \left( f_{n,1}(x,y) a^\ast_x a_y + f_{n,2}(x,y) a_x a^\ast_y \right) \right] \\ 1207 +& \ad^{n+1}_B(\dot B)\\ 1208 +& = \left[ \frac{1}{2}\int \di x \di y \left( k(x,y)a^\ast_x a^\ast_y - \cc{k(x,y)} a_x a_y \right) , \frac{1}{2}\int \di x\di y \left( f_{n,1}(x,y) a^\ast_x a_y + f_{n,2}(x,y) a_x a^\ast_y \right) \right] \\ 1270 1209  & = \frac{1}{2} \int \di x\di z \left( a^\ast_x a^\ast_z f_{n+1,1}(x,z) + a_x a_z f_{n+1,2}(x,z) \right) 1271 1210  \end{align*} 1272 1211  where @@ -1274,9 +1213,7 @@ \section{Estimates for the terms of the generator} 1274 1213  f_{n+1,1}(x,z) & = - \int \di y\, k(x,y)\left( f_{n,1}(z,y) + f_{n,2}(y,z) \right) \\ 1275 1214  f_{n+1,2}(x,z) & = - \int \di y\, \cc{k(x,y)}\left( f_{n,1}(y,z) + f_{n,2}(z,y) \right),  1276 1215  \end{align*} 1277 -so, as $n+1$ is even, we have the correct expression for $\ad^{n+1}_B(\dot B)$. Validity of the estimate for the $\norm{f_{n+1},1}$ and $\norm{f_{n+1},2}$ again follows by invoking Hoelder's inequality and the inductive hypothesis.\newline 1278 - 1279 -The induction is now complete. 1216 +so, as $n+1$ is even, we have the correct expression for $\ad^{n+1}_B(\dot B)$. Validity of the estimate for $\norm{f_{n+1,1}}$ and $\norm{f_{n+1,2}}$ again follows by invoking Hoelder's inequality and the inductive hypothesis. 1280 1217  \end{proof} 1281 1218   1282 1219   @@ -1284,7 +1221,7 @@ \section{Estimates for the terms of the generator} 1284 1221  \label{lm:timederivative} 1285 1222  There exists a constant $C$ independent of $N$ and $t$ such that the following estimate holds for all $t \in \Rbb$ and all $\psi \in \fock$: 1286 1223  \bd 1287 - \lvert \scal{\psi}{(\partial_t T^*_t)T_t \psi} \rvert \leq 2\norm{\dot k} \ev{\Ncal+1} e^{2\norm{k}} \leq C \norm{\ph}_{H^2}^2 \ev{\Ncal+1}. 1224 + \lvert \scal{\psi}{(\partial_t T^*_t)T_t \psi} \rvert \leq 2\norm{\dot k} e^{2\norm{k}} \ev{\Ncal+1} \leq C \norm{\ph}_{H^2}^2 \ev{\Ncal+1}. 1288 1225  \ed 1289 1226  \end{lem} 1290 1227  \begin{proof} @@ -1468,7 +1405,6 @@ \section{A-priori estimates} 1468 1405   1469 1406   1470 1407  \begin{proof}[Proof of Proposition \ref{p:formulae}] 1471 - \marginpar{This seems clear; we might be brief here.} 1472 1408  We give only an outline of the proof. Recall that $\phi(\varphi) = 1473 1409  a(\varphi) + a^*(\varphi)$. Then, parts (i), (ii) and (iii) follow easily by 1474 1410  a short calculation using parts (ii) and (iii) of Lemma \ref{l:W}. @@ -2309,172 +2245,44 @@ \subsection{$\varphi_t^{(N)}$ converges to $\varphi_t$ in $L^2$} 2309 2245  $\text{supp }V$ and $\| \varphi \|_{H^2}$. 2310 2246  \end{proof} 2311 2247   2312 -\marginpar{Strichartz version can be removed?} 2313 -\iffalse 2314 -\subsection{$\varphi_t^{(N)}$ converges to $\varphi_t$ in $L^2$ (using 2315 -Strichartz) (to be deleted)} 2316 - 2317 - 2318 -\begin{prp} 2319 - \label{p:httogp} 2320 - For $N > 0$, let $\varphi^{(N)}, \varphi \in H^2(\R^3)$ with $\| 2321 - \varphi^{(N)} \|_{L^2} = 1$ and $\| \varphi \|_{L^2} = 1$. Suppose that $f 2322 - \in L^1(\R^3)$ and $V \in C_c^\infty(\R^3)$ with $fV \ge 0$. Consider a 2323 - solution $\varphi_t^{(N)} \in H^1(\R^3)$ of the nonlinear Hartree equation 2324 - \begin{displaymath} 2325 - i \partial_t \varphi_t^{(N)} = - \Delta \varphi_t^{(N)} + (N f_N V_N * 2326 - |\varphi_t^{(N)}|^2) \varphi_t^{(N)} 2327 - \end{displaymath} 2328 - with initial data $\varphi^{(N)}_0 = \varphi^{(N)}$, where $f_N V_N(x) = N^2 2329 - f(Nx)V(Nx)$. Consider also a solution $\varphi_t \in H^1(\R^3)$ of the 2330 - nonlinear Gross-Pitaevskii equation 2331 - \begin{displaymath} 2332 - i \partial_t \varphi_t = - \Delta \varphi_t + 8 \pi a_0 |\varphi_t|^2 2333 - \varphi_t 2334 - \end{displaymath} 2335 - with initial data $\varphi_0 = \varphi$, where $8 \pi a_0 = \int f V$. Then, 2336 - there exist a real number $T > 0$ such that 2337 - \begin{displaymath} 2338 - \| \varphi_t - \varphi_t^{(N)} \|_{L^2} \le \| \varphi - \varphi^{(N)} 2339 - \|_{L^2} + \frac{C}{N}\| \varphi^{(N)} \|_{H^2} 2340 - \end{displaymath} 2341 - for all $t \in [0,T]$, where $C$ is a constant that depends only on $\| fV 2342 - \|_{L^1}$, $\text{supp }V$, $\| \varphi^{(N)} \|_{H^1}$ and $\| \varphi 2343 - \|_{H^1}$, and $T$ depends only on $\| fV \|_{L^1}$, $\| \varphi^{(N)} 2344 - \|_{H^1}$ and $\| \varphi \|_{H^1}$. 2345 -\end{prp} 2346 - 2347 - 2348 -\begin{proof} 2349 - Write 2350 - \begin{align*} 2351 - \varphi_t - \varphi_t^{(N)} & = e^{it\Delta}(\varphi - \varphi^{(N)}) + i 2352 - \int_0^t ds \, e^{i(t-s) \Delta} [ (N f_N V_N * |\varphi_s^{(N)}|^2) 2353 - \varphi_s^{(N)} - 8 \pi a_0 |\varphi_s|^2 \varphi_s ] \\ 2354 - & = e^{it\Delta}(\varphi - \varphi^{(N)}) + i \int_0^t ds \, e^{i(t-s) 2355 - \Delta} [ (N f_N V_N * |\varphi_s|^2 - 8 \pi a_0 |\varphi_s|^2) \varphi_s 2356 - \\ 2357 - & \quad + (N f_N V_N * |\varphi_s^{(N)}|^2)(\varphi_s^{(N)} - \varphi_s) + 2358 - (N f_N V_N * (|\varphi_s^{(N)}|^2 - |\varphi|^2)) \varphi_s]. 2359 - \end{align*} 2360 - As in the proof of Proposition \ref{p:reg1}, by Strichartz estimates, 2361 -  2362 - \begin{aligned} 2363 - & \| \varphi_{(\cdot)} - \varphi_{(\cdot)}^{(N)} \|_{L_t^\infty L_x^2} 2364 - \apprle \| \varphi - \varphi^{(N)} \|_{L^2} + \| (N f_N V_N * 2365 - |\varphi_{(\cdot)}|^2 - 8 \pi a_0 |\varphi_{(\cdot)}|^2) 2366 - \varphi_{(\cdot)} \\ 2367 - & \qquad \qquad + (N f_N V_N * 2368 - |\varphi_{(\cdot)}^{(N)}|^2)(\varphi_{(\cdot)}^{(N)} - 2369 - \varphi_{(\cdot)})+ (N f_N V_N * (|\varphi_{(\cdot)}^{(N)}|^2 - 2370 - |\varphi_{(\cdot)}|^2) \varphi_{(\cdot)} \|_{L_t^2 L_x^{6/5}}. 2371 - \end{aligned} 2372 - \label{diff} 2373 -  2374 - We next estimate each term in this inequality. 2375 - 2376 - 2377 - By H\"older's inequality and conservation of mass, 2378 - \begin{displaymath} 2379 - \| (N f_N V_N * |\varphi_{(\cdot)}|^2 - 8 \pi a_0 |\varphi_{(\cdot)}|^2) 2380 - \varphi_{(\cdot)} \|_{L_t^2 L_x^{6/5}} \le T^{1/2} \sup_{t \in [0,T]} \| N 2381 - f_N V_N * |\varphi_t|^2 - 8 \pi a_0 |\varphi_t|^2 \|_{L^3}. 2382 - \end{displaymath} 2383 - By Lemma \ref{l:interp} and Sobolev's inequality, 2384 - \begin{align*} 2385 - & \| (N f_N V_N * |\varphi_{(\cdot)}^{(N)}|^2)(\varphi_{(\cdot)}^{(N)} - 2386 - \varphi_{(\cdot)}) \|_{L_t^2 L_x^{6/5}} \\ 2387 - & \qquad \le \| fV \|_{L^1} T^{1/2} \sup_{t \in [0,T]} \| 2388 - |\varphi_t^{(N)}|^2 \|_{L^3} \sup_{t \in [0,T]} \| \varphi_t^{(N)} - 2389 - \varphi_t \|_{L^2} \\ 2390 - & \qquad \apprle \| fV \|_{L^1} T^{1/2} \sup_{t \in [0,T]} \| 2391 - \varphi_t^{(N)} \|_{H^1}^2 \sup_{t \in [0,T]} \| \varphi_t^{(N)} - 2392 - \varphi_t \|_{L^2}, 2393 - \end{align*} 2394 - and also by H\"older's inequality and triangle inequality, 2395 - \begin{align*} 2396 - & \| (N f_N V_N * (|\varphi_{(\cdot)}^{(N)}|^2 - |\varphi_{(\cdot)}|^2)) 2397 - \varphi_{(\cdot)}) \|_{L_t^2 L_x^{6/5}} \\ 2398 - & \qquad \le \| fV \|_{L^1} T^{1/2} \sup_{t \in [0,T]} \| 2399 - |\varphi_t^{(N)}|^2 - |\varphi_t|^2 \|_{L^{3/2}} \sup_{t \in [0,T]} \| 2400 - \varphi_t \|_{L^6} \\ 2401 - & \qquad \le \| fV \|_{L^1} T^{1/2} \sup_{t \in [0,T]} \| 2402 - \varphi_t^{(N)} - \varphi_t \|_{L^2} \sup_{t \in [0,T]} (\| 2403 - \varphi_t^{(N)} \|_{H^1} + \| \varphi_t \|_{H^1}) \| \varphi_t \|_{H^1}. 2404 - \end{align*} 2405 - Recall that $V \ge 0$. By conservation of mass and energy, and by 2406 - Proposition \ref{p:energy}, 2407 - \begin{align*} 2408 - \| \varphi_t^{(N)} \|_{H^1}^2 & \le \| \varphi_t^{(N)} \|_{L^2}^2 + 2409 - \mathcal{E}_N(\varphi_t^{(N)}) = 1 + \mathcal{E}_N(\varphi^{(N)}) \le C_1, 2410 - \\ 2411 - \| \varphi_t \|_{H^1}^2 & \le \| \varphi_t \|_{L^2}^2 + 2412 - \mathcal{E}_{GP}(\varphi_t) = 1 + \mathcal{E}_{GP}(\varphi) \le C_2, 2413 - \end{align*} 2414 - where $C_1$ and $C_2$ are constants that depend only on $\| fV \|_{L^1}$, 2415 - and $\| \varphi^{(N)} \|_{H^1}$ and $\| \varphi \|_{H^1}$, respectively. 2416 - Thus, substituting all this into \eqref{diff}, and choosing $T > 0$ 2417 - sufficiently small, but depending only on $\| fV \|_{L^1}$, $\| 2418 - \varphi^{(N)} \|_{H^1}$ and $\| \varphi \|_{H^1}$, we obtain 2419 -  2420 - \sup_{t \in [0,T]} \| \varphi_t - \varphi_t^{(N)} \|_{L^2} \le \| \varphi 2421 - - \varphi^{(N)} \|_{L^2} + C \sup_{t \in [0,T]} \| N f_N V_N * 2422 - |\varphi_t|^2 - 8 \pi a_0 |\varphi_t|^2 \|_{L^3}, 2423 - \label{endproof} 2424 -  2425 - where $C$ is a constant that depends only on $\| fV \|_{L^1}$, $\| 2426 - \varphi^{(N)} \|_{H^1}$ and $\| \varphi \|_{H^1}$. We are left to estimating 2427 - the right hand side of \eqref{endproof}. 2428 - 2429 - 2430 - Write 2431 - \begin{align*} 2432 - N f_N V_N * |\varphi_t|^2(x) - 8 \pi a_0 |\varphi_t|^2(x) & = \int dy 2433 - \big( |\varphi_t(x-y)|^2 - |\varphi_t(x)|^2 \big) N^3 fV(Ny) \\ 2434 - & = \int dz \big( |\varphi_t(x-z/N)|^2 - |\varphi_t(x)|^2 \big) fV(z). 2435 - \end{align*} 2436 - Let $R$ be such that $\text{supp }V \subset \{ x \in \R^3 \; | \;\; |x| \le 2437 - R \}$. By Minkowski's, H\"older's, and Sobolev's inequalities, 2438 - \begin{align*} 2439 - \| N f_N V_N * |\varphi_t|^2 - 8 \pi a_0 |\varphi_t|^2 \|_{L^3} & \le \int 2440 - dz \, \| |\varphi_t(\, \cdot \, -z/N)|^2 - |\varphi_t|^2 \|_{L^3} |fV(z)| 2441 - \\ 2442 - & \le \| fV \|_{L^1} \sup_{|z| \le R} \| |\varphi_t(\, \cdot \, - z/N)|^2 2443 - - |\varphi_t|^2 \|_{L^3}. 2444 - \end{align*} 2445 - Given $\varepsilon = 1/N$, there exists $\psi_t \in C^\infty(\R^3)$ such 2446 - that $\| \varphi_t - \psi_t \|_{H^2} < 1/N$. Hence, by H\"older's 2447 - inequality, Sobolev's inequality and an $\varepsilon/3$-argument, the mean 2448 - value theorem (with some constant $0 \le c \le 1$), and Sobolev's inequality 2449 - again, 2450 - \begin{align*} 2451 - \| |\varphi_t(\, \cdot \, - z/N)|^2 - |\varphi_t|^2 \|_{L^3} & \le 2 \| 2452 - \varphi_t \|_{L^6} \| |\varphi_t(\, \cdot \, - z/N)| - |\varphi_t| 2453 - \|_{L^6} \\ 2454 - & \apprle \| \varphi_t \|_{H^1} \big( 1/N + \| |\psi_t(\, \cdot \, - z/N)| 2455 - - |\psi_t| \|_{L^6} \big) \\ 2456 - & \apprle \| \varphi_t \|_{H^1} \big( 1/N + |z|/N \| \nabla |\psi_t(\, 2457 - \cdot \, - c z/N)| \|_{L^6} \big) \\ 2458 - & \apprle \| \varphi_t \|_{H^1} \big( 1/N + |z|/N \| \psi_t \|_{H^2} \big) 2459 - \\ 2460 - & \apprle \| \varphi_t \|_{H^1} \big( 1/N + |z|/N^2 + |z|/N \| \varphi_t 2461 - \|_{H^2} \big). 2462 - \end{align*} 2463 - Therefore, substituting this into the above inequality, and applying 2464 - Proposition \ref{p:reg1} (possibly after choosing a smaller $T$), we obtain 2465 - \begin{align*} 2466 - & \sup_{t \in [0,T]} \| N f_N V_N * |\varphi_t|^2 - 8 \pi a_0 2467 - |\varphi_t|^2 \|_{L^3} \\ 2468 - & \apprle \| fV \|_{L^1} \| \varphi_t \|_{H^1} \Big( \frac{1}{N} + 2469 - \frac{R}{N^2} + \frac{R}{N} \sup_{t \in [0,T]} \| \varphi_t \|_{H^2} \Big) 2470 - \le \frac{C}{N}(1 + \| \varphi \|_{H^2}) \le \frac{2C}{N}\| \varphi 2471 - \|_{H^2}, 2472 - \end{align*} 2473 - where $C$ is a constant that depends only on $\| fV \|_{L^1}$, $\text{supp 2474 - }V$, $\| \varphi^{(N)} \|_{H^1}$ and $\| \varphi \|_{H^1}$. In view of 2475 - \eqref{endproof}, this completes the proof. 2248 +\subsection{Regularity of $\phdot$} 2249 +\begin{lem}[$L^2$-bound for $\phdot$] 2250 +\label{lm:phdotregularity} 2251 +Let $\ph$ be a solution of the modified Hartree equation. Then 2252 + \bd 2253 +\norm{\phdot}_{L^2} \leq \norm{\ph}_{H^2} + 8\pi a_0 \norm{\ph}_{H^2}^2. 2254 +\ed 2255 +For $k(x,y) = -N w_N(x-y) \ph(x) \ph(y)$, we have the bound 2256 +\bd 2257 +\norm{\dot k} \leq 4 \max\{R,a_0\} \norm{\ph}_{H^1} \left( \norm{\ph}_{H^2} + 8\pi a_0 \norm{\ph}_{H^2}^2\right) \leq C \norm{\ph}_{H^2}^2, 2258 +\ed 2259 +where $C$ is a constant independent of $N$ and $t$. 2260 +\end{lem} 2261 +\begin{proof} The modified Hartree equation reads 2262 +\bd 2263 +i \phdot = -\Delta \ph + \left(N f_N V_N \ast \lvert \ph \rvert^2 \right) \ph. 2264 +\ed 2265 +It follows that 2266 +\bd 2267 +\norm{\phdot}_{L^2} \leq \norm{\ph}_{H^2} + \norm{\left(N f_N V_N \ast \lvert \ph\rvert^2 \right)\ph}_{L^2}. 2268 +\ed 2269 +Now we calculate 2270 +\begin{align*} 2271 +& \norm{\left(N f_N V_N \ast \lvert \ph\rvert^2 \right)\ph}_{L^2} \\ 2272 +& = \int \di x\, \lvert \ph(x)\rvert^2 \left\lvert \int \di y\, N f_NV_N(x-y) \lvert \ph(y)\rvert^2 \right\rvert^2 \\ 2273 +& \leq \int \di x\, \lvert \ph(x)\rvert^2 \left\lvert \norm{\ph}_\infty^2 \int \di y\, N f_N V_N(x-y) \right\rvert^2 \\ 2274 +& \leq \norm{\ph}_{L^2}^2 \norm{\ph}_\infty^4 (8\pi a_0)^2. 2275 +\end{align*} 2276 +We now prove the bound for $\norm{\dot k}$ using Hardy's inequality. 2277 +\begin{align*} 2278 +\norm{\dot k}^2 & = \int \di x\di y\, \left\lvert \frac{\di}{\di t} \left( -N w_N(x-y) \ph(x)\ph(y) \right) \right\rvert^2 \\ 2279 +& \leq 2 \int \di x\di y\, \lvert N w_N(x-y) \phdot(x) \ph(y) \rvert^2 + 2 \int \di x \di y\, \lvert N w_N(x-y) \ph(x) \phdot(y) \rvert^2 \\ 2280 +& \leq 4 \left( \max\{R,a_0\} \right)^2 \int \di x\di y\, \lvert \phdot(x)\rvert^2 \frac{\lvert\ph(y)\rvert^2}{\lvert x-y\rvert^2} \\ 2281 +& \leq 16 \left( \max\{R,a_0\} \right)^2 \int \di x\di y\, \lvert \phdot(x)\rvert^2 \lvert \nabla_y \ph(y)\rvert^2 \\ 2282 +& \leq \left( 4 \max\{R,a_0\} \right)^2 \norm{\phdot}_{L^2}^2 \norm{\ph}_{H^1}^2. 2283 +\end{align*} 2284 +Now use the estimate from the first part of the lemma to estimate $\norm{\phdot}_{L^2}^2$ and the proof is complete. 2476 2285  \end{proof} 2477 -\fi 2478 2286   2479 2287  \bibliographystyle{plain} 2480 2288  \bibliography{gross-pitaevskii}