\magnification \magstep1 \input amstex \documentstyle{amsppt} \NoRunningHeads %\NoPageNumbers %\TagsOnRight \baselineskip 15pt \pagewidth{6.4 truein} \pageheight{8.6 truein} \define\Dt#1{{1\over i}{\partial\hfill\over \partial t}\,#1} \define\bra#1{\langle #1 \rangle} \define\r{{(\rho)}} \topmatter \title Regulated Smoothing for Schr\"odinger Evolution \endtitle \author Lev Kapitanski~\footnotemark\ \footnotetext{On leave from St. Petersburg Branch of Steklov Mathematical Institute, St. Petersburg, Russia\ \ \ \ \ \ \ }% \ {\tenrm and\/}\ \ Igor Rodnianski~\footnotemark \footnotetext{On leave from Department of Physics, St. Petersburg State University, St. Petersburg, Russia\ \ \ \ \ \ \ \ \ }% \endauthor \affil Department of Mathematics\\ Kansas State University\\ Manhattan, Kansas 66506\\ U S A \endaffil \curraddr{Department of Mathematics, Kansas State University, Manhattan, Kansas 66506, USA} \endcurraddr \endtopmatter In the case of the {\it free\/} Schr\"odinger evolution, the wave function $\,\psi(t,x)\,=\,(e^{i t \Delta}\,\psi^0)(x)\,$ is infinitely differentiable with respect to $\,t>0\,$ and $\,x\in\Bbb R^n$, if $\,\psi^0\,$ is merely a distribution, but of compact support. The proof is evident, since $$S_{{}_{-\Delta}}(t,x,y)= e^{-in\frac \pi4} (4\pi t)^{-\frac n2}\, e^{i\,{|x-y|^2\over 4t}},$$ the kernel of $\,e^{i t \Delta}$, is infinitely differentiable in $\,t>0,\,x$ and $\,y$. For a more general evolution operator $\,e^{- i t \Cal A}$, where $\,\Cal A = -\Delta + V\,$ and $\,V = V(x)\,$ is a potential, the explicit form of the kernel $\,S_{{}_{\Cal A}}(t,x,y)\,$ is not usually available. Nevertheless, a few results on the regularity of $\,S_{{}_{\Cal A}}(t,x,y)\,$ are already known. In particular, if $\,V\,$ is infinitely differentiable and bounded, together with all its derivatives, i.e., $$\sup\limits_{x\in\Bbb R^n}\,|\partial^{\alpha}_x\,V(x)| \,<\,\infty, \qquad\forall \alpha\in\Bbb Z^n_+, \tag 1$$ then $\,S_{{}_{\Cal A}}(t,x,y)\,$ is $\,C^{\infty}\,$ in $\,\Bbb R^{>0}_{t}\times\Bbb R^n_x\times \Bbb R^n_y$. This was proved by S. Zelditch \cite{Zel}; see also \cite{OF}, \cite{Kit}. The same regularity of $\,S_{{}_{\Cal A}}(t,x,y)\,$ has been shown recently by W. Craig, T. Kappeler and W. Strauss \cite{CKS}, under the assumption that for all $\,x\in\Bbb R^n$, $$V(x)\;\text{is real,\ \ and}\;\quad |\partial^{\alpha}_x\,V(x)| \le c_{\alpha} \bra{x}^{\rho - |\alpha|}, \qquad\forall \alpha\in\Bbb Z^n_+, \tag 2$$ with some $\,\rho$, $\,0\le\rho < 1\,$ (as usual, $\,\bra{x}=\sqrt{1+|x|^2}$). In fact, they prove the smoothness of $\,S_{{}_{\Cal A}}(t,x,y)\,$ in the more general case of operators $\,\Cal A\,$ with variable coefficients that stabilize at infinity and satisfy a non-trapping condition. The regularity of $\,S_{{}_{\Cal A}}(t,x,y)\,$ for some other types of Schr\"odinger equations and systems with variable coefficients and, possibly, boundary conditions, has been shown by L. Kapitanski and Yu. Safarov \cite{KS}, but their main concern is not in enlarging the class of admissible potentials. Closely related to what we discuss are the constructions of the local (in time) parametrices for the Schr\"odinger equation suggested by D. Fujiwara \cite{Fuj}, H. Kitada \cite{Kit}, F. Treves \cite{Tre}. The results of F. Treves, \cite{Tre}, imply that $\,S_{{}_{\Cal A}}(t,x,y)\,$ is smooth at least for small $\,t$, $\,00$. One of the facts that we will establish in the present paper is that if the potential grows slower than quadratically, then the corresponding fundamental solution is smooth for $\,t>0$. When the first draft of our paper was already written we learned that the smoothness of $\,S_{{}_{-\Delta + V}}(t,x,y)\,$ for the potentials $\,V\,$ satisfying (2) with $\,\rho<2\,$ had been proved independently by K.~Yajima \cite{Yaj 4}. He starts by constructing an approximate propagator - a parametrix - for the evolution equation, and then shows that both the approximate propagator and the complementary error-term are smooth. In the second part of his very interesting preprint \cite{Yaj 4}, Yajima shows that if the spatial dimension $\,n=1$, then for certain potentials with {\it superquadratic\/} growth, the fundamental solution is nowhere smooth in $\,t>0$, $\,x\,$ and $\,y$. This is a remarkable result that still has to be better understood. \vglue 1pc In the present paper we deal with the {\it subquadratic\/} potentials though of more general class than (2) (see Hypotheses ($\bold V$) below). Our approach is entirely different from that of Yajima. We not only prove the $\,C^{\infty}\,$ regularity of $\,S_{{}_{\Cal A}}(t,x,y)\,$ but also describe more precisely how the decay of the initial data converts into the improvement of the local regularity of the wave function. An important result in this spirit was proven by A. Jensen \cite{Jen}: if the potential $\,V(x)\,$ is real-valued and bounded, together with its derivatives up to some order $\,k\ge 1$, then there exists a constant $\,C>0\,$ such that the following estimate holds: $$\|\bra{x}^{-k}\,e^{-it(-\Delta + V)}\,f\|_{H^k(\Bbb R^n)}\, \,\le\,C\,(|t|^k + |t|^{-k})\,\|\bra{x}^k\,f\|_{L^2(\Bbb R^n)},\tag 3$$ for all $\,f\in \bra{x}^{-k}\,L^2(\Bbb R^n)$. This result was extended by T. Ozawa in \cite{Oza 1} to include the potentials of the sort $\,V(x)=\sum\limits_{j=1}^m\,c_j |x|^{-\mu_j}$, where each $\,\mu_j>0\,$ and where these exponents obey certain restrictions involving the amount of smoothing, $\,k$, and the space dimension $\,n$. Note that because of the singularities of $\,V$, the value of $\,k\,$ cannot be arbitrarily large. Some ideas behind the proofs in \cite{Jen} and \cite{Oza 1} go back to the works of C. Wilcox \cite{Wil}, W. Hunziker \cite{Hun}, C. Radin and B. Simon \cite{RS}, P. Perry \cite{Per}, and others, where the weighted Sobolev spaces {\it invariant\/} under the Schr\"odinger group $\,e^{-it(-\Delta + V)}\,$ were sought. \vglue 1pc In what follows, we generalize Jensen's result in two directions: first, we allow the potentials to grow at infinity as $\,\bra{x}^{\rho}\,$ with $\,0\le\rho<2$, and second, we can include a magnetic field with appropriate growth conditions also. The proof is quite simple, and to illustrate the basic idea, let us discuss at first the situation with no magnetic field involved. We make the following assumption on the potential $\,V$: \newline{\bf Hypotheses ($\bold V$)\/}: \roster \item"" The potential $\,V(\cdot)\,$ is infinitely differentiable, its imaginary and real parts satisfy $$|\text{Im}\, V(x)|\,\le\,c_{{}_0},\quad |\text{Re}\, V(x)|\,\le\, c_{{}_0}\,\bra{x}^{\rho}\,,\quad\forall x\in\Bbb R^n,\tag 4a$$ while $$|\partial_x^{\alpha} V(x)|\,\le\,c_{{}_{|\alpha|}}\, \bra{x}^{{\rho\over 2}\,\nu|\alpha|}\,, \quad\alpha\in\Bbb Z^n_+,\;|\alpha|>0, \quad\forall x\in\Bbb R^n,\tag 4b$$ with $\,\rho\,$ and $\,\nu\,$ such that $$0\le\rho<2,\qquad 0\,\le\,\nu\,<\,1.\tag 4c$$ \endroster Consider now the initial-value problem for the Schr\"odinger equation in $\,\Bbb R^n$,\ $\,n\ge 1$, \align \Dt &\psi(t,x)\,-\,\Delta \psi(t,x)\,+\,V(x) \psi(t,x)\,=\,0, \qquad t>0,\;x\in\Bbb R^n \tag{5a} \\ &\psi(0,x)\,=\,\psi^0(x). \tag{5b} \endalign To measure the regularity of solutions and handle the fact that the potential is growing, we need some special function spaces. For this purpose, define the scale $\,\{ H^s_{(\rho)}\,,\;s\in\Bbb R\}\,$ of Hilbert spaces generated from $\,H^0_{(\rho)}\,=\,L^2(\Bbb R^n)\,$ by the powers of the operator $\,\Lambda_\rho\,=\,\sqrt{-\,\Delta\,+\,{\bra{x}}^{\rho}}\,$ in the same way as the Sobolev spaces $\,H^s\,$ are generated by the powers of $\,\Lambda_0\,=\,\sqrt{-\,\Delta\,+\,1}$. We are in position now to state our result. \proclaim{Theorem 1} Assume that $\,V\,$ satisfies the hypotheses {\rm($\bold V$)\/}. Let $\,\psi^0\in H^s_\r$, for some $\,s$, and, in addition, $\,\bra{x}^{\ell}\,\psi^0\in H^{s-\nu\ell}_\r\,$ for $\,\ell = 1,\dots, m$, some integer $\,m\ge 1$, where $\,\nu\,$ is the same as in (4b). Then for any $\,t>0\,$ the solution $\,\psi\,$ of problem (5) has the following regularity properties: $$\bra{x}^{- \ell}\,\psi(t,\cdot)\,\in\, H^{s+\ell (1-\nu)}_\r\,,\quad \ell = 0,\,1,\dots,\,m.\tag 6a$$ Moreover, given $\,T>0$, there exists a constant $\,c=c(T, m, V)>0\,$ such that the following estimate holds: $$\|\bra{x}^{-m}e^{-i t(-\Delta+V)}\,\psi^0\|_{H^{s+(1-\nu)m}}\, \le\,c\,\sum\limits_{\ell=0}^{m} (1+{1\over t^{\ell}})\, \|\bra{x}^{\ell}\psi^0\|_{H^{s-\ell\nu}}, \;\forall t,\;00\, independent of \,\psi^0. This is a particular case of the more general result that we need later anyway: for any real \,r, for any \,u_0\in H^r_{(\rho)}\, and for any \,h(\cdot_t,\cdot_x)\in L^1_{\text{loc}}(\Bbb R\to H^r_{(\rho)}), there exists a unique solution \,u(t,x)\, of the inhomogeneous problem,$$ \Dt{u(t,x)}\,-\,\Delta u(t,x)\,+\,V(x) u(t,x)\,=\,h(t,x), \quad u(0,x) = u_0(x), $$such that the corresponding mapping \,u:\,\Bbb R\,\owns t\,\mapsto\,u(t,\cdot)\in H^r_\r\, is (strongly) continuous. Moreover, for every \,T>0, there exists a constant \,\tilde{c}_T>0\, such that the following energy estimate prevails:$$ \|u(t,\cdot)\|_{H^r_\r}\,\le\,\tilde{c}_T\, \{\|u_0\|_{H^r_\r}\,+ \,\int\limits_0^t \|h(\tau,\cdot)\|_{H^r_\r}\,d\tau\}, \quad 0\le t\le T. \tag 8 $$For the proof see, e.g., \cite{Kap}. In order to be able to apply Lemma 2.1 of \cite{Kap} in our circumstances, one has to check that for every real \,k,$$ \Lambda_{\rho}^k\,V \,\Lambda_{\rho}^{-k}\,-\, \Lambda_{\rho}^{-k}\,V^*\,\Lambda_{\rho}^{k}\,:\, L^2\to L^2 \tag 9a $$(i.e., the operator maps \,L^2\, into itself and is bounded), where \,V\, is identified with the operator of multiplication by \,V(x), and \,V(x)^*\, is the complex conjugate of \,V(x). Notice also that$$ \Lambda_{\rho}^k\,\bra{x}^{\rho}\,\Lambda_{\rho}^{-k}\,-\, \Lambda_{\rho}^{-k}\,\bra{x}^{\rho}\,\Lambda_{\rho}^{k}\,:\, L^2\to L^2,\quad \forall k\in \Bbb R. \tag 9b $$Our assumptions on the potential \,V\, guarantee (9a), but the rigorous justification would involve some basic calculus for certain classes of pseudodifferential operators in \,\Bbb R^n. Those pseudodifferential operators (PDOs) are defined by their global symbols from the corresponding classes of symbols \,S^{\mu}=S^{\mu}(\Bbb R^n_x\times\Bbb R^n_{\xi}), \,\mu\in\Bbb R, where \,a(x,\xi)\, belongs to \,S^{\mu}\, if \,a\in C^{\infty}(\Bbb R^n_x\times\Bbb R^n_{\xi}), and for all \,\alpha,\;\beta\in\Bbb Z^n_+,$$ |\partial_{x}^{\alpha}\partial_{\xi}^{\beta}\, a(x,\xi)|\,\le\,c_{\alpha,\beta}\, \left\{ \aligned (\lambda_{\rho}(x,\xi))^{{}^{\mu - |\beta|}}, \qquad\qquad\quad\; &|\alpha|=0,\\ (\lambda_{\rho}(x,\xi))^{{}^{\mu -1-|\beta|}}\, \bra{x}^{{}^{\rho - |\alpha|}},\quad &|\alpha|\ge 1, \endaligned \right. $$where \,\lambda_{\rho}(x,\xi)=\sqrt{|\xi|^2+\bra{x}^{\rho}}. In fact, not only the justification of (9) but also several other steps in our proof require certain standard facts, such as the composition formula, parametrices, etc., for the operators of the classes \,\text{Op}\,S^{\mu}. In addition, we have to know such facts as that \,\text{Op}\,a\, maps \,H^r_\r\, continuously into \,H^{r-\mu}_\r, for all \,r, for \,a\in S^{\mu}. It is also useful to have in mind that an operator \,\Lambda_{\rho}^{\mu} = (-\Delta + \bra{x}^{\rho})^{\mu/2}\, belongs to \,\text{Op}\,S^{\mu}, and its `principal' symbol is \,(|\xi|^2 + \bra{x}^{\rho})^{\mu/2}. Although we couldn't find the references where the appropriate calculus is developed exactly in the form we need it, we refer the reader to the papers \cite{Bea}, \cite{Fei}, \cite{KT}, \cite{Par}, where similar classes of PDOs are considered. \vglue 1pc Returning to the proof of the theorem, denote, for brevity, \,\Cal L\,:=\,\Dt\,-\,\Delta\,+\,V\, and \,\Upsilon_j = x_j + 2 i t {\partial\hfill\over\partial x_j}, \,j=1,\dots,n. Note, that$$ [\ \Cal L,\ \Upsilon_j\ ]\,=\,- \,2\,i\,t\,{\partial V(x)\over\partial x_j}\,.\tag 10 $$Let us first prove the theorem in the case \,m=1. Assume that \,\psi^0\in H^s_\r\, and, in addition, \,\bra{x} \psi^0\in H^{s-\nu}_\r. Let \,\psi\, be the solution of (5), and denote \,\psi_j := \Upsilon_j \psi. Then \,\psi_j\, is the solution of the inhomogenious problem$$\align \Cal L\,\psi_j\quad &=\,-\, 2\,i\,t\,{\partial V(x)\over\partial x_j}\,\psi,\tag 11a \\ \psi_j\big |_{t=0}\,&=\,x_j\,\psi^0.\tag 11b \endalign $$When \,t>0, the commutator (10), as an operator, has order at most \,\nu\, in the Hilbert scale \,\{H^r_\r\}. In other words, \,[\Cal L,\Upsilon_j]:\,H^r_\r\to H^{r-\nu}_\r, or, equivalently, \,\Lambda_{\rho}^{r-\nu}[\Cal L,\Upsilon_j] \Lambda_{\rho}^{-r}\,:\,L^2\to L^2, for any \,r. This follows from the composition formula for PDOs, and some simple inequalities, the inspiring example of which is the following triviality (see (4b)):$$ |\nabla V|\,\le\, c_{{}_1}\,\bra{x}^{{\rho\over 2}\,\nu}\,\le\, c_{{}_1}\,\big(\sqrt{|\xi|^2 + \bra{x}^{\rho}}\,\big)^{\nu}. $$Since \,\psi\in C_{\text{loc}}(\Bbb R;\, H^s_\r), the right-hand side of (11a) is in \,C_{\text{loc}}(\Bbb R;\, H^{s-\nu}_\r). Also, \,\psi_j(0,\cdot)\in H^{s-\nu}_\r. Hence, \,\psi_j\in C_{\text{loc}}(\Bbb R;\, H^{s-\nu}_\r), and (see (8))$$ \|\psi_j(t)\|_{H^{s-\nu}_\r}\,\le\, \tilde{c}\,\{ \| x_j \psi^0\|_{H^{s-\nu}_\r}\,+\, \int\limits_0^t 2\tau \| {\partial V(x)\over\partial x_j}\, \psi(\tau)\|_{H^{s-\nu}_\r}\,d\tau\}. $$We also know that$$\| {\partial V(x)\over\partial x_j}\, \psi(\tau)\|_{H^{s-\nu}_\r}\le c\, [\nabla V]_{s,\nu}\, \sup\limits_{0\le\tau\le T}\|\psi(\tau)\|_{H^s_\r}\, \le\,c\,[\nabla V]_{s,\nu} \,\|\psi^0\|_{H^s_\r}, $$where \,[\nabla V]_{s,\nu}\, is the maximum of the norms of the operators \,{\partial V(x)\over\partial x_j}:\,H^s_\r\to H^{s-\nu}_\r. Thus,$$ \|\psi_j(t)\|_{H^{s-\nu}_\r}\,\le\, \tilde{c}\,\{ \| x_j \psi^0\|_{H^{s-\nu}_\r}\,+ \,t^2\,\|\psi^0\|_{H^s_\r}\}.\tag 12 $$Assuming that \,t>0\, is fixed, we rearrange the equality \,\bra{x}^{-1} \Upsilon_j \psi = \bra{x}^{-1} \psi_j as follows:$$ \,{\partial\hfill\over\partial x_j}\, \big({1\over \bra{x}}\psi\big)\, =\, {1\over 2\,i\,t\,\bra{x}}\,\psi_j\,-\, {x_j\over\bra{x}}\, \big( {1\over 2\,i\,t}\,+\,{1\over\bra{x}^2}\big)\,\psi. \tag 13a $$Taking the \,H^{s-\nu}_\r-norm of both sides, we obtain the estimate$$\multline \| {\partial\hfill\over\partial x_j}\, \big({1\over \bra{x}}\psi(t)\big) \|_{H^{s-\nu}_\r}\,\le\, c\,\{ {1\over t} \|\psi_j(t)\|_{H^{s-\nu}_\r}+ (1+{1\over t}) \|\psi(t)\|_{H^{s-\nu}_\r}\}\\ \hfill\le\,c\,(1+{1\over t})\,\{ \|\psi^0\|_{H^{s}_\r} + \| \bra{x}\,\psi^0\|_{H^{s-\nu}_\r}\},\hfill \endmultline \tag 13b $$where we have used (7), (12), and the fact that the operators of multiplication by \,x_j\bra{x}^{-1}\, and \,\bra{x}^{-k}, \,k>0, are of order \,\le 0. Now, for any \,r\in\Bbb R,$$ \big(\|\nabla_x f\|^2_{H^r_\r} + \|\bra{x}^{{}^{\rho\over 2}} f\|^2_{H^r_\r}\big)^{1\over 2} \tag 14 $$is an equivalent norm in \,H^{r+1}_\r. The estimate (13b) then yields,$$ \|{1\over \bra{x}}\psi(t)\|_{H^{s+1-\nu}_\r}\,\le\, c\,(1+{1\over t})\,\{ \|\psi^0\|_{H^{s}_\r} + \| \bra{x}\,\psi^0\|_{H^{s-\nu}_\r}\},\tag 15 $$which proves the theorem in the case \,m=1. Assume next that, in addition to \,\psi^0\in H^s_\r\, and \,\bra{x} \psi^0\in H^{s-\nu}_\r, we also have \,\bra{x}^2 \psi^0\in H^{s-2\nu}_\r. Denote \,\psi_{kj} = \Upsilon_k \psi_j =\Upsilon_k\,\Upsilon_j\,\psi. The function \,\psi_{kj}(t,x)\, solves the problem$$\align \Cal L\,\psi_{kj}\quad &=\,-\, 2\,i\,t\,{\partial V(x)\over\partial x_k}\,\psi_j \, -\,2\,i\,t\,{\partial V(x)\over\partial x_j}\,\psi_k \,+\,4\,t^2\,{\partial^2 V\over\partial x_k\partial x_j}\, \psi,\tag 16a \\ \psi_{kj}\big |_{t=0}\,&=\,x_k\,x_j\,\psi^0.\tag 16b \endalign $$Taking into account the information obtained in the first step, and the fact that multiplication by \,{\partial^2 V\over\partial x_k\partial x_j}\, is a bounded operator from \,H^s_\r\, into \,H^{s-2\nu}_\r, we see that the right-hand side of (16a) is in \,C_{\text{loc}}(\Bbb R;\, H^{s-2\nu}_\r), and its \,H^{s-2\nu}_\r-norm is bounded by$$ \le\,const\,\{ t\,\|\bra{x}\psi^0\|_{H^{s-\nu}_\r}\,+\, t^2\,\|\psi^0\|_{H^{s}_\r}\}. $$Hence, \,\psi_{kj}\in C_{\text{loc}}(\Bbb R;\, H^{s-2\nu}_\r), and$$ \|\psi_{kj}(t)\|_{H^{s-2\nu}_\r}\,\le\, \tilde{c}\,\{ \| x_k\,x_j \psi^0\|_{H^{s-\nu}_\r}\,+\, t^2\,\|\bra{x}\psi^0\|_{H^{s-\nu}_\r}\,+ \,t^3\,\|\psi^0\|_{H^s_\r}\}.\tag 17a $$Since \,\bra{x}^{-1} \Upsilon_k \psi_j = \bra{x}^{-1} \psi_{kj}, we obtain, in exactly the same fashion as we obtained (13b), an estimate for the \,H^{s-2\nu}_\r-norm of \,\nabla_x\,(\bra{x}^{-1} \psi_j(t))\, that, in turn, implies$$ \|{1\over \bra{x}}\psi_j(t) \|_{H^{s + 1 -2\nu}_\r}\,\le\, \,c\,(1+{1\over t})\,\{ \|\psi^0\|_{H^{s-\nu}_\r} + \|\bra{x}\psi^0\|_{H^{s-\nu}_\r} + \| \bra{x}^2\,\psi^0\|_{H^{s-2\nu}_\r}\}.\tag 17b $$With this new information at hand, we write the equality \,\bra{x}^{-2}\Upsilon_j \psi = \bra{x}^{-2} \psi_j\, in the form:$$ \,{\partial\hfill\over\partial x_j}\, \big({1\over \bra{x}^2}\psi\big)\, =\, {1\over 2\,i\,t\,\bra{x}}\,{1\over\bra{x}}\psi_j\,-\, {x_j\over\bra{x}}\, \big( {1\over 2\,i\,t}\,+\,{2\over\bra{x}^2}\big)\, {1\over\bra{x}}\psi, \tag 17c $$and take the \,H^{s + 1 -2\nu}_\r-norm of both sides. Using (15), (17) and (14), we obtain the estimate$$ \|{1\over \bra{x}^2}\psi(t) \|_{H^{s + 2 -2\nu}_\r}\,\le\, \,c\,(1+{1\over t^2})\,\{ \|\psi^0\|_{H^{s}_\r} + \|\bra{x}\psi^0\|_{H^{s-\nu}_\r} + \| \bra{x}^2\,\psi^0\|_{H^{s-2\nu}_\r}\}.\tag 17d $$and, thus, prove the theorem in the case \,m=2. If \,m>2, one should repeat the above argument several more times.\qed \enddemo An immediate corollary of Theorem 1 is the following result. \proclaim{Corollary 2} Assume that the potential \,V\, satisfies Hypotheses (\bold V). Then the fundamental solution \,S_{{}_{-\Delta + V}}(t,x,y)\, is \,C^{\infty}\, in \,t>0\, and \,x,\,y\in\Bbb R^n. If \,\psi^0\, is a distribution of compact support, then the solution \,\psi(t,x)\, of (5) is \,C^{\infty}\, in \,t>0\, and \,x\in\Bbb R^n.\qed \endproclaim \vglue 1pc As we already mentioned, the method we presented above works for the Schr\"odinger equation with magnetic field as well. Let us consider the following equation describing a nonrelativistic quantum spinless particle in time-independent electromagnetic field,$$\align (\Dt - A_0(x)) \psi(t,x) &+ \sum\limits_{k=1}^n ({1\over i}\,{\partial\hfill\over\partial x_k}- A_k(x))^2\,\psi(t,x) = 0,\tag 18a\\ \psi(0,x) &= \psi^0(x).\tag 18b \endalign $$Here \,A_0,\,A_1,\dots A_n\, are some given real-valued functions of \,x, \,A_0\, the electric and \,A_j\, (j=1,\dots,n)\, the magnetic potentials. To the best of the authors' knowledge, the regularity of the Schwartz kernel of the solution operator \,S(t):\,\psi^0(\cdot)\mapsto \psi(t,\cdot)\, for (18) has not been discussed in the literature. [We should mention, however, that the smoothing effect in the sense of space-time integrability properties of solutions (Strichartz-type estimates) for the Schr\"odinger equation with magnetic field was discussed in \cite{Yaj 1,2}.] Comparing (18) with (5), we may expect that smoothing takes place if the \,A_j's grow at infinity slower than \,|x|, and \,A_0 grows slower than \,|x|^2. Essentially, this turns out to be true. The linear growth of the \,A_j's is critical: if \,n=3, \,A_0=0, and \,\vec{A}(x) = \frac12\,\vec{B}_0 \times \vec{x}, so that the magnetic field \,\vec{B}(x) = curl\,\vec{A}(x)=\vec{B}_0\, is constant, then the {\sl magnetic\/} analogue of Mehler's formula (see \cite{AHB, {\rom I}}, formula (3.5); and \cite{FH}) shows that the fundamental solution \,S(t,x,y)\, develops singularities in finite time. But, if the growth rate of \,\vec{A}(x)\, is less than \,1, then, {\sl as we prove below\/}, there are no singularities for \,t>0. \vglue .5pc Our assumptions on \,A_0,\,A_1,\dots A_n\, are as follows. \newline{\bf Hypotheses (\bold A)\/} The potentials \,A_0,\,A_1,\dots A_n\, are infinitely smooth, \,A_1,\dots A_n\, are real-valued, and \roster \item"1)" \,A_0(x)\, satisfies Hypotheses (\bold V) with some \,\rho\, and \,\nu\, obeying (4c); \item"2)" Each \,A_j,\;j=1,\dots,n, satisfies$$ |A_j(x)|\le \tilde{c}_0\,\bra{x}^{{}^{\rho\over 2}},\quad\quad |\partial_x^{\alpha}\,A_j(x)|\le \tilde{c}_{|\alpha|}\, \bra{x}^{{}^{{\rho\over 2}(|\alpha|-1)}},\quad \forall \alpha\in\Bbb Z_+^n,\quad|\alpha|>0,\tag 19a $$for all \,x\in\Bbb R^n, with some constants \,\tilde{c}; \item"3)" For \,j,\,k = 1,\dots, n, \,j\ne k, and for any \,\alpha\in\Bbb Z_+^n,$$ |\partial_x^{\alpha}\,F_{jk}(x)|\,\le\, c'_{|\alpha|}\,\bra{x}^{{}^{{\rho\over 2}\,\nu\,|\alpha|}}.\tag 19b $$Also, for any \,k=1,\dots,n,$$ |\partial_x^{\alpha}\,\sum_{j=1}^n x_j\,F_{jk}(x)|\,\le\, c'_{|\alpha|}\,\bra{x}^{{}^{{\rho\over 2}\,\nu\,|\alpha|}}.\tag 19c $$[Here and further on we use the notation \,F_{jk}(x)= \partial_{x_j}\,A_k(x)- \partial_{x_k}\,A_j(x).] \endroster \proclaim{Theorem 1'} Assume that the potentials \,A_0,\dots,A_n\, satisfy the hypotheses {\rm (\bold A)\/}. If \,\bra{x}^{\ell}\,\psi^0\in H^{s-\nu\ell}_\r\, for \,\ell = 0,\,1,\dots, m, some integer \,m\ge 1, then the assertions of Theorem~1 hold for the solution \,\psi(t,x)\, of (18). \endproclaim \demo{Proof} We use the same idea as in the proof of Theorem~1. As there, for the scale of Hilbert spaces we choose \,\{H^r_\r\}, generated by the powers of the operator \,\Lambda_\r=\sqrt{-\Delta + \bra{x}^{\rho}}. The hypotheses 1) and 2) in (\bold A) are sufficient for showing that for every real \,k,$$ \Lambda_\r^k\, ({\partial\hfill\over\partial x_k}\, A_k\,+\, \,A_k\,{\partial\hfill\over\partial x_k})\Lambda_\r^{-k}\, -\, \Lambda_\r^{-k}\, ({\partial\hfill\over\partial x_k}\, A_k\,+\, \,A_k\,{\partial\hfill\over\partial x_k})\Lambda_\r^{k}\, :\;L^2\,\to\,L^2, $$and$$ \Lambda_\r^k\,(A_0-\sum_{1\le j\le n} |A_j|^2)\, \Lambda_\r^{-k}\, -\, \Lambda_\r^{-k}\, (A_0^{*}-\sum_{1\le j\le n} |A_j|^2)\, \Lambda_\r^{k}\, :\;L^2\,\to\,L^2, $$as well. This and (9b), according to \cite{Kap}, Lemma~2.1, guarantee that the problem$$ (\Dt - A_0(x)) u(t,x) + \sum\limits_{k=1}^n ({1\over i}\,{\partial\hfill\over\partial x_k}- A_k(x))^2\,u(t,x) = h(t,x),\quad u(0,x)=u_0(x), $$has a (unique) solution in \,C_{\text{loc}}(\Bbb R;\,H^r_\r)\,for every \,u_0\in H^r_\r\, and \,h\in L^1_{\text{loc}}(\Bbb R;\,H^r_\r), with the energy estimate (8); \,r\, being arbitrary real. Denote \,\Cal L := \Dt + \sum_{k=1}^n D_k^2 - A_0, where \,D_k = {1\over i}\,{\partial\hfill\over\partial x_k}- A_k(x). The operators \,\Upsilon_j\, used in the proof of Theorem~1 will be modified in a natural way to \,\Upsilon_j=x_j+2it({\partial\hfill\over\partial x_j} - i\,A_j(x)) = x_j - 2t D_j. Let \,\psi^0\, satisfy the regularity assumption of the theorem and \,\psi\, be the corresponding solution of (18). Denote \,\psi_j := \Upsilon_j \psi. Then the \,\psi_j's satisfy$$\align \Cal L \psi_j + \frac2i \sum_{k=1}^n F_{kj}(x)\psi_k &= \frac2i \sum_{k=1}^n \big( x_k F_{kj}(x) - i{\partial F_{kj}(x)\over\partial x_k} \big)\psi + 2it {\partial A_0(x)\over\partial x_j} \psi\,, \tag 20a \\ \psi_j\big|_{{}_{t=0}} &= x_j\,\psi^0.\tag 20b \endalign $$We now view (20) as a {\it system\/} of equations for \,n\, functions \,\psi_1,\dots, \psi_n. Taking into account the hypotheses 1) and 2) of (\bold A), we see that the Cauchy problem$$ \Cal L u_j + \frac2i \sum_{k=1}^n F_{kj}(x)\, u_k = h_j, \quad u_j\big|_{{}_{t=0}} = u_j^0,\quad j=1,\dots,n, $$is well posed in the scale of Hilbert spaces \,H^r_\r\times\cdots\times H^r_\r\, (n\, factors), \,r\in\Bbb R. Because of the hypotheses 3) of (\bold A), and since \,\psi\in C_{\text{loc}}(\Bbb R;\,H^s_\r), the right-hand sides of (20a) lie in \,H^{s-\nu}_\r. This implies that \,\psi_j\in C_{\text{loc}}(\Bbb R;\,H^{s-\nu}_\r), and an estimate of the form (12) holds. From the equality \,\psi_j=(x_j+2t A_j) \psi + 2it{\partial\hfill\over\partial x_j}\psi, we arrive at estimate (15), in complete analogy with the proof of Theorem~1. This proves Theorem~1' in the case \,m=1. For larger \,m, we consider consecutively \,\psi_{kj}=\Upsilon_k\psi_j, \,\psi_{\ell kj}=\Upsilon_\ell \psi_{kj},\dots, use the systems of equations they obey to show that \,\psi_{kj}\in C_{\text{loc}}(\Bbb R;\,H^{s-2\nu}_\r), \,\psi_{\ell kj}\in C_{\text{loc}}(\Bbb R;\,H^{s-3\nu}_\r),\dots, and then return, step by step, to \,\psi, gaining regularity at the expense of weight. We skip the details.\qed \enddemo The generalisation of Corollary 2 is straightforward and we do not discuss it. \vglue 1pc An important property of problem (18) is its invariance with respect to the gauge transformations,$$ \psi\to\psi^g=e^{-i g} \psi, \; A_0\to A_0^g=A_0-{\partial g\over \partial t},\; \; A_j\to A_j^g=A_j-{\partial g\over \partial x_j},\; j=1,\dots,n, \tag 19a $$where \,g\, is an arbitrary function of \,t\, and \,x. Obviously, the relation between the corresponding fundamental solutions is as follows:$$ S{{}_{\Cal A}}(t,x,y)\,=\, e^{ig(t,x)}\,S{{}_{\Cal A^g}}(t,x,y)\,e^{-ig(t,y)}.\tag 19b $$This can be a source of information about smoothing in the situations where the magnetic potentials \,\{A_j\}\, do not initially satisfy Hypotheses (\bold A), but after the gauge transformation with an appropriate gauge \,g(x), the potentials \,\{A_j^g\} do satisfy those hypotheses. For example, we have the following result. \proclaim{Corollary 3} Assume that the potentials \,A_0,\dots, A_n\, are smooth, \,A_0\, satisfies condition 1) of Hypotheses (\bold A), and there exists a \,C^{\infty}\, gauge \,g(x)\, such that the potentials \,A_j^g\, satisfy conditions 2) and 3) of Hypotheses (\bold A). If the initial datum \,\psi^0\, is such that$$ \bra{x}^\ell\,\psi^0\,\in L^2,\quad \ell=0,\,1,\dots, m, \tag 20 $$then the solution \,\psi\, of (18) has the following regularity properties:$$ {1\over M_\ell(x)}\,\partial^\ell_x\,\psi(t,x)\,\in L^2, \quad \ell=0,\,1,\dots, m, \tag 21 $$uniformly in \,t\, on every finite interval \,[a,b], \,a>0. The weights \,M_\ell(x)\, are defined as follows$$ M_\ell(x)\,:=\, \max_{\alpha,\beta\in\Bbb Z^n_+,\,|\alpha+\beta|\le\ell} \quad\bra{x^\alpha\,\partial_x^\beta\,g(x)}\,.  \endproclaim \vglue 1pc {\bf ACKNOWLEDGEMENTS.\/} The authors are grateful to M. S. Birman, V. S. Buslaev and I. W. Herbst for valuable discussions. We thank K.~Yajima for sending us his very recent preprints. The work of the second author was partially supported by International Soros Science Education Program, grant No. 120\_S. \Refs \widestnumber\key{CFKS} \ref\key AHS\by J. Avron, I. Herbst and B. Simon \paper Schr\"odinger operator with magnetic fields. {\rom I}. \jour Duke Math. J. \vol 45\pages 847-881 \yr 1978 \moreref {\rom II}\jour Ann. Phys. (NY)\vol 114 \yr 1978\pages 431-451 \moreref {\rom III} \jour Comm. Math. Phys. \vol 79 \yr 1981\pages 529-572 \endref \ref\key Bea \by R. Beals \paper A general calculus of pseudodifferential operators \jour Duke Math. J. \vol 42 \pages 1-42 \yr 1975 \endref \ref\key CKS\by W. Craig, T. Kappeler \& W. Strauss \paper Microlocal dispersive smoothing for the Schr\"odinger equation \paperinfo draft of a paper\yr 1994 \endref \ref\key CFKS\by H. L. Cycon, R. G. Froese, W. Kirsch, and B. Simon \book Schr\"odinger operators \publ Springer-Verlag \publaddr Berlin $\cdot$ Heidelberg \yr 1987 \endref \ref\key Fei\by V. I. Feigin \paper Some classes of pseudodifferential operators in $\Bbb R^n$ and some applications \jour Trudy Moscow. Mat. Obsch. Trans. Moscow Math. Soc. \vol 36 \yr 1978 \pages 155-194 \paperinfo (Russian) \moreref English translation in: Trans. Moscow Math. Soc. \vol 36 \yr 1978 \endref \ref\key FH\by R. P. Feynman and A. Hibbs \book Quantum mechanics and path integrals \publ McGraw Hill \publaddr Hightstown, New Jersey \yr 1965 \endref \ref\key Fuj 1\by D. Fujiwara \paper A construction of the fundamental solution for the Schr\"odinger equation \jour Journ. d'Analyse Math. \vol 35\pages 41-96\yr 1979 \endref \ref\key Fuj 2\by D. Fujiwara \paper Remarks on convergence of the Feynman path integrals \jour Duke Math. J.\vol 47\pages 559-600 \yr 1980 \endref \ref\key Hun \by W. Hunziker \paper On the space-time behavior of Schr\"odinger wavefunctions \jour J. Math. Phys. \vol 7\issue 2 \yr 1966 \pages 300-304 \endref \ref\key IK\by T. Ikebe and T. Kato \paper Uniqueness of the self-adjoint extension of singular elliptic differential operators \jour Arch. Rational Mech. Anal. \vol 9\yr 1962 \pages 77-92 \endref \ref\key Jen \by A. Jensen \paper Commutator methods and smoothing property of the Scgr\"odinger evolution group \jour Math. Z. \vol 191 \yr 1986 \pages 53-59 \endref \ref\key Kap\by L. Kapitanski \paper Some generalizations of the Strichartz-Brenner inequality \jour Leningrad Math. J. \vol 1\issue 3\yr 1990 \endref \ref\key KS\by L. Kapitanski and Yu. G. Safarov \paper Dispersive smoothing for Schr\"odinger evolution \paperinfo Preprint \yr 1995 \endref \ref\key Kit 1\by H. Kitada \paper On a construction of the fundamental solution for Schr\"odinger equations \jour J. Fac. Sci. Univ. Tokyo \vol IA 27 \pages 193-226 \yr 1980 \endref \ref\key Kit 2\by H. Kitada \paper Fundamental Solutions and eigenfunction expansions for Schr\"odinger operators: I. Fundamental solutions \jour Math. Z.\vol 198\pages 181-190\yr 1988 \endref \ref\key KK\by H. Kitada and H. Kumano-go \paper A family of Fourier integral operators and the fundamental solution for a Schr\"odinger equation \jour Osaka J. Math. \vol 18 \pages 291-360 \yr 1981 \endref \ref\key KT\by H. Kumano-go and K. Taniguchi \paper Oscillatory integrals of symbols of pseudo-differential operators on $R^n$ and operators of Fredholm type \jour Proc. Japan Acad. \vol 49\issue 6\yr 1973\pages 397-402 \endref \ref\key MF\by V. P. Maslov and M. V. Fedoriuk \book Semi-classical approximation in quantum mechanics \publ D. Reidel Publ. Co. \publaddr Dordrecht, Holland \yr 1981 \endref \ref\key OF\by T. A. Osborn and Y. Fujiwara \paper The evolution kernels: uniform asymptotics expansions \jour J. Math. Phys. \vol 24 \issue 5 \yr 1983 \pages 1093-1103 \endref \ref\key Oza 1\by T. Ozawa \paper Smoothing effects and dispersion of singularities for the Schr\"odinger evolution group \jour Archive for Rational Mechanics and Analysis \vol 110 \yr 1990 \pages 165-186 \endref \ref\key Oza 2\by T. Ozawa \paper Invariant subspaces for the Schr\"odinger evolution group \jour Ann. Inst. Henri Poincar\'e, Physique th\'eorique \vol 54 \issue 1 \yr 1991\pages 43-57 \endref \ref\key Par\by C. Parenti \paper Operatori pseudo-differenziali in $R^n$ e applicazioni \jour Annali di Matematica Pura ed Applicata \vol 93 \yr 1972 \pages 359-389 \endref \ref\key Per\by P. Perry \paper Propagation of states in dilation-analytic potentials and asymptotic completeness \jour Commun. Math. Phys. \vol 81 \yr 1981 \pages 243-259 \endref \ref\key RS\by C. Radin and B. Simon \paper Invariant domains for the time-dependent Schr\"odinger equation \jour J. Diff. Equations \vol 29 \yr 1978 \pages 289-296 \endref \ref\key Sim 1\by B. Simon \book Functional integration and quantum physics \publ Academic Press \publaddr New York \yr 1979 \endref \ref\key Sim 2\by B. Simon \paper Schr\"odinger semigroups \jour Bull. Amer. Math. Soc. \vol 7\yr 1982 \pages 447-526 \endref \ref\key Tre\by F. Treves \paper Parametrices for a class of Schr\"odinger equations \jour Commun. Pure Appl. Math. \vol 48 \yr 1995 \pages 13-78 \endref \ref\key Wei\by A. Weinstein \paper A symbol class for some Schr\"odinger equations on $\,\Bbb R^n$ \jour Amer. Journ. Math.\vol 107\pages 1-23\yr 1985 \endref \ref \key Wil\by C. Wilcox \paper Uniform asymptotic estimates for wave packets in the quantum theory of scattering \jour J. Math. Phys. \vol 6 \yr 1965 \pages 611-620 \endref \ref\key Yaj 1\by K. Yajima \paper On smoothing property of Schr\"odinger propagators \jour Lecture Notes in Math. \vol 1450 \yr 1990 \pages 20-35 \endref \ref\key Yaj 2\by K. Yajima \paper Schr\"odinger evolution equations and associated smoothing effect \inbook Rigorous results in quantum dynamics \eds J. Dittrich and P. Exner \publ World Scientific \publaddr Singapore \yr 1991 \pages 167-185 \endref \ref\key Yaj 3\by K. Yajima \paper Schr\"odinger evolution equation with magnetic field \jour J. d'Analyse Math. \vol \yr \pages \endref \ref\key Yaj 4\by K. Yajima \paper Smoothness and non-smoothness of the fundamental solution for initial value problem for time dependent Schr\"odinger equations \paperinfo Preprint \yr 1995 \endref \ref\key Zel\by S. Zelditch \paper Reconstruction of singularities for solutions of Schr\"odinger equations \jour Comm. Math. Phys. \vol 90\yr 1983\pages 1-26 \endref \endRefs \end