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Analyticity of the Schrödinger propagator on the Heisenberg group. (English) Zbl 1256.22004

The authors consider the Schrödinger equation on \(\mathbb R^k: \) \(iu_t =\mathcal P u(x,t), \- u(x,0)=f(x).\) They analyze the Hilbert-continuity and the nature of the image of the generalized Segal-Bargmann transform \( f \rightarrow u(x,t):=e^{-ti\mathcal P}f, \) which formally solves the Schrödinger equation.
More precisely, the authors consider operators \(\mathcal P \in \{ H:=-\Delta + | x |^2, \mathcal L_\lambda= -\Delta + \frac{\lambda}{4} (| x |^2+| u |^2)-i\lambda \sum_{j=1}^{j=n}(x_j \partial_{u_j}-u_j \partial_{x_j}), \mathcal L =-\sum_{j=1}^{j=n} (X_j^2 + U_j^2 ), \mathcal L_\alpha=-(\frac{\partial^2}{\partial r^2 }+\frac{2\alpha +1}{r} \frac{\partial}{\partial r } +\frac{r^2}{4} \frac{\partial^2 }{\partial \xi^2})\}.\) Here. \(X_j=\frac{\partial}{\partial x_j} -\frac{u_j}{2} \frac{\partial}{\partial \xi}, U_j= \frac{\partial}{\partial u_j} +\frac{x_j}{2} \frac{\partial}{\partial \xi} .\)
For each operator, the authors define a convenient \(L^2-\)weighted space of functions \(f\) so that \(u(?,t)\) extends to an entire function on \(\mathbb C^k\) and determine the image of the map \(f \rightarrow u.\) For the first two operators the result is a straightforward generalization of the theorem of Bargmann-Segal, for the last two operators the image is a direct integral of spaces of entire functions. They also show: For the operator \(\mathcal L\) the straightforward generalization of the theorem of Bargmann-Segal (\(\natural\)) does not hold.
To be concrete we state two of the theorems in the note.
i)
Set \( \omega_t^\lambda (\xi, \eta)=e^{(\lambda \mathrm{cot} \lambda t ) \xi .\eta } q_{sin^2 t\lambda /\lambda^2}(\eta).\) Let \(S_t^\lambda (\mathbb C^{2n})\) denote the space of entire functions for which \( \int_{\mathbb C^{2n}} | F(\xi +i \eta)|^2 \omega_t^\lambda (\xi, \eta)d\xi d\eta < \infty . \) The authors show: The operator \(e^{it\mathcal L_\lambda} \) is an isometric isomorphism between \(L^2(\mathbb R^{2n}, e^{| \xi |^2})\) and the Hilbert space \(S_t^\lambda (\mathbb C^{2n}). \) Moreover \( \int_{\mathbb C^{2n}} | e^{it\mathcal L_\lambda} f(\xi +i \eta)|^2 \omega_t^\lambda (\xi, \eta)d\xi d\eta = \int_{\mathbb R^{2n}} | f(x)|^2 e^{| x |^2} dx (\natural) \) and the function \(\omega_t^\lambda\) is unique.
ii)
For a function \(f\) on the Heisenberg group \(H^n\) set \(f^\lambda(x,u)=\int_\mathbb R f(x,u,\xi)e^{i\lambda \xi} d\xi.\) For \(R>0 \) let \(\mathcal H_R\) stand for the subspace of \(f \in L^2(H^n) \) such that \(f^\lambda\) is supported in \(| \lambda | \leq R, \int_{\mathbb R^{2n+1}} | f(x,u,\xi)|^2 e^{| x |^2 +| u |^2} dx du d\xi. \) Let \(T_t\) stand for the map which takes \(f \in L^2(H^n)\) into \((e^{it\mathcal L_\lambda} f^\lambda)_{\lambda \in [-R,R]}.\) Then for \( | t | <\frac{\pi}{R},\) \(T_t : \mathcal H_R \rightarrow \int_{-R}^R S_t^\lambda (\mathbb C^{2n}) d\lambda \) is an isometric isomorphism.

MSC:

22E30 Analysis on real and complex Lie groups
35G10 Initial value problems for linear higher-order PDEs
47A63 Linear operator inequalities
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