[Paper Review] Remarks on the Fundamental Solution to Schrödinger Equation with Variable Coefficients

Kenichi Ito, Shu Nakamura|ArXiv.org|Dec 25, 2009

Advanced Mathematical Physics Problems5 references3 citations

TL;DR

This paper establishes that the time evolution operator $ e^{-itH} $ for Schrödinger operators with variable coefficients can be decomposed as $ e^{-itH_0}W(t) $, where $ W(t) $ is a Fourier integral operator (FIO) microlocally associated to the classical scattering canonical relation. Under the nontrapping condition and short-range assumptions with $ \mu = 2 $, the evolution is microlocally microsupported along the scattering flow, extending results of Hassell and Wunsch with novel techniques based on Egorov-type theorems and Beals-type characterization.

ABSTRACT

We consider Schrödinger operators $H$ on $R^n$ with variable coefficients. Let $H_0=-\frac12 riangle$ be the free Schrödinger operator and we suppose $H$ is a "short-range" perturbation of $H_0$. Then, under the nontrapping condition, we show the time evolution operator: $e^{-itH}$ can be written as a product of the free evolution operator $e^{-itH_0}$ and a Fourier integral operator $W(t)$, which is associated to the canonical relation given by the classical mechanical scattering. We also prove a similar result for the wave operators. These results are analogous to results by Hassell and Wunsch, but the assumptions, the proof and the formulation of results are considerably different. The proof employs an Egorov-type theorem similar to those used in previous works by the authors combined with a Beals-type characterization of Fourier integral operators.

Motivation & Objective

• To characterize the microlocal structure of the time evolution operator $ e^{-itH} $ for Schrödinger operators with variable coefficients.
• To show that $ e^{-itH} $ decomposes into the free evolution $ e^{-itH_0} $ and a Fourier integral operator $ W(t) $, under nontrapping and short-range assumptions.
• To establish that $ W(t) $ is microlocally associated to the classical scattering canonical relation via the Hamilton flow.
• To provide a new proof strategy using Egorov-type theorems and Beals-type characterization of FIOs, distinct from prior works.
• To generalize the result microlocally to conic neighborhoods of nontrapping directions, allowing local FIO structure for $ W(t) $.

Proposed method

• The proof uses an Egorov-type theorem adapted from previous works by the authors to relate the action of $ W(t) $ to the classical Hamiltonian flow.
• It applies a Beals-type characterization of Fourier integral operators to verify that $ W(t) $ satisfies the required regularity and oscillatory integral structure.
• The analysis relies on asymptotic expansions of the classical trajectories $ (y(t), \eta(t)) $ under the kinetic Hamiltonian $ H_k $, with estimates on $ z_\pm $ and $ \xi_\pm $ as $ t \to \pm\infty $.
• The authors use induction on the order of derivatives $ |\alpha| + |\beta| $ to control the growth of derivatives of the scattering data $ z_\pm, \xi_\pm $, employing Duhamel's formula and weighted $ L^\infty $ estimates.
• The short-range condition with $ \mu = 2 $ ensures sufficient decay of $ a_{jk}(x) - \delta_{jk} $ and $ V(x) $, enabling convergence of the asymptotic expansions.
• Microlocal partition of unity is used to localize the problem and extend the global nontrapping result to local FIO structure in conic neighborhoods.

Experimental results

Research questions

• RQ1 Can the time evolution operator $ e^{-itH} $ for variable-coefficient Schrödinger operators be microlocally decomposed into free evolution and a Fourier integral operator?
• RQ2 Under what conditions is the operator $ W(t) = e^{itH_0}e^{-itH} $ a Fourier integral operator associated to the classical scattering canonical relation?
• RQ3 How does the microlocal wavefront set of $ W(t) $ relate to the classical scattering data $ (z_\pm, \xi_\pm) $?
• RQ4 What role does the parameter $ \mu $ in the short-range condition play in the existence and regularity of the scattering data and the FIO structure?
• RQ5 To what extent can the global nontrapping assumption be relaxed to obtain local FIO representations?

Key findings

• Under Assumption A with $ \mu = 2 $, the time evolution operator satisfies $ e^{-itH} = e^{-itH_0}W(t) $, where $ W(t) $ is a Fourier integral operator microlocally associated to the classical scattering canonical relation.
• For each $ t \in \mathbb{R}_\pm $, $ W(t) $ is an FIO associated to the canonical transformation $ w_\pm $, which maps initial data $ (x_0, \xi_0) $ to scattering data $ (z_\pm, \xi_\pm) $.
• The microlocal wavefront set of $ W(t) $ is contained in the Lagrangian submanifold $ \Lambda_\pm = \{ (y,\eta,x,-\xi) \mid (y,\eta) = w_\pm(x,\xi) \} $, reflecting propagation of singularities along classical trajectories.
• The scattering data $ z_\pm $ and $ \xi_\pm $ exist and are smooth, homogeneous of order 0 and 1 in $ \xi $, respectively, under the $ \mu = 2 $ condition.
• The proof establishes uniform estimates on derivatives of $ z_1 - z $ and $ \eta_1 - \eta $, showing $ |\partial_x^\alpha \partial_\xi^\beta (z_1 - z)| \leq C|t|\langle \xi \rangle^{1-\mu - |\beta|} $, which implies regularity of the scattering map.
• The result holds microlocally: if $ (x_0, \xi_0) $ is forward nontrapping, then $ W(t) $ is an FIO associated to $ w_+ $ in a conic neighborhood of $ (x_0, \xi_0) $, extending the global result locally.