Abstract
We study and develop the stationary scattering theory for a class of one-body Stark Hamiltonians with short-range potentials, including the Coulomb potential, continuing our study in Adachi et al. (JDE 268: 5179–5206, 2020; Stationary scattering theory for 1-body Stark operators). The classical scattering orbits are parabolas parametrized by asymptotic orthogonal momenta, and the kernel of the (quantum) scattering matrix at a fixed energy is defined in these momenta. We show that the scattering matrix is a classical type pseudodifferential operator and compute the leading order singularities at the diagonal of its kernel. Our approach can be viewed as an adaption of the method of Isozaki-Kitada (Tokyo Univ. 35: 81–107, 1985) used for studying the scattering matrix for one-body Schrödinger operators without an external potential. It is more flexible and more informative than the more standard method used previously by Kvitsinsky-Kostrykin (Teoret. Mat. Fiz. 75(3): 416-430, 1988) for computing the leading order singularities of the kernel of the scattering matrix in the case of a constant external field (the Stark case). Our approach relies on Sommerfeld’s uniqueness result in Besov spaces, microlocal analysis as well as on classical phase space constructions.
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Acknowledgements
K.I. is supported by JSPS KAKENHI Grant No. 17K05325. E.S. is supported by the Research Institute for Mathematical Sciences, a Joint Usage/Research Center located in Kyoto University, and by DFF Grant No. 4181-00042. K.I. and E.S. are supported by the Swedish Research Council Grant No. 2016-06596 (residing at Institut Mittag-Leffler in Djursholm, Sweden, during the Spring semester of 2019).
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Appendices
Appendix A. Proof of (2.6)
For convenience we only consider \(\phi ^+_{\lambda ,{\tilde{a}}}[\xi ]\) and \(\lambda =0\), whence we need to consider the asymptotics of
of a symbol \({\tilde{a}}={\tilde{a}}(x,y;\eta ,\zeta )\) obeying (2.2b). Recall that there is exactly one relevant stationary point for \(x>R\) and \({|{y}|}< C\sqrt{2x}\), cf. (2.5), say denoted \(z(h,y)=(\eta ^+,\zeta ^+)\). This is given by
Let similarly \(z=(\eta ,\zeta )\). To obtain the asymptotics as \(h\rightarrow 0\) uniformly in y we write the phase \({\tilde{\theta }}=h \theta \) as
and \(\phi \) is a diffeomorphism in z from an open neighborhood U of z(h, y) onto an open neighborhood V of 0 with \(\phi _{|z=z(h,y)}=0\) and derivative \(\nabla _z\phi _{|z=z(h,y)}=I\). The existence of such map \(\phi \) follows as in the proof of [11, Lemma 4.2] and the computation
Indeed we can introduce \(\breve{\theta }(\breve{z})={\tilde{\theta }}(z)\) by substituting \(z=\breve{z}+z(h,y) \), write
and use the inverse function theorem to solve
for a unique real symmetric \(d\times d\) matrix \(\Gamma =\Gamma (\breve{z})=\Gamma (\breve{z}, h,y)\) near \(A=A(h,y)\). Note that \(\Phi (A)=A=B(0)\). Then, \( \phi (z):=A^{-1}\Gamma (\breve{z})\breve{z}\) works in (A.1), in fact with \(U=z(h,y)+B_r(0)\) where \(B_r(0)\) is the open ball centered at 0 with radius \(r>0\) being independent of (h, y) (seen conveniently by using [20, Lemma 1.18]). Fix such r. One easily checks that \(\phi \) has bounded derivatives with bounds being independent of (h, y) (using the same property of \(\Gamma \)).
The inverse map \(\psi :V\rightarrow U\) has derivatives which similarly are bounded uniformly in (h, y) (seen inductively by the Faà di Bruno formula). We change variable \(z\rightarrow \phi \) and write, possibly at this point taking \(\epsilon >0\) smaller and \(R=R(\epsilon )>2\) larger,
and compute
(Note for the latter bounds that \( \int 1_V \mathrm{d}\phi <\infty \).) By the Plancherel theorem and [9, Theorem 7.6.1]
By the inversion formula
On the other hand, by using the bound
we can estimate for any integer \(n> 2+d/2\)
Finally, by invoking \( A=-I+{{\mathcal {O}}}(h)\) (uniformly in y) and (2.5), the asymptotics (2.6) follows. \(\square \)
Remark A.1
We used above only the zeroth order Taylor expansion of the Gaussian function of \(\breve{\zeta }\) at zero. If the symbol \({\tilde{a}}\) vanishes to any order at the stationary point \((\eta ^+,\zeta ^+)\), then higher order Taylor expansion yields that the integral is \({{\mathcal {O}}}(h^\infty )\) rather than \({{\mathcal {O}}}(h^{d/2})\) as proved above.
Appendix B. Borel Construction for (6.5)
We consider \(c^\pm :=\sum _0^\infty \chi _k b^\pm _k\), where \(\chi _k=\chi (f/C_k>1)\) needs to be determined. Fix \(\epsilon >0\) (for example take \(\epsilon =1\)). Thanks to (6.4), we can for any \(k\in {{\mathbb {N}}}_0\) find a sufficiently big \(C_k\ge 2\) such that
and clearly we can take \(C_0=2\) and assume that \(C_k>1+C_{k-1}\) for \(k\ge 1\).
Note that for all \(l\in {{\mathbb {N}}}_0\) there exists \(C(l)>0\) such for all \(k\in {{\mathbb {N}}}_0\)
By combining (B.1) and (B.2) with the product rule we conclude that for all \(l\in {{\mathbb {N}}}_0\), there exists \(\breve{C}(l)>0\) such for all \(k\ge l\)
By summing up we conclude that \(c^\pm \) are well-defined smooth functions in the regions \({\{{\pm a>-\varepsilon }\}}\), respectively, with bounds
Since \(a_B^\pm =\chi ^\pm _\varepsilon c^\pm \), these bounds and the product rule yields the first bound of (6.6) (however being only locally uniform in \(\zeta \)). Note at this point that
For the second assertion of (6.6), the term \({{\mathcal {O}}} \big ( f^{-\infty }\big )\) is given explicitly as
By using (B.1)–(B.3) one easily checks that indeed the right-hand side is bounded along with all derivatives by any inverse power of f with a bounding constant being locally uniform in \(\zeta \). This completes the proof of (6.6). The related bounds (7.1) easily follow too.
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Ito, K., Skibsted, E. Stationary Scattering Theory for One-body Stark Operators, II. Ann. Henri Poincaré 23, 513–548 (2022). https://doi.org/10.1007/s00023-021-01101-9
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DOI: https://doi.org/10.1007/s00023-021-01101-9