Skip to main content
Log in

Stationary Scattering Theory for One-body Stark Operators, II

  • Original Paper
  • Published:
Annales Henri Poincaré Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Adachi, T.: Asymptotic observables for \(N\)-body Stark Hamiltonians. Ann. Inst. H. Poincaré 68(3), 247–283 (1998)

    MathSciNet  MATH  Google Scholar 

  2. Avron, J.E., Herbst, I.W.: Spectral and scattering theory of Schrödinger operators related to the Stark effect. Commun. Math. Phys. 52(3), 239–254 (1977)

    Article  ADS  Google Scholar 

  3. Adachi, T., Itakura, K., Ito, K., Skibsted, E.: Spectral theory for the Stark Hamiltonian. JDE 268, 5179–5206 (2020)

    Article  ADS  Google Scholar 

  4. Adachi, T., Itakura, K., Ito, K., Skibsted, E.: Stationary scattering theory for \(1\)-body Stark operators, I. arXiv:1905.03539, to appear in Pure and Applied Functional Analysis

  5. Adachi, T., Itakura, K., Ito, K., Skibsted, E.: New methods in spectral theory of \(N\)-body Schrödinger operators, Rev. Math. Phys. 33(5), 2150015(2021)

  6. Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions. Dover Publications, New York (1972)

    MATH  Google Scholar 

  7. Gérard, C., Isozaki, H., Skibsted, E.: \(N\)-body resolvent estimates. J. Math. Soc. Jpn. 48(1), 135–160 (1996)

    Article  MathSciNet  Google Scholar 

  8. Herbst, I.W.: Unitary equivalence of stark Hamiltonians. Math. Z. 155(1), 55–70 (1977)

    Article  MathSciNet  Google Scholar 

  9. Hörmander, L.: The Analysis of Linear Partial Differential Operators. Springer, Berlin (1990)

    MATH  Google Scholar 

  10. Hörmander, L.: The Analysis of Linear Partial Differential Operators II–IV. Springer, Berlin (1983)

    MATH  Google Scholar 

  11. Ikebe, T., Isozaki, H.: A stationary approach to the existence and completeness of long-range operators. Int. Equ. Oper. Theory 5, 18–49 (1982)

    Article  MathSciNet  Google Scholar 

  12. Isozaki, H.: Differentiability of generalized Fourier transforms associated with Schrödinger operators. J. Math. Kyoto Univ. 25(4), 789–806 (1985)

    MathSciNet  MATH  Google Scholar 

  13. Isozaki, H., Kitada, H.: Scatttering matrices for two-body Schrödinger operators. Scientific papers of the college of arts and sciences. Tokyo Univ. 35, 81–107 (1985)

    Google Scholar 

  14. Ito, K., Skibsted, E.: Time-dependent scattering theory on manifolds. J. Funct. Anal. 277, 1423–1468 (2019)

    Article  MathSciNet  Google Scholar 

  15. Jensen, A.: Propagation estimates for Schrödinger operators. Trans. AMS 291(1), 129–144 (1985)

    MATH  Google Scholar 

  16. Kvitsinsky, A., Kostrykin, V.: Potential scattering in homogeneous external electrostatic field, Teoret. Mat. Fiz. 75 no. 3 (1988), 416–430; translation in Theoret. and Math. Phys. 75 (1988), no. 3, 619–629

  17. Kvitsinsky, A., Kostrykin, V.: \(S\)-matrix and Jost functions of Schrödinger Hamiltonian related to the Stark effect. J. Math. Phys. 31, 2731–2736 (1990)

    Article  ADS  MathSciNet  Google Scholar 

  18. Mourre, E.: Absence of singular continuous spectrum for certain selfadjoint operators. Commun. Math. Phys. 78(3), 391–408 (1980)

    Article  ADS  MathSciNet  Google Scholar 

  19. Nakamura, S.: Remarks on scattering matrices for Schrödinger operators with critically long-range perturbations. Ann. Inst. H. Poincaré 68(21), 3119–3139 (2020)

    Article  ADS  Google Scholar 

  20. Schwartz, J. T.: Nonlinear Functional Analysis. New York, London (1969)

  21. Stein, E.M.: Harmonic Analysis: Real-Variable Methods, Orthogonality and Oscillatory Integrals. Princeton University Press, Princeton (1993)

    MATH  Google Scholar 

  22. Titchmarsh, E.C.: Eigenfunction expansions associated with second-order differential equations. Vol. 2. Oxford, at the Clarendon Press (1958)

  23. White, D.A.W.: The Stark effect and long range scattering in two Hilbert spaces. Ind. Univ. Math. J. 39(2), 517–546 (1990)

    Article  MathSciNet  Google Scholar 

  24. Yajima, K.: Spectral and scattering theory for Schrödinger operators with Stark effect. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 26(3), 377–390 (1979)

    MathSciNet  MATH  Google Scholar 

  25. Yajima, K.: Spectral and scattering theory for Schrödinger operators with Stark effect II. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 28(1), 1–15 (1981)

    MathSciNet  MATH  Google Scholar 

Download references

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).

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to K. Ito.

Additional information

Communicated by Jan Derezinski.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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

$$\begin{aligned} c\int \mathrm{d}\zeta \,\xi (\zeta )\int \mathrm{e}^{\mathrm{i}h^{-1}{\tilde{\theta }}}\,\tilde{a}\,\chi _\epsilon \big ( {\eta }-h^{-1}\big )\chi _\epsilon \big ( \big |\zeta - hy\big |\big )\,\mathrm{d}\eta . \end{aligned}$$

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

$$\begin{aligned} \eta ^+= \sqrt{x+ \big ( x^2 - y^2\big )^{1/2}}\text { and }\zeta ^+=y/\eta ^+. \end{aligned}$$

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

$$\begin{aligned} {\tilde{\theta }}={\tilde{\theta }}_{|z=z(h,y)}+ \tfrac{1}{2} \left\langle {\phi ,A\phi } \right\rangle , \text { where } A=A(h,y)=\nabla ^2_z {\tilde{\theta }}_{|z=z(h,y)} \end{aligned}$$
(A.1)

and \(\phi \) is a diffeomorphism in z from an open neighborhood U of z(hy) 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

$$\begin{aligned} \nabla ^2_z\theta =-\eta \big ( I+{{\mathcal {O}}}(\zeta /\eta )\big ),\text { yielding roughly }\nabla ^2_z{\tilde{\theta }}\approx -I. \end{aligned}$$

Indeed we can introduce \(\breve{\theta }(\breve{z})={\tilde{\theta }}(z)\) by substituting \(z=\breve{z}+z(h,y) \), write

$$\begin{aligned} \breve{\theta }(\breve{z})={\tilde{\theta }}_{|z=z(h,y)}+\tfrac{1}{2}\left\langle {\breve{z},B(\breve{z})\breve{z}} \right\rangle ;\quad B=B(\breve{z})=2\int _0^1(1-\tau ) \nabla ^2_z \breve{\theta } \big ( \tau \breve{z}+z(h,y)\big )\,\mathrm{d}\tau , \end{aligned}$$

and use the inverse function theorem to solve

$$\begin{aligned} \Phi (\Gamma ):=\Gamma A^{-1}\Gamma =B \end{aligned}$$

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 (hy) (seen conveniently by using [20, Lemma 1.18]). Fix such r. One easily checks that \(\phi \) has bounded derivatives with bounds being independent of (hy) (using the same property of \(\Gamma \)).

The inverse map \(\psi :V\rightarrow U\) has derivatives which similarly are bounded uniformly in (hy) (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,

$$\begin{aligned}&\int \mathrm{d}\zeta \,\xi (\zeta )\int \mathrm{e}^{\mathrm{i}h^{-1}{\tilde{\theta }}}\,\tilde{a}\,\chi _\epsilon \big ( {\eta }-h^{-1}\big )\chi _\epsilon \big ( \big |\zeta - hy\big |\big )\,\mathrm{d}\eta \\ {}&=\mathrm{e}^{\mathrm{i}\theta _{|z=z(h,y)}}\int _V \mathrm{e}^{\mathrm{i}h^{-1}2^{-1} \left\langle {\phi ,A\phi } \right\rangle }\,f(\phi )\,\mathrm{d}\phi ;\\ {}&\quad \quad f(\phi )= \Big ( \xi (\zeta )\tilde{a}(\cdot )\,\chi _\epsilon {(\eta -h^{-1})}\chi _\epsilon \big ( \big |\zeta - hy\big |\big )\Big ) \big ( \psi (\phi )\big ){|{\det (\psi ')(\phi )}|}, \end{aligned}$$

and compute

$$\begin{aligned} \forall \alpha :&\quad \partial ^\alpha _\phi f={{\mathcal {O}}}(\left\langle {x,y} \right\rangle ^0),\\ \forall \alpha :&\quad {||{\partial ^\alpha _\phi f}||}_2={{\mathcal {O}}}(\left\langle {x,y} \right\rangle ^0), \end{aligned}$$

(Note for the latter bounds that \( \int 1_V \mathrm{d}\phi <\infty \).) By the Plancherel theorem and [9, Theorem 7.6.1]

$$\begin{aligned} \int \mathrm{e}^{\mathrm{i}h^{-1}2^{-1} \left\langle {\phi ,A\phi } \right\rangle }\,f(\phi )\,\mathrm{d}\phi&=h^{d/2}\mathrm{e}^{\mathrm{i}\pi \,\mathrm{sgn}(A)/4}{|{\det (A)}|}^{-1/2}\\&\quad \times \int _{{{\mathbb {R}}}^d} \mathrm{e}^{-\mathrm{i}h2^{-1} \left\langle {\breve{\zeta },A^{-1}\breve{\zeta }} \right\rangle }{{\hat{f}}}(\breve{\zeta }) \,\mathrm{d}\breve{\zeta }. \end{aligned}$$

By the inversion formula

$$\begin{aligned} \int {{\hat{f}}}(\breve{\zeta }) \,\mathrm{d}\zeta =(2\pi )^{d/2}f(0). \end{aligned}$$

On the other hand, by using the bound

$$\begin{aligned} {|{\mathrm{e}^{-\mathrm{i}h2^{-1} \left\langle {\breve{\zeta },A^{-1} \breve{\zeta }} \right\rangle }-1}|}\le h2^{-1}{|{\left\langle {\breve{\zeta },A^{-1}\breve{\zeta }} \right\rangle }|}, \end{aligned}$$

we can estimate for any integer \(n> 2+d/2\)

$$\begin{aligned}&\Big |\int _{{{\mathbb {R}}}^d} \big ( \mathrm{e}^{-\mathrm{i}h2^{-1} \left\langle {\breve{\zeta },A^{-1}\breve{\zeta }} \right\rangle }-1\big ){{\hat{f}}}(\breve{\zeta }) \,\mathrm{d}\breve{\zeta }\Big |\\ {}&\le h C_1\max _{{|{\alpha }|}\le n}\,{||{\partial _z^\alpha f}||}_2 \\&\quad \le C_2h. \end{aligned}$$

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

$$\begin{aligned} \begin{aligned}&\big |\partial _{\eta ,\zeta }^\alpha \,\partial _{x}^\beta \,\partial _{y}^\gamma b^\pm _k\big |\le 2^{-k}f^{\epsilon -(2k\delta +{|{\alpha }|} +2{|{\beta }|}+2{|{\gamma }|})} \\&\quad \text { for }{|{\alpha }|}+{|{\beta }|}+{|{\gamma }|}\le k, \quad \pm a>-\varepsilon \quad \text {and}\quad f>C_k, \end{aligned} \end{aligned}$$
(B.1)

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 \(k\in {{\mathbb {N}}}_0\)

$$\begin{aligned} \big |\partial _{x}^\beta \,\partial _{y}^\gamma {\chi _k}\big |\le C(l) f^{ -2{|{\beta }|}} \, \min \big ( f^2,\left\langle {y} \right\rangle _m\big )^{-{|{\gamma }|}}\text { for }{|{\beta }|}+{|{\gamma }|}\le l. \end{aligned}$$
(B.2)

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 \(k\ge l\)

$$\begin{aligned} \big |\partial _{\eta ,\zeta }^\alpha \,&\partial _{x}^\beta \,\partial _{y}^\gamma \big ( \chi _kb^\pm _k\big )\big |\le \breve{C}(l)2^{-k}f^{\epsilon -(2k\delta +{|{\alpha }|} +2{\beta })} \, \min \big ( f^2,\left\langle {y} \right\rangle _m\big )^{-{|{\gamma }|}}\\ {}&\text { for }{|{\alpha }|}+{|{\beta }|}+{|{\gamma }|}\le l \text { and for }\pm a>-\varepsilon . \end{aligned}$$

By summing up we conclude that \(c^\pm \) are well-defined smooth functions in the regions \({\{{\pm a>-\varepsilon }\}}\), respectively, with bounds

$$\begin{aligned} \big |\partial _{\eta ,\zeta }^\alpha \,&\partial _{x}^\beta \,\partial _{y}^\gamma c^\pm \big |\le C_{\alpha ,\beta ,\gamma }f^{-({|{\alpha }|} +2{\beta })} \, \min \big ( f^2,\left\langle {y} \right\rangle _m\big )^{-{|{\gamma }|}};\quad \pm a>-\varepsilon . \end{aligned}$$

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

$$\begin{aligned} \big |\partial _{\eta ,\zeta }^\alpha \,&\partial _{x}^\beta \,\partial _{y}^\gamma \chi ^\pm _\varepsilon \big |\le C_{\alpha ,\beta ,\gamma }f^{-({|{\alpha }|} +2{\beta })} \, \min \big ( f^2,\left\langle {y} \right\rangle _m\big )^{-{|{\gamma }|}};\quad f>C_0. \end{aligned}$$
(B.3)

For the second assertion of (6.6), the term \({{\mathcal {O}}} \big ( f^{-\infty }\big )\) is given explicitly as

$$\begin{aligned} {{\mathcal {O}}} \big ( f^{-\infty }\big )= \sum ^\infty _{0}\chi ^\pm _\varepsilon (\chi _{k+1}-\chi _k) \big ( qb^\pm _{k}-\tfrac{1}{2} (\Delta _{(x,y)} b^\pm _{k})\big ). \end{aligned}$$

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.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

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

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00023-021-01101-9

Navigation