\(L^p\)-Boundedness of Wave Operators for 2D Schrödinger Operators with Point Interactions

Abstract

For two dimensional Schrödinger operator H with point interactions, we prove that wave operators of scattering for the pair \((H,H_0)\), \(H_0\) being the free Schrödinger operator, are bounded in the Lebesgue space \(L^p({{\mathbb {R}}}^2)\) for \(1<p<\infty \) if and only if there are no generalized eigenfunctions of \(Hu(x)=0\) which satisfy \(u(x)= C|x|^{-1}+ o(|x|^{-1})\) as \(|x|\rightarrow \infty \), \(C\not =0\). Otherwise they are bounded for \(1<p\le 2\) and unbounded for \(2<p<\infty \).

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References

  1. 1.

    Agmon, S.: Spectral properties of Schrödinger operators and scattering theory. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 2, 151–218 (1975)

    MathSciNet  MATH  Google Scholar 

  2. 2.

    Albeverio, S., Gesztesy, F., Høegh-Krohn, R., Holden, H.: Solvable Models in Quantum Mechanics, 2nd edn. AMS Chelsea Publishing, Providence (2005)

    Google Scholar 

  3. 3.

    Beceanu, M., Schlag, W.: Structure formulas for wave operators under a small scaling invariant condition. J. Spectr. Theory 9(3), 967–990 (2019)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Cornean, H.D., Michelangeli, A., Yajima, K.: Two dimensional Schrödinger operators with point interactions, threshold expansions and \(L^p\)-boundedness of wave operators. Rev. Math. Phys. 31(4), 32 (2019)

    Article  Google Scholar 

  5. 5.

    Cornean, H.D., Michelangeli, A., Yajima, K.: Errata: Two dimensional Schrödinger operators with point interactions, threshold expansions and \(L^p\)-boundedness of wave operators. Rev. Math. Phys. 32(4), 5 (2020)

    Article  Google Scholar 

  6. 6.

    Dancona, P., Fanelli, L.: \(L^p\)-boundedness of the wave operator for the one dimensional Schrödinger operator. Commun. Math. Phys. 268(2), 415–438 (2006)

    ADS  Article  Google Scholar 

  7. 7.

    Digital Library of Mathematical Functions. https://dlmf.nist.gov/

  8. 8.

    Dell’Antonio, G., Michelangeli, A., Scandone, R., Yajima, K.: The \(L^p\)-boundedness of wave operators for the three-dimensional multi-centre point interaction. Ann. Inst. H. Poincaré 19, 283–322 (2018)

    ADS  Article  Google Scholar 

  9. 9.

    Duchêne, V., Marzuola, J.L., Weinstein, M.I.: Wave operator bounds for one-dimensional Schrödinger operators with singular potentials and applications. J. Math. Phys. 52, 013505, 17 (2011)

    Article  Google Scholar 

  10. 10.

    Erdoğan, M.B., Green, W.R.: Dispersive estimates for Schrödinger operators in dimension two with obstructions at zero energy. Trans. Am. Math. Soc. 365, 6403–6440 (2013)

    Article  Google Scholar 

  11. 11.

    Erdoğan, M.B., Goldberg, M., Green, W.R.: On the \(L^p\) boundedness of wave operators for two-dimensional Schrödinger operators with threshold obstructions. J. Funct. Anal. 274, 2139–2161 (2018)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Finco, D., Yajima, K.: The \(L^p\) boundedness of wave operators for Schrödinger operators with threshold singularities. II. Even dimensional case. J. Math. Sci. Univ. Tokyo 13(3), 277–346 (2006)

    MathSciNet  MATH  Google Scholar 

  13. 13.

    Galtbayar, A., Yajima, K.: The \(L^p\)-continuity of wave operators for one dimensional Schrödinger operators. J. Math. Sci. Univ. Tokyo 7(2), 221–240 (2000)

    MathSciNet  MATH  Google Scholar 

  14. 14.

    Goldberg, M., Green, W.R.: The \(L^p\) boundedness of wave operators for Schrödinger operators with threshold singularities. Adv. Math. 303, 360–389 (2016)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Jensen, A., Kato, T.: Spectral properties of Schrödinger operators and time-decay of the wave functions. Duke Math. J. 46(3), 583–611 (1979)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Jensen, A., Nenciu, G.: A unified approach to resolvent expansions at thresholds. Rev. Math. Phys. 13(6), 717–754 (2001)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Jensen, A., Yajima, K.: On \(L^p\) boundedness of wave operators for \(4\)-dimensional Schrödinger operators with threshold singularities. Proc. Lond. Math. Soc. (3) 96(1), 136–162 (2008)

    MathSciNet  Article  Google Scholar 

  18. 18.

    Jensen, A., Yajima, K.: A remal on the \(L^p\)-boundedness of wave operators for two domensional Schrödinger operators. Commun. Math. Phys. 225(3), 633–637 (2002)

    ADS  Article  Google Scholar 

  19. 19.

    Kato, T.: Perturbation of Linear Operators. Springer, Heidelberg (1966)

    Google Scholar 

  20. 20.

    Kuroda, S.T.: Introduction to Scattering Theory, Lecture Notes, Matematisk Institut, Aarhus University (1978)

  21. 21.

    Murata, M.: Asymptotic expansions in time for solutions of Schrödinger-type equations. J. Funct. Anal. 49(1), 10–56 (1982)

    MathSciNet  Article  Google Scholar 

  22. 22.

    Peral, J.C.: \(L^p\) estimate for the wave equation. J. Funct. Anal. 36, 114–145 (1980)

    Article  Google Scholar 

  23. 23.

    Reed, M., Simon, B.: Methods of Modern Mathematical Physics II, Fourier Analysis, Selfadjointness. Academic Press, New York (1975)

    Google Scholar 

  24. 24.

    Schlag, W.: Dispersive estimates for Schrödinger operators in dimension two. Commun. Math. Phys. 257, 87–117 (2005)

    ADS  Article  Google Scholar 

  25. 25.

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

    Google Scholar 

  26. 26.

    Watson, G.N.: Theory of Bessel Functions. Cambridge Univ. Press, London (1922)

    Google Scholar 

  27. 27.

    Weder, R.: The \(W^{k, p}\)-continuity of the Schrödinger wave operators on the line. Commun. Math. Phys. 208(2), 507–520 (1999)

    ADS  Article  Google Scholar 

  28. 28.

    Yajima, K.: The \(W^{k, p}\)-continuity of wave operators for Schrödinger operators. J. Math. Soc. Jpn. 47(3), 551–581 (1995)

    Article  Google Scholar 

  29. 29.

    Yajima, K.: \(L^p\) boundedness of wave operators for two dimensional Schrödinger operators. Commun. Math. Phys. 208(1), 125–152 (1999)

    ADS  Article  Google Scholar 

  30. 30.

    Yajima, K.: \(L^1\) and \(L^\infty \)-boundedness of wave operators for three dimensional Schrödinger operators with threshold singularities. Tokyo J. Math. 41(2), 385–406 (2018)

    MathSciNet  Article  Google Scholar 

  31. 31.

    Yajima, K.: Remarks on \(L^p\)-boundedness of wave operators for Schrödinger operators with threshold singularities. Doc. Math. 21, 391–443 (2016)

    MathSciNet  MATH  Google Scholar 

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Correspondence to Kenji Yajima.

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Dedicated to Professor Arne Jensen on the occasion of his 70th birthday.

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Supported by JSPS Grant in aid for scientific research No. 19K03589.

Communicated by Alain Joye.

Appendix

Appendix

In this appendix we show the following lemma:

Lemma A

For any \(\varepsilon >0\) there exits a constant \(C_\varepsilon >0\) such that

$$\begin{aligned} \int _{{{\mathbb {R}}}^4}e^{ix\xi -ip y} \frac{\chi _{\le {\varepsilon }}(|\xi |)\chi _{\le \varepsilon }(|p|)}{|\xi |+ |p|} d\xi {dp} {\le _{|\, \cdot \, |}\, }\frac{C_\varepsilon }{({\langle x \rangle }+ {\langle y \rangle })^3}\log \left( \frac{({\langle x \rangle }+ {\langle y \rangle })^2}{{\langle x \rangle }{\langle y \rangle }}\right) .\nonumber \\ \end{aligned}$$
(4.55)

Proof

If we use the identity

$$\begin{aligned} \frac{1}{|\xi |+|p|} = \int _0^\infty e^{-t(|\xi |+ |p|)} \hbox {d}t \end{aligned}$$

and Fubini’s theorem, then the left side of (4.55) becomes

$$\begin{aligned} \int _0^\infty \left( \frac{1}{2\pi } \int _{{{\mathbb {R}}}^2}e^{ix\xi -t|\xi |} \chi _{\le {\varepsilon }}(|\xi |)\hbox {d}\xi \right) \left( \frac{1}{2\pi } \int _{{{\mathbb {R}}}^2}e^{-ip y-t|p|} \chi _{\le \varepsilon }(|p|){\hbox {d}p} \right) \hbox {d}t.\nonumber \\ \end{aligned}$$
(4.56)

The functions inside parentheses are convolutions of the Poisson kernel with bump functions \({{\mathcal {F}}}\chi _{\le {\varepsilon }}(x)\) and \({{\mathcal {F}}}\chi _{\le {\varepsilon }}(y)\), respectively, (see [25], p. 61). They are bounded by \(C_1 t({\langle x \rangle }^2+ t^2)^{-3/2}\) and \(C_2 t({\langle y \rangle }^2+ t^2)^{-3/2}\), respectively. It follows by changing variable t to \({\langle x \rangle }^{1/2}{\langle y \rangle }^{1/2} t \) that

$$\begin{aligned} (4.56)&{\le _{|\, \cdot \, |}\, }C \int _0^\infty \frac{t^2 \hbox {d}t}{({\langle x \rangle }^2+ t^2)^{3/2}({\langle y \rangle }^2+ t^2)^{3/2}} \nonumber \\&= \frac{C}{{\langle x \rangle }^{3/2}{\langle y \rangle }^{3/2}} \int _0^\infty \frac{t^2\hbox {d}t}{(t^4 + s^2 t^2 +1)^{3/2}}, \quad s^2= {\frac{{\langle x \rangle }^2+{\langle y \rangle }^2}{{\langle x \rangle }{\langle y \rangle }}}. \end{aligned}$$
(4.57)

We estimate the integral in the right hand side of (4.57) by slitting \((0,\infty )\) into intervals into (0, 1/s), (1/ss) and \((s,\infty )\) where the denominator is bounded from below by 1, \(s^3 t^3\) and \(t^6\), respectively. Then

$$\begin{aligned} \int _0^\infty \frac{t^2\hbox {d}t}{(t^4 + s^2 t^2 +1)^{3/2} } \le \int _0^{1/s}{t^2\hbox {d}t}+ \int _{1/s}^s \frac{dt}{s^3 t} + \int _{s}^\infty \frac{\hbox {d}t}{t^4} = \frac{2}{3s^3}(1+3 \log {s}) . \end{aligned}$$

Since \(s\ge \sqrt{2}\), the right side may be further estimated by \(C s^{-3}\log {s^2}\) and \(s^2\ge ({\langle x \rangle }+{\langle y \rangle })^2/2{\langle x \rangle }{\langle y \rangle }\). Combining this with (4.57), we obtain the lemma. \(\square \)

For applications in the text we need only the following weaker version which trivially follows from Lemma A.

Lemma B

For any \(\varepsilon >0\) there exits a constant \(C_\varepsilon >0\) such that

$$\begin{aligned} \frac{1}{(2\pi )^2} \int _{{{\mathbb {R}}}^4}e^{ix\xi -ip y} \frac{\chi _{\le {2\varepsilon }}(\xi )\chi _{\le 2 \varepsilon }(|p|)}{|\xi |+ |p|} \hbox {d}\xi {\hbox {d}p} {\le _{|\, \cdot \, |}\, }\frac{C_\varepsilon }{{\langle x \rangle }{\langle y \rangle }({\langle x \rangle }+ {\langle y \rangle })}. \end{aligned}$$
(4.58)

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Yajima, K. \(L^p\)-Boundedness of Wave Operators for 2D Schrödinger Operators with Point Interactions. Ann. Henri Poincaré (2021). https://doi.org/10.1007/s00023-021-01017-4

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