Volume 8 (2025)
Tadano, Yukihide
Construction of Isozaki–Kitada modifiers for discrete Schrödinger operators on general lattices
Annales Henri Lebesgue, Volume 8 (2025), pp. 255-276.

Metadata

DOI10.5802/ahl.235
Keywords long-range scattering theory, discrete Schrödinger operators, modified wave operators, time-independent modifiers

Abstract

We consider a scattering theory for difference operators on $\mathcal{H}=\ell ^2(\mathbb{Z}^d; \mathbb{C}^n)$ perturbed with a long-range potential $V:\mathbb{Z}^d\rightarrow \mathbb{R}^n$. One of the motivating examples is discrete Schrödinger operators on $\mathbb{Z}^d$-periodic graphs. We construct time-independent modifiers, so-called Isozaki–Kitada modifiers, and we prove that the modified wave operators with the above-mentioned Isozaki–Kitada modifiers exist and that they are complete.


References

[ABdMG96] Amrein, Werner O.; Boutet de Monvel, Anne; Georgescu, Vladimir C 0 -groups, commutator methods and spectral theory of N-body Hamiltonians, Progress in Mathematics, 135, Birkhäuser, 1996 | DOI | MR | Zbl

[AF78] Asada, Kenji; Fujiwara, Daisuke On some oscillatory integral transformations in L 2 (ℝ n ), Jpn. J. Math., New Ser., Volume 4 (1978) no. 2, pp. 299-361 | DOI | MR | Zbl

[AIM16] Ando, Kazunori; Isozaki, Hiroshi; Morioka, Hisashi Spectral properties of Schrödinger operators on perturbed lattices, Ann. Henri Poincaré, Volume 17 (2016) no. 8, pp. 2103-2171 | DOI | MR | Zbl

[DG97] Dereziński, Jan; Gérard, Christian Scattering Theory of Classical and Quantum N-Particle Systems, Texts and Monographs in Physics, Springer, 1997 | DOI | MR | Zbl

[IK85] Isozaki, Hiroshi; Kitada, Hitoshi Modified wave operators with time-independent modifiers, J. Fac. Sci., Univ. Tokyo, Sect. I A, Volume 32 (1985) no. 1, pp. 77-104 | MR | Zbl

[Nak14] Nakamura, Shu Modified wave operators for discrete Schrödinger operators with long-range perturbations, J. Math. Phys., Volume 55 (2014) no. 11, 112101, 8 pages | DOI | MR | Zbl

[Nak22] Nakamura, Shu Long-range scattering matrix for Schrödinger-type operators, Anal. PDE, Volume 15 (2022) no. 7, pp. 1725-1762 | DOI | MR | Zbl

[PR18] Parra, Daniel; Richard, Serge Spectral and scattering theory for Schrödinger operators on perturbed topological crystals, Rev. Math. Phys., Volume 30 (2018) no. 4, 1850009, 39 pages | DOI | MR | Zbl

[RS78] Reed, Michael; Simon, Barry The Methods of Modern Mathematical Physics. IV: Analysis of Operators, Academic Press Inc., 1978 | MR | Zbl

[RS79] Reed, Michael; Simon, Barry Methods of modern mathematical physics. III: Scattering theory, Academic Press Inc., 1979 | MR | Zbl

[RT09] Ruzhansky, Michael; Turunen, Ville Pseudo-differential operators and symmetries. Background analysis and advanced topics, Pseudo-Differential Operators. Theory and Applications, 2, Birkhäuser, 2009 | DOI | MR | Zbl

[Tad19a] Tadano, Yukihide Long-range scattering for discrete Schrödinger operators, Ann. Henri Poincaré, Volume 20 (2019) no. 5, pp. 1439-1469 | DOI | MR | Zbl

[Tad19b] Tadano, Yukihide Long-range scattering theory for discrete Schrödinger operators on graphene, J. Math. Phys., Volume 60 (2019) no. 5, 052107, 11 pages | DOI | MR | Zbl

[Yaf92] Yafaev, Dimitri R. Mathematical Scattering Theory. General Theory, Mathematical Surveys and Monographs, 105, American Mathematical Society, 1992 | DOI | Zbl

[Yaf10] Yafaev, Dimitri R. Mathematical scattering theory. Analytic theory, Mathematical Surveys and Monographs, 158, American Mathematical Society, 2010 | DOI | Zbl