Hokkaido Mathematical Journal

Low energy spectral and scattering theory for relativistic Schroedinger operators

Serge RICHARD and Tomio UMEDA

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Spectral and scattering theory at low energy for the relativistic Schr\"odinger operator are investigated. Some striking properties at thresholds of this operator are exhibited, as for example the absence of 0-energy resonance. Low energy behavior of the wave operators and of the scattering operator are studied, and stationary expressions in terms of generalized eigenfunctions are proved for the former operators. Under slightly stronger conditions on the perturbation the absolute continuity of the spectrum on the positive semi axis is demonstrated. Finally, an explicit formula for the action of the free evolution group is derived. Such a formula, which is well known in the usual Schr\"odinger case, was apparently not available in the relativistic setting.

Article information

Hokkaido Math. J., Volume 45, Number 2 (2016), 141-179.

First available in Project Euclid: 2 August 2016

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Zentralblatt MATH identifier

Primary: 81U05: $2$-body potential scattering theory [See also 34E20 for WKB methods] 35Q40: PDEs in connection with quantum mechanics 47F05: Partial differential operators [See also 35Pxx, 58Jxx] (should also be assigned at least one other classification number in section 47)

relativistic Schr\"odinger operators low energy scattering theory wave operators dilation group


RICHARD, Serge; UMEDA, Tomio. Low energy spectral and scattering theory for relativistic Schroedinger operators. Hokkaido Math. J. 45 (2016), no. 2, 141--179. doi:10.14492/hokmj/1470139399. https://projecteuclid.org/euclid.hokmj/1470139399

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