Validity of lorentz invariance at extremely high energies

Experimental data on the distribution of times of arrival of electrons (and muons) in extensive air showers produced by cosmic rays in the energy range 10¹²–10¹⁹ eV have been examined to see if there is any evidence for a departure of the velocity of ultra-high energy particles from that of light as suggested by the model proposed by Pavlopoulos. No evidence for such a departure has been found. An upper limit to the “fundamental length” occurring in the theory is obtained as ～10^?21 cm.     

Instability in the Gel'fand inverse problem at high energies

We give an instability estimate for the Gel'fand inverse boundary value problem at high energies. Our instability estimate shows an optimality of several important preceding stability results on inverse problems of such a type.     

Eikonalization of the scattering amplitude in some field theory models at high energies

Conclusions The various examples of interactions considered above, which are more complicated than processes described by generalized ladder diagrams, show that in this case too the eikonal representation holds, but with a very different phase. Comparing the eikonal formula obtained in §1 with the asymptotic behavior of the individual diagrams of perturbation theory for Fig. 3, we conclude that, as in the case of the theory ³, there is probably a compensation of the contributions of the asymptotic behaviors of the individual diagrams. For example, the diagrams of Fig. 3 have a higher power of ln s [the diagram of order (g²h)⁴ is proportional to (ln s)³], whereas the corresponding terms of the expansion of (1.10) in a series have no logarithmic terms in s at all.Institute of High-Energy Physics. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 9, No. 3, pp. 343–354, December, 1971.     

Quantum scattering at low energies

For a class of negative slowly decaying potentials, including V(x):=−γ|x|^−μ with 0<μ<2, we study the quantum mechanical scattering theory in the low-energy regime. Using appropriate modifiers of the Isozaki-Kitada type we show that scattering theory is well behaved on the whole continuous spectrum of the Hamiltonian, including the energy 0. We show that the modified scattering matrices S(λ) are well-defined and strongly continuous down to the zero energy threshold. Similarly, we prove that the modified wave matrices and generalized eigenfunctions are norm continuous down to the zero energy if we use appropriate weighted spaces. These results are used to derive (oscillatory) asymptotics of the standard short-range and Dollard type S-matrices for the subclasses of potentials where both kinds of S-matrices are defined. For potentials whose leading part is −γ|x|^−μ we show that the location of singularities of the kernel of S(λ) experiences an abrupt change from passing from positive energies λ to the limiting energy λ=0. This change corresponds to the behaviour of the classical orbits. Under stronger conditions one can extract the leading term of the asymptotics of the kernel of S(λ) at its singularities.     

On two-particle Anderson localization at low energies

We prove exponential spectral localization in a two-particle lattice Anderson model, with a short-range interaction and an external i.i.d. random potential, at sufficiently low energies. The proof is based on the multi-particle multi-scale analysis developed earlier in Chulaevsky and Suhov (2009) [4] in the case of high disorder. Our method applies to a larger class of random potentials than in Aizenman and Warzel (2009) [2] where dynamical localization was proved with the help of the fractional moment method.