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Register a new userJustifying Kubos formula for gapped systems at zero temperature: a brief review and some new results
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We first review the problem of a rigorous justification of Kubos formula for transport coefficients in gapped extended Hamiltonian quantum systems at zero temperature. In particular, the theoretical understanding of the quantum Hall effect rests on the validity of Kubos formula for such systems, a connection that we review briefly as well. We then highlight an approach to linear response theory based on non-equilibrium almost-stationary states (NEASS) and on a corresponding adiabatic theorem for such systems that was recently proposed and worked out by one of us in [51] for interacting fermionic systems on finite lattices. In the second part of our paper we show how to lift the results of [51] to infinite systems by taking a thermodynamic limit.
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We introduced some contact potentials that can be written as a linear combination of the Dirac delta and its first derivative, the $delta$-$delta$ interaction. After a simple general presentation in one dimension, we briefly discuss a one dimensional periodic potential with a $delta$-$delta$ interaction at each node. The dependence of energy bands with the parameters (coefficients of the deltas) can be computed numerically. We also study the $delta$-$delta$ interaction supported on spheres of arbitrary dimension. The spherical symmetry of this model allows us to obtain rigorous conclusions concerning the number of bound states in terms of the parameters and the dimension. Finally, a $delta$-$delta$ interaction is used to approximate a potential of wide use in nuclear physics, and estimate the total number of bound states as well as the behaviour of some resonance poles with the lowest energy.
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