A-Priori Decay for Eigenfunction of Perturbed Periodic Schrödinger Operators

Norman W. Bazley, Hans R. Fankhauser

M. Bertero, Giorgio Talenti, G. A. Viano

Martin Gebert

We prove the exact asymptotics of the scalar product of the ground states of two non-interacting Fermi gases confined to a 3-dimensional ball BL of radius L in the thermodynamic limit, where the underlying one-particle operators differ by a Dirac-δ perturbation. More precisely, we show the algebraic decay of the correlation determinant det〈φjL,ψkL〉j,k=1,…,N2=L−ζ(E)+o(1), as N, L → ∞ and N/BL→ρ>0, where φjL and ψkL denote the lowest-energy eigenfunctions of the finite-volume one-particle Schrödinger operators. The decay exponent is given in terms of the s-wave scattering phase shift ζ(E)≔1π2δ2(E). For an attractive Dirac-δ perturbation, we conclude that the decay exponent 1π2arcsinT(E)/2HS2 found in the work of Gebert et al. [J. Spect. Theory (to be published); e-print arXiv:1407.2512] does not provide a sharp upper bound on the decay of the correlation determinant.

M.A. Astaburuaga, V.H. Cortés, Claudio Fernández

Victor Dinu, Arne Jensen, G. Nenciu

We consider a two channel model of the form Hε=[Hop00E0]+ε[0W12W210] on H=Hop⊕C. The operator Hop is assumed to have the properties of a Schrödinger operator in odd dimensions, with a threshold at zero. As the energy parameter E0 is tuned past the threshold, we consider the survival probability |⟨Ψ0,e−itHεΨ0⟩|2, where Ψ0 is the eigenfunction corresponding to eigenvalue E0 for ε=0. We find nonexponential decay laws for ε small and E0 close to zero provided that the resolvent of Hop is not at least Lipschitz continuous at the threshold zero.

Leonid Andreevich Pastur, Mira Shamis

The paper deals with the asymptotic behavior of a widely used correlation characteristic in large quantum systems. The correlation is quantum entanglement, the characteristic is entanglement entropy, and the system is an ideal gas of lattice fermions. If the one-body Hamiltonian of fermions is an ergodic finite difference operator with an exponentially decaying spectral projection, then the large-block form of the entanglement entropy is the so-called area law. However, the only class of one-body Hamiltonians for which this spectral condition was verified consists of discrete Schrödinger operators with random potential. In this paper, we prove the area law for several classes of Schrödinger operators whose potentials are ergodic but not random. We begin with quasiperiodic and limit-periodic operators and then move to a highly non-trivial case of potentials generated by subshifts of finite type. These arose in the theory of dynamical systems when studying chaotic phenomena. The corresponding asymptotic study requires involved spectral analysis, which therefore constitutes the bulk of the paper. Specifically, we prove uniform localisation of the eigenfunctions for the Maryland model and exponential decay of the eigenfunction correlator for various models. We believe these properties are of significant independent interest.