Pointwise estimates of eigenfunctions for Schrödinger operators

Marius Măntoiu, Radu Purice

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

Leonid Andreevich Pastur, Mira Shamis

We review rigorous results concerning the Area Law for the entanglement entropy of free fermions whose one-body Hamiltonian is an ergodic Schrödinger operator with a strongly localised pure point spectrum. Particular attention is paid to operators with deterministic potentials (quasi-periodic ones and those generated by subshifts of finite type). Our approach relies on reducing the problem to obtaining exponential estimates for the eigenfunction correlator. This provides an effective route from localisation to the Area Law and allows for the identification of additional spectral estimates that need to be proven for various classes of ergodic operators. We realise this approach by using various localisation techniques, including dynamical localisation and the uniformly localised eigenfunction property for bounded and unbounded ergodic operators (e.g. the Maryland model), as well as a new method for obtaining the corresponding exponential bounds for Schrödinger operators generated by subshifts of finite type.

David Milton Fradkin

The classical solution is used to provide appropriate variables for the quantum-mechanical free particle time-dependent Schrödinger wave equation, which then becomes separable giving nonenergy eigenfunctions of closed form. Solutions derived in this fashion may fail to satisfy the homogeneous Schrödinger wave equation at one point in space/time; if this should occur, an inhomogeneous term is produced and a corresponding restricted solution reduces to the free particle propagator. In general, it is shown that the nonenergy eigenfunctions may also be interpreted as eigenfunctions of a time-independent transformed momentum operator. The similarity transformation, by means of which is constructed the transformed momentum operator, the transformed position operator, and the transformed Hamiltonian, are explicitly determined. The relationship of the nonenergy eigenfunctions to the energy eigenfunctions is also exhibited.

Rainer Dick