Testing of a georadar in tunnel drifting

Optimal Fluctuations and Tail States of Non-Hermitian Operators

“…We remark that, in higher dimensions, the spherical symmetry of the optimal potential reduces the problem to being effectively one-dimensional. In this case, one obtains the general result [30],…”

“…For example, in the case of the Passive Scalar problem, where the disorder has the form of a random velocity field, the analogue of Eq. ( 6) relates the optimal configuration of the random velocity field to spatial derivatives of the eigenfunctions ψ R , ψ L [30].…”

“…From the relation Ĥ † = Ĥ * it follows that one can constrain the complex wavefunctions to obey the relation ψ R (r) = ψ L * (r) = ψ(r). In doing so, leaving aside the lengthy but straightforward analysis [30], one can show that the following simplifications obtain: Firstly, Eq. ( 8) can be effectively disregarded; secondly, by performing a complex rotation of ψ(r), one finds that arbitrary non-zero values can be assigned to the Lagrange multipliers λ 1 and λ 2 .…”

“…Indeed, in the case of a Gaussian distributed incompressible (∇ • v = 0) flow, the tail states of Ĥps are found to be governed by equations which have the same form as Eqs. ( 9) and ( 10) [30].…”