The overlap integrals of scattering states in potentials of finite widths are expressed with their asymptotic behaviors and those of energies
E1 and
E2 consist of diagonal terms that are proportional to
δ(E1−E2) and nondiagonal terms. Owing to the composition of nondiagonal terms, superpositions of stationary states have time-dependent norms and finite probability currents. These do not represent isolate states. In various exceptional potentials and in free theory, nondiagonal terms do not exist, and the superpositions of states with different energies represent isolate particles that exactly describe scattering processes.
The goal of this thesis is a mathematical understanding of a phenomenon called Anderson's Orthogonality in the physics literature. Given two non-interacting fermionic systems which differ by an exterior short-range scattering potential, the scalar product of the corresponding ground states is predicted to decay algebraically in the thermodynamic limit. This decay is referred to as Anderson's orthogonality catastrophe in the physics literature and goes back to P.W.Anderson [Phys. Rev. Lett. 18:1049--1051] and is used to explain anomalies in the X-ray absorption spectrum of metals. We call this scalar product
SNL, where
N refers to the particle number and
L to the diameter of the considered fermionic system. This decay was proven in the works [Commun. Math. Phys. 329:979--998] and [arXiv:1407.2512] for rather general pairs of Schrödinger operators in arbitrary dimension
d∈N, i.e.
∣SNL∣2≤L−γ in the thermodynamic limit
N/Ld→ρ>0 approaching a positive particle density. In the general case, the biggest found decay exponent is given by $\gamma=\frac 1 {\pi^2} \norm{\arcsin|T/2|}_{\text{HS}}$, where T refers to the scattering T-matrix. In this thesis, we prove such upper bounds in more general situations than considered in both [Commun. Math. Phys. 329:979--998] and [arXiv:1407.2512]. Furthermore, we provide the first rigorous proof of the exact asymptotics Anderson predicted. We prove that in the
3-dimensional toy model of a Dirac-
δ perturbation that the exact decay exponent is given by
ζ:=δ2/π2. Here,
δ refers to the s-wave scattering phase shift. In particular, this result shows that the previously found decay exponent
γ does not provide the correct asymptotics of
§NL in general. Since the decay exponent is expressed in terms of scattering theory, these bounds depend on the existence of absolutely continuous spectrum of the underlying Schrödinger operators. We are able to deduce a different behavior in the contrary situation of Anderson localization. We prove the non-vanishing of the expectation value of the non-interacting many-body scalar product in the thermodynamic limit. Apart from the behavior of the scalar product of the non-interacting ground states, we also study the asymptotics of the difference of the ground-state energies. We show that this difference converges in the thermodynamic limit to the integral of the spectral-shift function up to the Fermi energy. Furthermore, we quantify the error for models on the half-axis and show that higher order error terms depend on the particular thermodynamic limit chosen.
The simple harmonic oscillator has a well-known normalizable, positive energy, bound state spectrum. We show that degenerate with each such positive energy eigenvalue there is a non-normalizable positive energy eigenstate whose eigenfunction is orthogonal to that of the standard energy eigenfunction. This class of states is not built on the vacuum that
a annihilates, but is instead built on the vacuum that
a†a annihilates. These non-normalizable but nonetheless stationary energy eigenstates are accompanied by yet another set of non-normalizable states, states whose wave functions however are not stationary but instead grow linearly in time. With these states not being energy eigenstates, the eigenbasis of the Hamiltonian is incomplete; with the full Hilbert space containing states that are not energy eigenstates. Thus each energy eigenvalue of the harmonic oscillator is an exceptional point at which the Hamiltonian becomes of non-diagonalizable, and thus manifestly non-Hermitian, Jordan-block form. Such non-Hermitian structures occur for Hamiltonians that have an antilinear
PT symmetry. As is characteristic of such systems, one can construct a probability conserving inner product that despite the linear in time growth is nonetheless time independent, and not only that, it leads to states with zero norm. In addition, as is again characteristic of
PT symmetry, these non-normalizable states can be made normalizable by a continuation into a so-called Stokes wedge domain in the complex plane. In this domain one has a completely consistent quantum theory, one that lives alongside the standard normalizable energy eigenspectrum sector. This thus not quite so simple harmonic oscillator provides an explicit realization of our general contention that antilinearity is more basic to quantum theory than Hermiticity.