Peter Otte

The exponent in the orthogonality catastrophe for Fermi gases

We quantify the asymptotic vanishing of the ground-state overlap of two non-interacting Fermi gases in d-dimensional Euclidean space in the thermodynamic limit. Given two one-particle Schrodinger operators in finite-volume which differ by a compactly supported bounded potential, we prove a power-law upper bound on the ground-state overlap of the corresponding non-interacting N-Fermion systems. We interpret the decay exponent γ in terms of scattering theory and find γ=π−2∥arcsin|TE/2|∥2HS, where TE is the transition matrix at the Fermi energy E. This exponent reduces to the one predicted by Anderson [Phys. Rev. 164, 352–359 (1967)] for the exact asymptotics in the special case of a repulsive point-like perturbation.

AN INTEGRAL FORMULA FOR SECTION DETERMINANTS OF SEMI-GROUPS OF LINEAR OPERATORS

We derive an integral formula that expresses the section determinants of semi-groups of linear operators through the solution to a linear integral equation. The solution theory of this integral equation is developed and for a special case a concrete solvability criterion is presented.

Anderson?s orthogonality catastrophe in one dimension induced by a magnetic field

According to Andersons orthogonality catastrophe, the overlap of the

Boundedness properties of fermionic operators

N

Enclosure results for second-order relative spectra by elementary means

-particle ground states of a free Fermi gas with and without an (electric) potential decays in the thermodynamic limit. For the finite one-dimensional system various boundary conditions are employed. Unlike the usual setup the perturbation is introduced by a magnetic (vector) potential. Although such a magnetic field can be gauged away in one spatial dimension there is a significant and interesting effect on the overlap caused by the phases. We study the leading asymptotics of the overlap of the two ground states and the two-term asymptotics of the difference of the ground-state energies. In the case of periodic boundary conditions our main result on the overlap is based upon a well-known asymptotic expansion by Fisher and Hartwig on Toeplitz determinants with a discontinuous symbol. In the case of Dirichlet boundary conditions no such result is known to us and we only provide an upper bound on the overlap, presumably of the right asymptotic order.

Deficiency indices of operator polynomials in creation and annihilation operators

The fermionic second quantization operator dΓ(B) is shown to be bounded by a power Ns/2 of the number operator N given that the operator B belongs to the rth von Neumann–Schatten class, s=2(r−1)/r. Conversely, number operator estimates for dΓ(B) imply von Neumann–Schatten conditions on B. Quadratic creation and annihilation operators are treated as well.

Section determinants of unitary operators and the operator-valued Riccati differential equation

Motivated by the general approach due to Shargorodsky we derive enclosure results for the second-order relative spectrum of bounded selfadjoint operators by studying quadratic operator pencils. The quality of the results is discussed by means of a simple example.

Anderson's Orthogonality Catastrophe for One-Dimensional Systems

We present a new method to compute deficiency indices of operators that are homogeneous polynomials in one pair of creation and annihilation operators. To this end we prove a classification theorem for special cubic forms by means of SU(1,1) transformations and derive new non-self-adjointness criteria for Jacobi-like matrices. The method presented illuminates and systemizes former results of Rabsztyn.

An adiabatic theorem for section determinants of spectral projections

A special initial value problem for the operator-valued Riccati differential equation (RDE) is used for studying section determinants of unitary operators. A solvability criterion for this initial value problem is derived by providing an a-priori estimate of the solution. Alternatively, the same estimate is derived by a method that is based entirely upon linear algebra.