Hagen Neidhardt

Weyl function and spectral properties of self-adjoint extensions

We characterize the spectra of self-adjoint extensions of a symmetric operator with equal deficiency indices in terms of boundary values of their Weyl functions. A complete description is obtained for the point and absolutely continuous spectrum while for the singular continuous spectrum additional assumptions are needed. The results are illustrated by examples.

Scattering matrices and Weyl functions

For a scattering system {A Θ , A 0 } consisting of self-adjoint extensions A Θ and A 0 of a symmetric operator A with finite deficiency indices, the scattering matrix {S Θ (λ)} and a spectral shift function ξ Θ are calculated in terms of the Weyl function associated with a boundary triplet for A*, and a simple proof of the Krein-Birman formula is given. The results are applied to singular Sturm-Liouville operators with scalar and matrix potentials, to Dirac operators and to Schrodinger operators with point interactions.

On Error Estimates for the Trotter–Kato Product Formula

We study the error bound in the operator-norm topology for the Trotter exponential product formula as well as for its generalization à la Kato. Within the framework of an abstract setting, we give a simple proof of error estimates which improve some recent results in this direction.

Density and current of a dissipative Schrödinger operator

We regard a current flow through an open one-dimensional quantum system which is determined by a dissipative Schrodinger operator. The imaginary part of the corresponding form originates from Robin boundary conditions with certain complex valued coefficients imposed on Schrodinger’s equation. This dissipative Schrodinger operator can be regarded as a pseudo-Hamiltonian of the corresponding open quantum system. The dilation of the dissipative operator provides a (self-adjoint) quasi-Hamiltonian of the system, more precisely, the Hamiltonian of the minimal closed system which contains the open one is used to define physical quantities such as density and current for the open quantum system. The carrier density turns out to be an expression in the generalized eigenstates of the dilation while the current density is related to the characteristic function of the dissipative operator. Finally a rigorous setup of a dissipative Schrodinger–Poisson system is outlined.

Fractional powers of self-adjoint operators and Trotter-Kato product formula

We study error bounds in the operator norm topology for the Trotter-Kato product formula. We prove that they depend on fractional power conditions (domains and relative boundedness) for operators involved in this formula.

The Trotter-Kato product formula for Gibbs semigroups

The trace-norm convergence of the Trotter-Lie product formula has recently been proved for particular classes of Gibbs semigroups. In the present paper we prove it for the whole generality including generalization of the product formula proposed by Kato.

A QUANTUM TRANSMITTING SCHRÖDINGER–POISSON SYSTEM

We study a stationary Schrodinger–Poisson system on a bounded interval of the real axis. The Schrodinger operator is defined on the bounded domain with transparent boundary conditions. This allows us to model a non-zero current through the boundary of the interval. We prove that the system always admits a solution and give explicit a priori estimates for the solutions.

Macroscopic current induced boundary conditions for Schrödinger-type operators

We describe an embedding of a quantum mechanically described structure into a macroscopic flow. The open quantum system is partly driven by an adjacent macroscopic flow acting on the boundary of the bounded spatial domain designated to quantum mechanics. This leads to an essentially non-selfadjoint Schrödinger-type operator, the spectral properties of which will be investigated.

The effect of time-dependent coupling on non-equilibrium steady states

Abstract.Consider (for simplicity) two one-dimensional semi–infinite leads coupled to a quantum well via time dependent point interactions. In the remote past the system is decoupled, and each of its components is at thermal equilibrium. In the remote future the system is fully coupled. We define and compute the non equilibrium steady state (NESS) generated by this evolution. We show that when restricted to the subspace of absolute continuity of the fully coupled system, the state does not depend at all on the switching. Moreover, we show that the stationary charge current has the same invariant property, and derive the Landau–Lifschitz and Landauer–Büttiker formulas.

Trotter–Kato product formula and fractional powers of self-adjoint generators

Let A and B be non-negative self-adjoint operators in a Hilbert space such that their densely defined form sum H=A+·B obeys dom(Hα)⊆dom(Aα)∩dom(Bα) for some α∈(1/2,1). It is proved that if, in addition, A and B satisfy dom(A1/2)⊆dom(B1/2), then the symmetric and non-symmetric Trotter–Kato product formula converges in the operator norm: ||(e−tB/2ne−tA/ne−tB/2n)n−e−tH||=O(n−(2α−1))||(e−tA/ne−tB/n)n−e−tH||=O(n−(2α−1)) uniformly in t∈[0,T], 0<T<∞, as n→∞, both with the same optimal error bound. The same is valid if one replaces the exponential function in the product by functions of the Kato class, that is, by real-valued Borel measurable functions f(·) defined on the non-negative real axis obeying 0⩽f(x)⩽1, f(0)=1 and f′(+0)=−1, with some additional smoothness property at zero. The present result improves previous ones relaxing the smallness of Bα with respect to Aα to the milder assumption dom(A1/2)⊆dom(B1/2) and extending essentially the admissible class of Kato functions.