arXiv subjects

Hendrik Vogt

Publications and source records attributed to Hendrik Vogt.

23 records · Page 2Linked to original sources

arXiv2015

$L_1$-estimates for eigenfunctions and heat kernel estimates for semigroups dominated by the free heat semigroup

We investigate selfadjoint positivity preserving $C_0$-semigroups that are dominated by the free heat semigroup on $\mathbb R^d$. Major examples are semigroups generated by Dirichlet Laplacians on open subsets or by Schrödinger operators with absorption potentials. We show explicit global Gaussian upper bounds for the kernel that correctly reflect the exponential decay of the semigroup. For eigenfunctions of the generator that correspond to eigenvalues below the essential spectrum we prove estimates of their $L_1$-norm in terms of the $L_2$-norm and the eigenvalue counting function. This estimate is applied to a comparison of the heat content with the heat trace of the semigroup.

math.AP↗

arXiv2014

A generalisation of the form method for accretive forms and operators

The form method as popularised by Lions and Kato is a successful device to associate m-sectorial operators with suitable elliptic or sectorial forms. McIntosh generalised the form method to an accretive setting, thereby allowing to associate m-accretive operators with suitable accretive forms. Classically, the form domain is required to be densely embedded into the Hilbert space. Recently, this requirement was relaxed by Arendt and ter Elst in the setting of elliptic and sectorial forms. Here we study the prospects of a generalised form method for accretive forms to generate accretive operators. In particular, we work with the same relaxed condition on the form domain as used by Arendt and ter Elst. We give a multitude of examples for many degenerate phenomena that can occur in the most general setting. We characterise when the associated operator is m-accretive and investigate the class of operators that can be generated. For the case that the associated operator is m-accretive, we study form approximation and Ouhabaz type invariance criteria.

math.FA↗

arXiv2013

Secret-key generation from wireless channels: Mind the reflections

Secret-key generation in a wireless environment exploiting the randomness and reciprocity of the channel gains is considered. A new channel model is proposed which takes into account the effect of reflections (or re-radiations) from receive antenna elements, thus capturing an physical property of practical antennas. It turns out that the reflections have a deteriorating effect on the achievable secret-key rate between the legitimate nodes at high signal-to-noise-power-ratio (SNR). The insights provide guidelines in the design and operation of communication systems using the properties of the wireless channel to prevent eavesdropping.

cs.IT↗

arXiv2013

A weak Gordon type condition for absence of eigenvalues of one-dimensional Schrödinger operators

We study one-dimensional Schrödinger operators with complex measures as potentials and present an improved criterion for absence of eigenvalues which involves a weak local periodicity condition. The criterion leads to sharp quantitative bounds on the eigenvalues. We apply our result to quasiperiodic measures as potentials.

math.SP↗

arXiv2012

Note on basic features of large time behaviour of heat kernels

Large time behaviour of heat semigroups (and more generally, of positive selfadjoint semigroups) is studied. Convergence of the semigroup to the ground state and of averaged logarithms of kernels to the ground state energy is shown in the general framework of positivity improving selfadjoint semigroups. This framework includes Laplacians on manifolds, metric graphs and discrete graphs.

math.FA↗