arXiv subjects

Oliver Fürst

Publications and source records attributed to Oliver Fürst.

6 recordsLinked to original sources

arXiv2025

Higher order spectral shift of Euclidean Callias operators

We consider Dirac-Schrödinger operators over odd-dimensional Euclidean space. The conditions for the potential are based on those of C. Callias in his famous paper on the corresponding index problem. However, we treat the case where the potential can take values in unbounded operators of a separable Hilbert space, and crucially, we also do not assume that the potential needs to be invertible outside a compact region. Hence, the Dirac-Schrödinger operator is not necessarily Fredholm. In the setup we discuss, it however still admits a related trace formula in terms of the underlying potential. In this paper we express the trace formula for these Callias-type operators in terms of higher order spectral shift functions, leading to a functional equation which generalizes a known functional equation found first by A. Pushnitski. To the knowledge of the author, this paper presents the first multi-dimensional non-Fredholm extension of the Callias index theorem involving higher order spectral shift functions. More precisely, we also show that under a Lebesgue point condition on the higher order spectral shift function associated to the potential, the Callias-type operator admits a regularized index, even in non-Fredholm settings. This corresponds to a known Witten index result in the one-dimensional case shown by A. Carey et al. The regularized index that we introduce is a minor extension of the classical Witten index, and we present an index formula, which generalizes the classical Callias index theorem. As an example, we treat the case of $(d+1)$-massless Dirac-Schrödinger operators, for which we calculate the associated higher order spectral shift functions.

math.SP↗

arXiv2024

Trace and Index of Dirac-Schrödinger Operators on Open Space with Operator Potentials

We develop a principal trace and generalized index formula for a Dirac-Schrödinger operator $D$ on open space of odd dimension $d\geq 3$ with a potential given by a family of self-adjoint unbounded operators acting on a infinite dimensional Hilbert space $H$. The presented results generalize formulas surrounding the Callias index theorem to to the case of unbounded operator potentials, for which the operator $D$ is not necessarily Fredholm. This is the principal novelty of this paper. As application, we include examples where the trace formula is used to calculate the Witten index of non-Fredholm massless $(d+1)$-Dirac-Schrödinger operators acting in $L^2\left(\mathbb{R}^{d+1},H\right)$.

math.FA↗

arXiv2024

Commutative C* algebras and Gelfand theory through phase space methods

We show how the Gelfand spectrum of certain commutative operator algebras can be studied based on the theorem of Stone and von Neumann. The method presented is a natural addition to the tools of quantum spectral synthesis, which were recently used to characterize certain commutative Toeplitz algebras on the Fock space. Our method applies to this setting and also to more general abelian phase spaces. Besides characterizing Gelfand spectra of such commutative operator algebras, we also prove an extension of this result to the operator-valued case.

math.FA↗

arXiv2024

The Witten Index of massless $(d+1)$-Dirac-Schrödinger Operators

We calculate the Witten index of a class of (non-Fredholm) Dirac-Schrödinger operators over $\mathbb{R}^{d+1}$ for $d\geq 3$ odd, and thus generalize known results for the case $d=1$. For a concrete example of the potential, we give a more explicit index formula, showing that the Witten index assumes any real number on this class of operators.

math.FA↗

arXiv2022

Trace Class Properties of Resolvents of Callias Operators

We present conditions for a family $\left(A\left(x\right)\right)_{x\in\mathbb{R}^{d}}$ of self-adjoint operators in $H^{r}=\mathbb{C}^{r}\otimes H$ for a separable complex Hilbert space $H$, such that the Callias operator $D=ic\nabla+A\left(X\right)$ satisfies that $\left(D^{\ast}D+1\right)^{-N}-\left(DD^{\ast}+1\right)^{-N}$ is trace class in $L^2\left(\mathbb{R}^{d},H^{r}\right)$. Here, $c\nabla$ is the Dirac operator associated to a Clifford multiplication $c$ of rank $r$ on $\mathbb{R}^{d}$, and $A\left(X\right)$ is fibre-wise multiplication with $A\left(x\right)$ in $L^2\left(\mathbb{R}^{d},H^{r}\right)$.

math.FA↗

arXiv2021

Hölder estimates for magnetic Schrödinger semigroups in $\mathbb{R}^{d}$ from mirror coupling

We use the mirror coupling of Brownian motion to show that under a $β\in (0,1)$-dependent Kato type assumption (which is satisfied under a suitable $L^q$-assumption on the electro-magnetic potential, where $q$ depends on $β$ and the dimension $d$) on the possibly nonsmooth electro-magnetic potential, the corresponding magnetic Schrödinger semigroup in $\mathbb{R}$ has a global $L^{p}$-to-$C^{0,β}$ Hölder smoothing property for all $p\in [1,\infty]$, in particular all eigenfunctions are uniformly $β$-Hölder continuous. This result shows that the eigenfunctions of the Hamilton operator of a molecule in a magnetic field are uniformly $β$-Hölder continuous under weak $L^q$-assumptions on the magnetic potential.

math-ph↗