arXiv subjects

Hermann Schulz-Baldes

Publications and source records attributed to Hermann Schulz-Baldes.

At least 19 recordsLinked to original sources

arXiv2026

Analysis of inverse stochastic resonance: Effects of neural excitability and timescale separation

We analyze inverse stochastic resonance (ISR) in a bistable FitzHugh--Nagumo neuron driven by additive noise in the voltage variable, focusing on how neural excitability and timescale separation regulate the noise-induced modulation of spiking activity. A codimension-two bifurcation analysis identifies a narrow bistable region in which a stable fixed point and a stable limit cycle coexist, separated by an unstable periodic orbit. Finite-time Monte Carlo simulations show that the occupation of the limit-cycle basin may appear to depend on the initial basin when rare transitions are not fully resolved. We prove that this dependence is not asymptotic: the stochastic system admits a unique invariant probability measure, so long-time firing statistics are independent of the initial basin of attraction. The parameter dependence of ISR is characterized by quasi-potential barriers computed with a geometric minimum action method for the degenerate noise. The difference between the limit-cycle and fixed-point quasi-potentials partitions the bistable wedge into two escape-dominated regimes. A reduced metastable two-state Markov approximation yields a weak-noise formula for the limit-cycle basin occupation probability, a sign criterion for genuine ISR, and a semiquantitative prediction of the ISR-minimizing noise amplitude. In the fitted regime with a negative effective exponent, a genuine ISR minimum occurs only when the limit-cycle quasi-potential exceeds that of the fixed point. These results provide an escape-balance mechanism linking intrinsic neuronal parameters to asymptotic noise-induced spike suppression.

nlin.AO↗

arXiv2025

Limit absorption and Green function estimates for matrix-valued periodic operators

The boundary value of the resolvent of a generic periodic tight-binding Hamiltonian with matrix symbols is shown to satisfy a limit absorption principle which is continuous in energy in dimensions $d=3$, and in dimension $d=2$ away from critical points of the energy bands corresponding to van Hove singularities. The analysis away from critical points of the energy bands is based on the coarea formula, while at the critical points it involves a parametric Morse lemma and stationary phase arguments. In particular, at Weyl points a new type of oscillatory integrals is dealt with.

math-ph↗

arXiv2025

Detecting local topology with the spectral localizer

The spectral localizer is a predictive framework for the computation of topological invariants of natural and artificial materials. Here, three crucial improvements on the criterion for the validity of the framework are reported: first, merely a properly defined local spectral gap of the Hamiltonian is required, second, only relative bounds on the Hamiltonian and its noncommutative derivative are relevant, and, third, the numerical constant in a tapering estimate is improved. These developments further stress the local nature of the spectral localizer framework, enabling more precise predictions in heterostructures, aperiodic, and disordered systems. Moreover, these results strengthen the bounds on the spectral localizer's spectral flow when crossing topological phase boundaries.

math-ph↗

arXiv2025

Local topology for periodic Hamiltonians and fuzzy tori

A variety of local index formulas is constructed for quantum Hamiltonians with periodic boundary conditions. All dimensions of physical space as well as many symmetry constraints are covered, notably one-dimensional systems in Class DIII as well as two- and three-dimensional systems in Class AII. The constructions are based on several periodic variations of the spectral localizer and are rooted in the existence of underlying fuzzy tori. For these latter, a general invariant theory is developed.

math-ph↗

arXiv2025

Scaling of the Lyapunov exponent at a balanced hyperbolic critical point

In both the random hopping model and at topological phase transitions in one-dimensional chiral systems, the Lyapunov exponent vanishes at zero energy, but is here shown to have an inverse logarithmic increase with a coefficient that is computed explicitly. This is the counterpart of the Dyson spike in the density of states. The argument also transposes to the free energy density of the random field Ising model, and more generally to many so-called balanced hyperbolic critical points. It is based on the fact that the Furstenberg measure in rescaled logarithmic Dyson-Schmidt variables can be well-approximated by an absolutely continuous measure with trapezoidal density.

math-ph↗

arXiv2025

Local perturbations of block Toeplitz matrices

This work is about the asymptotic spectral theory of tridiagonal Toeplitz matrices with matrix entries, with periodicity broken on a finite number of entries. Varying the ranks of these perturbations allow to interpolate between open boundary and circulant Toeplitz matrices. While the continuous part of the limit spectrum only depends on these ranks and no other aspect of the perturbation, the outliers of the spectrum depend continuously on the local perturbation. The proof is essentially based on a new generalized Widom formula for the characteristic polynomial. All this holds for Lebesgue almost all perturbed Toeplitz matrices, a fact that constitutes another important extension of Widom's work. The mathematical results are illustrated by numerics.

math-ph↗

arXiv2024

Transfer matrix analysis of non-hermitian Hamiltonians: asymptotic spectra and topological eigenvalues

Transfer matrix techniques are used to provide a new proof of Widom's results on the asymptotic spectral theory of finite block Toeplitz matrices. Furthermore, a rigorous treatment of the skin effect, spectral outliers, the generalized Brillouin zone and the bulk-boundary correspondence in such systems is given. This covers chiral Hamiltonians with topological eigenvalues close to zero, but no line-gap.

math-ph↗

arXiv2024

The generators of the K-groups of the sphere

This note presents an elementary iterative construction of the generators for the complex $K$-groups $K_i(C(\SM^d))$ of the $d$-dimensional spheres. These generators are explicitly given as the restrictions of Dirac or Weyl Hamiltonians to the unit sphere. Connections to solid state physics are briefly elaborated on.

math-ph↗

arXiv2024

Topological indices in condensed matter

This contribution describes the mathematical theory of topological indices in solid state systems composed of non-interacting Fermions. In particular, this covers the spectral localizer and the bulk-boundary correspondence.

math-ph↗

arXiv2024

Local markers for crystalline topology

Over the last few years, crystalline topology has been used in photonic crystals to realize edge- and corner-localized states that enhance light-matter interactions for potential device applications. However, the band-theoretic approaches currently used to classify bulk topological crystalline phases cannot predict the existence, localization, or spectral isolation of any resulting boundary-localized modes. While interfaces between materials in different crystalline phases must have topological states at some energy, these states need not appear within the band gap, and thus may not be useful for applications. Here, we derive a class of local markers for identifying material topology due to crystalline symmetries, as well as a corresponding measure of topological protection. As our real-space-based approach is inherently local, it immediately reveals the existence and robustness of topological boundary-localized states, yielding a predictive framework for designing topological crystalline heterostructures. Beyond enabling the optimization of device geometries, we anticipate that our framework will also provide a route forward to deriving local markers for other classes of topology that are reliant upon spatial symmetries.

physics.optics↗

arXiv2023

Footprint of a topological phase transition on the density of states

For a generalized Su-Schrieffer-Heeger model the energy zero is always critical and hyperbolic in the sense that all reduced transfer matrices commute and have their spectrum off the unit circle. Disorder driven topological phase transitions in this model are characterized by a vanishing Lyapunov exponent at the critical energy. It is shown that the integrated density of states away from a transition has a pseudogap with an explicitly computable Hölder exponent, while it has a characteristic divergence (Dyson spike) at the transition points. The proof is based on renewal theory for the Prüfer phase dynamics and the optional stopping theorem for martingales of suitably constructed comparison processes.

math-ph↗

arXiv2023

Spectral localizer for line-gapped non-hermitian systems

Short-ranged and line-gapped non-hermitian Hamiltonians have strong topological invariants given by an index of an associated Fredholm operator. It is shown how these invariants can be accessed via the signature of a suitable spectral localizer. This numerical technique is implemented in an example with relevance to the design of topological photonic systems, such as topological lasers.

math-ph↗

arXiv2022

Analysis of Sparse Recovery Algorithms via the Replica Method

This manuscript goes through the fundamental connections between statistical mechanics and estimation theory by focusing on the particular problem of compressive sensing. We first show that the asymptotic analysis of a sparse recovery algorithm is mathematically equivalent to the problem of calculating the free energy of a spin glass in the thermodynamic limit. We then use the replica method from statistical mechanics to evaluate the performance in the asymptotic regime. The asymptotic results have several applications in communications and signal processing. We briefly go through two instances of these applications: Characterization of joint sparse recovery algorithms used in distributed compressive sensing, and tuning of receivers employed for detection of spatially modulated signals.

cs.IT↗

arXiv2022

Spectral localization for semimetals and Callias operators

A semiclassical argument is used to show that the low-lying spectrum of a selfadjoint operator, the so-called spectral localizer, determines the number of Dirac or Weyl points of an ideal semimetal. Apart from the IMS localization procedure, an explicit computation for the local toy models given by a Dirac or Weyl point is the key element of proof. The argument has numerous similarities to Witten's reasoning leading to the strong Morse inequalities. The same techniques allow to prove a spectral localization for the Callias operator in terms of a multi-parameter spectral flow of selfadjoint Fredholm operators.

math-ph↗

arXiv2022

Partially hyperbolic random dynamics on Grassmannians

A sequence of invertible matrices given by a small random perturbation around a fixed diagonal partially hyperbolic matrix induces a random dynamics on the Grassmann manifolds. Under suitable weak conditions it is known to have a unique invariant (Furstenberg) measure. The main result gives concentration bounds on this measure showing that with high probability the random dynamics stays in the vicinity of stable fixed points of the unperturbed matrix, in a regime where the strength of the random perturbation dominates the local hyperbolicity of the diagonal matrix. As an application, bounds on sums of Lyapunov exponents are obtained.

math-ph↗

arXiv2022

Harmonic analysis in operator algebras and its applications to index theory and topological solid state systems

This monograph develops the theory of Besov spaces for abelian group actions on semifinite von Neumann algebras and then proves Peller criteria for traceclass properties of associated Hankel operators. This allows to extend known index theorems to symbols lying in Sobolev or Besov spaces. The duality theory for pairings over the smooth Toeplitz extension is developed in detail. Numerous applications to solid state systems are presented. In particular, a bulk-boundary correspondence is obtained for insulators with edges of irrational angles and for chiral semimetals having a pseudogaps. The latter implies the existence of flat bands of edge for tight-binding graphene models and shows how the density of surface states is expressed in terms of weak Chern numbers of the system without boundaries.

math-ph↗

arXiv2022

Levinson theorem for discrete Schrödinger operators on the line with matrix potentials having a first moment

This paper proves new results on spectral and scattering theory for matrix-valued Schrödinger operators on the discrete line with non-compactly supported perturbations whose first moments are assumed to exist. In particular, a Levinson theorem is proved, in which a relation between scattering data and spectral properties (bound and half bound states) of the corresponding Hamiltonians is derived. The proof is based on stationary scattering theory with prominent use of Jost solutions at complex energies that are controlled by Volterra-type integral equations.

math-ph↗

arXiv2022

Reduced transfer operators for singular difference equations

For tridiagonal block Jacobi operators, the standard transfer operator techniques only work if the off-diagonal entries are invertible. Under suitable assumptions on the range and kernel of these off-diagonal operators which assure a homogeneous minimal coupling between the blocks, it is shown how to construct reduced transfer operators that have the usual Krein space unitarity property and also a crucial monotonicity in the energy variable. This allows to extend the results of oscillation theory to such systems.

math-ph↗