arXiv subjects

Horia D. Cornean

Publications and source records attributed to Horia D. Cornean.

At least 19 recordsLinked to original sources

arXiv2026

The fibre operators in the Bloch-Floquet decomposition of periodic magnetic pseudo-differential operators

We study the structure of the fibre operators corresponding to periodic magnetic pseudo-differential operators having periodic magnetic potentials. We obtain explicit formulas for their distribution kernel, both when these fibres are seen as operators on the $d$-dimensional torus, and also when they are seen as infinite matrices acting on a discrete $\ell^2$ space via a discrete Fourier transform. Moreover, using these distribution kernels we prove that the fibre operators are toroidal pseudo-differential operators.

math.AP↗

arXiv2026

Spectral and dynamical results related to certain non-integer base expansions on the unit interval

We consider certain non-integer base $β$-expansions of Parry's type and we study various properties of the transfer (Perron-Frobenius) operator $\mathcal{P}:L^p([0,1])\mapsto L^p([0,1])$ with $p\geq 1$ and its associated composition (Koopman) operator, which are induced by a discrete dynamical system on the unit interval related to these $β$-expansions. We show that if $f$ is Lipschitz, then the iterated sequence $\{\mathcal{P}^N f\}_{N\geq 1}$ converges exponentially fast (in the $L^1$ norm) to an invariant state corresponding to the eigenvalue $1$ of $\mathcal{P}$. This "attracting" eigenvalue is not isolated: for $1\leq p\leq 2$ we show that the point spectrum of $\mathcal{P}$ also contains the whole open complex unit disk and we explicitly construct some corresponding eigenfunctions.

math.SP↗

arXiv2026

Magnetic Weyl Super Calculus: Schatten-class properties, commutator criterion, and complete positivity

We combine our previous results on magnetic pseudo-differential operators for Hörmander symbols dominated by tempered weights [arXiv:2511.07184] with the magnetic Weyl super calculus of Lee and Lein [arXiv:2201.11487, arXiv:2405.19964]. This allows us to extend some previous results on the semi-super and super Moyal algebra, as well as to prove boundedness, compactness, and Schatten-class properties of super operators. Moreover, we prove a Beals-type commutator criterion for super operators and we also formulate sufficient conditions on super symbols in order to give rise to completely positive and trace preserving maps. For most of the proofs we use decompositions of operators and super operators based on Parseval frames of smoothing operators.

math-ph↗

arXiv2026

Universality of the Hall conductivity for a weakly interacting magnetic fermionic gas in the Hartree-Fock approximation

We consider a two-dimensional gas of interacting fermions in presence of an external constant magnetic field: the system is extended and homogeneous, and thus assumed to be invariant under magnetic translations. Working within the Hartree-Fock approximation, we analyze the system directly in the thermodynamic limit by solving a self-consistent fixed-point equation for the one-particle density matrix. We prove that, provided that the interactions among fermions are sufficiently weak, there exists a unique one-particle density matrix that solves the self-consistency condition. By choosing the Fermi-Dirac distribution as the function in the fixed-point equation, this approach can describe both positive and zero-temperature cases. At zero temperature and when the chemical potential of the non-interacting system lies in a spectral gap of the free Landau operator, our self-consistent solution is an orthogonal projection (an "interacting" effective Fermi projection). We prove that its integrated density of states varies linearly with the external magnetic field, provided the interaction is weak enough: the slope of this variation is quantized and independent of the interaction. According to the Středa formula, this can be seen as yet another expression of the universality of the quantum Hall effect in weakly interacting systems, at least within the Hartree-Fock approximation.

math-ph↗

arXiv2026

On the magnetic perturbation theory for Chern insulators

The gauge covariant magnetic perturbation theory is tailored for one-body Schrödinger operators perturbed by long-range magnetic fields. In this work we present a self-contained exposition of the method, by outlining its technical foundations and discussing the physical heuristics behind the proofs. We apply it in order to prove the stability of spectral gaps and to study the location of the discrete spectrum. We also analyze the (lack of) continuity with respect to the magnetic field of spectral projections corresponding to finite spectral islands, which is a particularly important situation for systems modelling Chern insulators. Finally, we show how to construct approximate projections that have an explicit dependence with respect to the magnetic field parameter.

math-ph↗

arXiv2026

A fresh look at the Peierls-Onsager substitution

We formulate a general version of the Peierls-Onsager substitution for a finite family of Bloch eigenvalues under a local spectral gap hypothesis, via strongly localized tight-frames and magnetic matrices. This extends the existing results to long-range magnetic fields without any slow-variation hypothesis and without any triviality assumption for the associated Bloch sub-bundle. Moreover, our results cover a large class of periodic, elliptic pseudo-differential operators. We also prove the existence of an approximate time evolution for initial states supported inside the range of the isolated Bloch family, with a precise error control.

math-ph↗

arXiv2025

A rigorous Peierls-Onsager effective dynamics for semimetals in long-range magnetic fields

We consider periodic (pseudo)differential {elliptic operators of Schrödinger type} perturbed by weak magnetic fields not vanishing at infinity, and extend our previous analysis in \cite{CIP,CHP-2,CHP-4} to the case {of a semimetal having a finite family of Bloch eigenvalues whose range may overlap with the other Bloch bands but remains isolated at each fixed quasi-momentum.} We do not make any assumption of triviality for the associated Bloch bundle. In this setting, we formulate a general form of the Peierls-Onsager substitution {via strongly localized tight-frames and magnetic matrices. We also} prove the existence of an approximate time evolution for initial states supported inside the range of the isolated Bloch family, with a precise error control.

math-ph↗

arXiv2025

Green's functions for magnetic Dirac operators and bulk-edge correspondence

We study the resolvent kernel (Green's function) of magnetic Dirac operators on a half-plane with boundary conditions interpolating between infinite mass and zigzag cases, excluding the latter. We show that these kernels have all the required properties so that the proof of bulk-edge correspondence from the infinite mass case shown recently can be repeated ad litteram. The zigzag case exhibits qualitatively different behavior and will be addressed in future work.

math-ph↗

arXiv2025

Sharp iteration asymptotics for transfer operators induced by greedy $β$-expansions

We consider base-$β$ expansions of Parry's type, where $a_0 \geq a_1 \geq 1$ are integers and $a_0<β<a_0+1$ is the positive solution to $β^2 = a_0β+ a_1$ (the golden ratio corresponds to $a_0=a_1=1$). The map $x\mapsto βx-\lfloor βx\rfloor$ induces a discrete dynamical system on the interval $[0,1)$ and we study its associated transfer (Perron-Frobenius) operator $\mathscr{P}$. Our main result can be roughly summarized as follows: we explicitly construct two piecewise affine functions $u$ and $v$ with $\mathscr{P}u=u$ and $\mathscr{P}v=β^{-1} v$ such that for every sufficiently smooth $F$ which is supported in $[0,1]$ and satisfies $\int_0^1 F \; \mathrm{d} x=1$, we have $\mathscr{P}^kF= u +β^{-k}\big ( F(1)-F(0)\big )v +o(β^{-k})$ in $L^\infty$. This is also compared with the case of integer bases, where more refined asymptotic formulas are possible.

math.DS↗

arXiv2024

Spectral regularity with respect to dilations for a class of pseudodifferential operators

We continue the study of the perturbation problem discussed in \cite{CP3} and get rid of the 'slow variation' assumption by considering symbols of the form $a\big(x+δ\,F(x),ξ\big)$ with $a$ a real Hörmander symbol of class $S^0_{0,0}(\mathbb{R}^d\times\mathbb{R}^d)$ and $F$ a smooth function with all its derivatives globally bounded, with $|δ|\leq1$. We prove that while the Hausdorff distance between the spectra of the Weyl quantization of the above symbols in a neighbourhood of $δ=0$ is still of the order $\sqrt{|δ|}$, the distance between their spectral edges behaves like $|δ|^ν$ with $ν\in[1/2,1)$ depending on the rate of decay of the second derivatives of $F$ at infinity.

math.AP↗

arXiv2024

From orbital magnetism to bulk-edge correspondence

By extending the gauge covariant magnetic perturbation theory to operators defined on half-planes, we prove that for $2d$ random ergodic magnetic Schrödinger operators, the zero-temperature bulk-edge correspondence can be obtained from a general bulk-edge duality at positive temperature involving the bulk magnetization and the total edge current. Our main result is encapsulated in a formula, which states that the derivative of a large class of bulk partition functions with respect to the external constant magnetic field, equals the expectation of a corresponding edge distribution function of the velocity component which is parallel to the edge. Neither spectral gaps, nor mobility gaps, nor topological arguments are required. The equality between the bulk and edge indices, as stated by the conventional bulk-edge correspondence, is obtained as a corollary of our purely analytical arguments by imposing a gap condition and by taking a ``zero-temperature" limit.

math-ph↗

arXiv2024

On the self-consistent Landauer-Büttiker formalism

We provide sufficient conditions such that the time evolution of a mesoscopic tight-binding open system with a local Hartree-Fock non-linearity converges to a self-consistent non-equilibrium steady state, which is independent of the initial condition from the "small sample". We also show that the steady charge current intensities are given by Landauer-Büttiker-like formulas, and make the connection with the case of weakly self-interacting many-body systems.

math-ph↗

arXiv2024

Matrix representation of Magnetic pseudo-differential operators via tight Gabor frames

In this paper we use some ideas from \cite{FG-97, G-06} and consider the description of Hörmander type pseudo-differential operators on $\mathbb{R}^d$ ($d\geq1$), including the case of the magnetic pseudo-differential operators introduced in \cite{IMP-1, IMP-19}, with respect to a tight Gabor frame. We show that all these operators can be identified with some infinitely dimensional matrices whose elements are strongly localized near the diagonal. Using this matrix representation, one can give short and elegant proofs to classical results like the Calder{ó}n-Vaillancourt theorem and Beals' commutator criterion, and also establish local trace-class criteria.

math.AP↗

arXiv2024

Magnetic Dirac systems: Violation of bulk-edge correspondence in the zigzag limit

We consider a Dirac operator with constant magnetic field defined on a half-plane with boundary conditions that interpolate between infinite mass and zigzag. By a detailed study of the energy dispersion curves we show that the infinite mass case generically captures the profile of these curves, which undergoes a continuous pointwise deformation into the topologically different zigzag profile. Moreover, these results are applied to the bulk-edge correspondence. In particular, by means of a counterexample, we show that this correspondence does not always hold true in the zigzag case.

math-ph↗

arXiv2023

Sharp spectral stability for a class of singularly perturbed pseudo-differential operators

Let $a(x,ξ)$ be a real Hörmander symbol of the type $S_{0,0}^0(\mathbb{R}^{d}\times \mathbb{R}^d)$, let $F$ be a smooth function with all its derivatives globally bounded, and let $K_δ$ be the self-adjoint Weyl quantization of the perturbed symbols $a(x+F(δ\, x),ξ)$, where $|δ|\leq 1$. First, we prove that the Hausdorff distance between the spectra of $K_δ$ and $K_{0}$ is bounded by $\sqrt{|δ|}$, and we give examples where spectral gaps of this magnitude can open when $δ\neq 0$. Second, we show that the distance between the spectral edges of $K_δ$ and $K_0$ (and also the edges of the inner spectral gaps, as long as they remain open at $δ=0$) are of order $|δ|$, and give a precise dependence on the width of the spectral gaps.

math-ph↗

arXiv2023

Bulk-edge correspondence for unbounded Dirac-Landau operators

We consider two-dimensional unbounded magnetic Dirac operators, either defined on the whole plane, or with infinite mass boundary conditions on a half-plane. Our main results use techniques from elliptic PDEs and integral operators, while their topological consequences are presented as corollaries of some more general identities involving magnetic derivatives of local traces of fast decaying functions of the bulk and edge operators. One of these corollaries leads to the so-called Středa formula: if the bulk operator has an isolated compact spectral island, then the integrated density of states of the corresponding bulk spectral projection varies linearly with the magnetic field as long as the gaps between the spectral island and the rest of the spectrum are not closed, and the slope of this variation is given by the Chern character of the projection. The same bulk Chern character is related to the number of edge states which appear in the gaps of the bulk operator.

math-ph↗

arXiv2022

Optimal Profile Design for Acoustic Black Holes using Timoshenko beam Theory

We revisit the problem of constructing one-dimensional acoustic black holes. Instead of considering the Euler-Bernoulli beam theory, we use Timoshenko's approach instead, which is known to be more realistic at higher frequencies. Our goal is to minimize the reflection coefficient under a constraint imposed on the normalized wave number variation. We use the calculus of variations in order to derive the corresponding Euler-Lagrange equation analytically and then use numerical methods to solve this equation in order to find the optimal height profile for different frequencies. We then compare these profiles to the corresponding ones previously found using the Euler-Bernoulli beam theory and see that in the lower range of the dimensionless frequency $Ω$ (defined using the largest height of the plate), the optimal profiles almost coincide, as expected. For higher such frequencies, even for values where Euler-Bernoulli theory should still be marginally valid, the profiles predicted using Euler-Bernoulli differ substantially from the correct ones predicted by Timoshenko theory. One explanation for this phenomenon is that unlike in the constant height case, in our setting the wave numbers also depend on the ratio between the smallest and the largest heights.

physics.class-ph↗

arXiv2022

Discrete approximations to Dirac operators and norm resolvent convergence

We consider continuous Dirac operators defined on $\mathbf{R}^d$, $d\in\{1,2,3\}$, together with various discrete versions of them. Both forward-backward and symmetric finite differences are used as approximations to partial derivatives. We also allow a bounded, Hölder continuous, and self-adjoint matrix-valued potential, which in the discrete setting is evaluated on the mesh. Our main goal is to investigate whether the proposed discrete models converge in norm resolvent sense to their continuous counterparts, as the mesh size tends to zero and up to a natural embedding of the discrete space into the continuous one. In dimension one we show that forward-backward differences lead to norm resolvent convergence, while in dimension two and three they do not. The same negative result holds in all dimensions when symmetric differences are used. On the other hand, strong resolvent convergence holds in all these cases. Nevertheless, and quite remarkably, a rather simple but non-standard modification to the discrete models, involving the mass term, ensures norm resolvent convergence in general.

math-ph↗