arXiv subjects

Baruch Schneider

Publications and source records attributed to Baruch Schneider.

At least 19 recordsLinked to original sources

arXiv2026

Discrete q-Hermitian Clifford analysis

We develop a discrete $q$-Hermitian Clifford calculus based on coordinatewise Jackson differences on multiplicative $q$-lattices. The resulting Hermitian Jackson--Dirac operators are nilpotent, factor a $q$-Laplacian, and have occupancy-dependent Euler anticommutators. These give explicit Fischer projectors and homotopies, but the local calculus is not controlled by total degree. We prove a two-face Cauchy--Kovalevskaya theorem for polynomials and extend it to a $q$-analytic Jackson--Fischer class. A divided-power conjugation recovers scalar Euler relations and transfers the classical joint Fischer decomposition and dimension formulas. The Jackson--Fischer completion is a vector-valued $q$-Fock space whose polarized nullspaces have projected reproducing kernels; the joint kernel is obtained by alternating projections. We also determine the linear symmetry group of the multiplicative lattice.

math.CV↗

arXiv2026

Optimal location of small favourable regions for Robin eigenvalues with indefinite weights

We consider the positive principal eigenvalue of an elliptic problem with a Robin boundary condition and bang--bang indefinite weight $κ\mathbf 1_D-\mathbf 1_{Ω\setminus D}$, and ask where a favourable region $D$ of prescribed small volume $|D|=δ$ should be located. Put $\varepsilon=δ^{1/N}$ and $τ_δ=α_δ\varepsilon$. We prove that there is a finite threshold $τ_*=τ_*(N,κ)$, independent of the ambient domain, such that $δ^{2/N}Λ_δ(α_δ)\toΛ_{\mathbb H}(τ)$ whenever $τ_δ\toτ<\infty$. If $τ<τ_*$, optimal small regions concentrate at the boundary. If $τ>τ_*$, their concentration centres move to distances much larger than $\varepsilon$ from the boundary, and after recentring and rescaling the favourable sets converge in measure to the whole-space optimal ball. At $τ=τ_*$, the half-space problem admits a compact boundary optimiser as well as minimising sequences escaping to infinity. In the finer regime $τ_δ=τ_*+σ\varepsilon+o(\varepsilon)$, we determine the first-order competition between the boundary and interior configurations. Tangential symmetry of compact threshold optimisers reduces the geometric correction to mean curvature. In particular, every fixed finite Robin coefficient is asymptotically in the boundary regime. Numerical experiments illustrate the boundary--interior transition, the three cases in the first-order selection law, and the curvature-dependent boundary location; for $N=2$ and $κ=1$ they suggest a transition near $τ_*\approx3.2$.

math.AP↗

arXiv2026

Intrinsic {Right} $q$-Radial Vector Derivatives and Localized Fischer Decompositions on Radial Algebras

We define intrinsic right $q$-radial vector derivatives on radial algebras of abstract vector variables. For a finite parameter set $Y$, a scalar Jackson calculus on $x^2$ and the mixed anticommutators $\{x,y_i\}$ is combined with the exterior decomposition relative to $x$. Relabelling covariance and compatibility under inclusions of parameter sets give a direct-limit operator on arbitrary radial algebras. {Under the dimension specialization $Q=q^M$, its classical limit agrees in every $M$-dimensional Clifford realization with the standard right Clifford derivative; for an exterior blade one obtains $[x\wedge y_I]\partial_x=(-1)^{|I|}(M-|I|)y_I$.} Two Fischer-type constructions are considered. The anticommutator with exterior creation is triangular and admits an explicit Green inverse after localization at its diagonal factors. {For right $q$-monogenic elements, the derivative is paired with right multiplication $R_xF=Fx$. The homogeneous operators $\partial^{Y,\mathrm R}_{x,q}R_x$ have generically nonzero determinants, which yields determinant-localized Fischer decompositions and explicit projectors.} The one- and two-vector determinants are computed explicitly. {After $Q=q^M$, the two-vector determinant is nonzero for every $0<q<1$ and every positive integer $M$. For arbitrary finite support, a support filtration factors the determinant into exact-support terms; in degree zero, support rank $p$ contributes the factor $[m-p]_q+p$.}

math.CV↗

arXiv2026

Spectral properties of the magnetic Robin Laplacian on curvilinear half-plane

We analyse magnetic Robin Laplacian in curvilinear half-plane with a smooth boundary. It is well known that the spectrum of the Robin Laplacian is unstable with respect to boundary deformations. This means that if the boundary is a straight line then the spectrum of the Robin Laplacian is purely essential. From the other hand, the perturbation of the boundary produces eigenvalues below the essential spectrum. In this paper, the Robin-Laplace operator with a compactly supported magnetic field is considered. We prove that the spectrum of the magnetic Robin Laplacian is stable under small and local deformations of the boundary.

math.SP↗

arXiv2026

Revisiting Fischer decompositions by inframonogenic functions

We explore Fischer decompositions from the perspective of inframonogenic functions in Clifford analysis, a non-commutative version of harmonic functions in Euclidean spaces. We study decomposition formulas for polynomial spaces involving inframonogenic and harmonic components, providing fresh insights into their algebraic structures and further important applications.

math.CV↗

arXiv2026

A Chain- and Diagram-Level Semantics for Morphological Calculus Refinement, monodromy, and bivector orbit decompositions

Morphological calculus represents decompositions of geometric objects by polynomial-like expressions in a symbol for the real line. Sommen's examples reveal two basic difficulties: a single space may admit several such expressions, and scalar addition and multiplication suppress the incidence and attachment maps needed to reconstruct the space. We formulate a finite semantics in terms of script chain complexes. The cell-count polynomial satisfies \[ \mathcal M_S(t)=\mathcal P_{S,\Bbbk}(t)+(1+t)\mathcal B_{S,\Bbbk}(t), \] where $\mathcal P$ is the Poincaré polynomial and $\mathcal B$ records boundary ranks. Over $\mathbb Z$, Smith labels distinguish unit pairs---which model refinement overhead in the explicit subdivisions treated here---from non-unit pairs carrying torsion. We separate Cartesian products from bundles, derive the exact monodromy defect for mapping tori, and replace scalar gluing by a finite bar construction for diagrams of script complexes. Mapping cones, joins, and double mapping cylinders appear as reduced models of this diagrammatic semantics. We apply this construction to the bivector discrepancies left open in Sommen's calculations, using finite CW models after unit-sphere normalization. In dimension four, Hodge decomposition identifies the unit bivector sphere with $S^2*S^2$; the angular excess in Sommen's calculation is the contractible relative complex produced by inserting the rank-two midpoint. In dimension five, the $SO(5)$-action on $S(Λ^2\mathbb R^5)=S^9$ has principal orbit $SO(5)/T^2$ and singular orbits $\widetilde G_2(\mathbb R^5)$ and $\mathbb{CP}^3$. For standard Bruhat cell structures, the reduced double-mapping-cylinder inventory differs from the sphere homology by seven unit-labelled pairs.

math.AG↗

arXiv2026

Stable $q$-Hermitian coordinate calculus: Capelli--PBW transport, mixed polarization algebra, and localization obstructions

We extend a finite-support $q$-Hermitian radial calculus to Clifford-valued coordinate polynomials. For $N$ bosonic Hermitian vector labels and finite fermionic support, the stable range $m\ge 2N$ admits a canonical Gram--harmonic normal form by Howe separation. A label--Capelli inverse normalizes the Clifford contractions, while a finite Wick--Chevalley conjugation lifts the divided-power scalar gauge to the full Clifford PBW module. Coupling the normalized contractions to contractible fermionic seed complexes gives two conjugate polynomial-preserving $q$-Hermitian coordinate families. They are square-zero, anticommute within each polarization, restrict exactly to the radial PBW calculus, have scalar boundary trace $[2m-2n]_q$, and recover the classical Hermitian super Dirac pair as $q\to1$. For the mixed polarizations we introduce a relative triangular transport. The Grassmann relations for the normalized Capelli contractions hold on the full coordinate module and yield a labelwise factorization. Hence all higher-filtration mixed terms are finite subset products of explicit local defects, and the mixed polarization algebra closes for arbitrary finite label sets and finite fermionic support. The classical complex structure is common to both polarizations, whereas the two full transported Clifford--Weyl products are distinct for $0<q<1$. We also show that the fermionic Cartan denominator is forced in the natural polynomial first-order Berezin--Weyl class and determine the minimal factorwise Ore localization needed for the conjugate all-charge homotopies. The $q$-dependent Cartan orbit is nonresonant for $m\ge\max\{2N,n\}$. Finally, fixed finite-order differential and finite nonzero-shift realizations on the undeformed coordinate algebra are ruled out.

math.CV↗

arXiv2026

Exceptional supersphere integration and logarithmic Pizzetti kernels

We study orthosymplectically invariant supersphere integration at the exceptional superdimensions $M=-2u$, where the harmonic Fischer structure becomes nonsemisimple and the Pizzetti pairing degenerates. For the meromorphically continued homogeneous inverse kernels we obtain the generating function \[ \mathscr G_μ(ρ;x,y) =\frac{Γ(μ/2)}{2π^{μ/2}} \bigl(1+ρ\{x,y\}+ρ^2x^2y^2\bigr)^{-μ/2}. \] At $μ=-2u$, its Laurent expansion has a polynomial residue and a logarithmic finite part. We prove that these coefficients recover the complete degreewise duality structure on a fixed superspace with nonzero bosonic dimension. In degrees $k\le u$, the residue inverts a canonical renormalized pairing on $\mathcal P_k$. In the collision range $u<k\le2u$, the ordinary pairing has radical $(x^2)^{k-u}\mathcal P_{2u-k}$; the finite part reproduces the quotient, while the residue reproduces the radical after transport from the reflected degree. For $k\ge2u+1$, the finite part is the ordinary inverse kernel. We also establish the nondegenerate head--socle pairing on the generalized harmonic modules. As an application, we derive covariant right--left radial $q$-monogenic zonal symbols and identify precise degree-one obstructions to transferring scalar Pizzetti reproduction through a one-sided $q$-Fischer projection.

math.CV↗

arXiv2026

A connection between quaternionic slice regular and Fueter hyperholormorphic functions

This paper presents some results for the slice regular functions and hyperholomorphic functions theories, as consequences of new relationships between the global operator $G$, Fueter operator $\mathcal D$ and Laplacian operator $Δ_{\mathbb{R}^{4}}$. Although our results are presented in the quaternionic setting, a fine interpretation in terms of vector calculus of the slice regular function theory yields an interesting relationship between this theory and harmonic analysis. Moreover, we introduce and study a submodule of the hyperholomorphic functions associated to the unit ball $\mathbb B^4(0,1)$ given in terms of new series expansions of the functions.

math.CV↗

arXiv2026

Global small-hole minimization of the first Dirichlet eigenvalue in a square with two hard obstacles

We study the global minimization of the first Dirichlet eigenvalue of a square containing two equal non-overlapping circular obstacles as their common radius $r$ tends to zero. A capacitary localization theorem first shows that every obstacle in a configuration whose eigenvalue excess above $π^2/2$ is $O(r^4)$ must lie in a fixed $O(r)$ corner layer; this excludes interior, open-side, and intermediate boundary scales. We then analyze the resulting corner cells. The correct leading functional is the full $u$-capacity, containing both the exterior harmonic corrector energy and the polynomial energy inside the reflected obstacles. Its monotonicity forces asymptotic true-corner tangency, while a direct quantitative capacity estimate excludes two-hole clustering at one corner. For holes near distinct corners, an exact polarization argument shows that a diagonally opposite placement is strictly improved by reflecting one obstacle to an adjacent corner. Hence every unrestricted global minimizer is asymptotic, up to the symmetries of the square and interchange of the holes, to a pair of disks tangent at adjacent corners, in the sense that the center errors are $o(r)$. We also obtain the leading $r^4$ expansion of the global minimum. A reproducible finite element study provides a qualitative validation of the representative branch ordering.

math.AP↗

arXiv2026

Right q-vector calculus at integral superdimension: localized decompositions and resonance

We specialize the intrinsic right $q$-vector derivative on radial algebras to integral superdimension. The formal dimension is encoded by an independent coefficient $Q$, and formal radial superspace of superdimension $M=m-2n$ is obtained by the coefficient specialization $Q\mapsto q^M$. This gives a rigorous universal calculus and, on finite blocks containing at most $m$ abstract vectors, a faithful coordinate realization on $\mathbb{R}^{m|2n}$. The localized exterior result is formulated as a Green decomposition by complementary projector images. Whenever the specialized finite determinant is nonzero, full left multiplication yields a determinant-localized right-monogenic Fischer decomposition. Beyond this base-change theory, we determine the exceptional one-vector calculus completely: for $M=-2\ell$ there is one additional singular monomial and one missing image monomial, whereas all other integral superdimensions give a surjective derivative with constants as its kernel. We then prove that the degree-zero Fischer operator is exactly diagonal on exterior blades, obtain its determinant explicitly, classify all support-resonance values in $0<q<1$, and give an exact kernel-rank formula as a sum of support multiplicities. On a block with $N$ auxiliary vectors, an even support rank $p$ has pure multiplicity $\binom Np$ on the support truncation. At an odd-support root, every lower odd factor is nonzero and any simultaneous lower resonance is unique, even, and characterized by one strictly monotone scalar equation. These results distinguish persistent nonpositive-even-superdimension defects from isolated support-dependent $q$-resonances. An appendix records that constant scalar projection of two independent orthogonal right $q$-vector derivatives does not descend to the Hermitian quotient.

math.CV↗

arXiv2026

Lieb-Thirring bounds for Melik-Adamyan canonical Hamiltonians

We study a class of positive matrix Hamiltonians arising from the canonical differential expressions of Melik-Adamyan and appearing in the appendix of Alpay--Gohberg. Let $J$ and $B$ be self-adjoint involutions on $\mathbb{C}^{2n}$ satisfying $JB=-BJ$, and let $\mathcal{H}>0$ satisfy $\mathcal{H} J\mathcal{H}=J$. For $m>0$ we consider \[ \mathcal A_{m,\mathcal{H}} = \mathcal{H}^{-1}\left(-iJ\frac d{dt}+mB\right) \] in the weighted space $L^2_{\mathcal{H}}$. A locally absolutely continuous $J$-unitary gauge $Θ$ representing $\mathcal{H}$ reduces this expression to the free massive Dirac operator plus the Hermitian coefficient \[ P_{m,Θ}=-iΘ^*JΘ'+m(Θ^*BΘ-B). \] Whenever this coefficient belongs to $L^2$, the corresponding self-adjoint realization, including its operator domain, is independent of the chosen representing gauge. Minimizing $\int{\rm Tr}|P_{m,Θ}|^2$ over the gauge fibre defines an intrinsic energy. A two-sided Birman--Schwinger decoupling, combined with a truncated pseudo-relativistic estimate proved here, gives a $3/2$-moment bound for all eigenvalues in the gap $(-m,m)$ in terms of this energy. The Dirac estimate applies to arbitrary Hermitian matrix coefficients in $L^2$ and requires no sign condition. On the half-line we treat every self-adjoint Lagrangian boundary condition. Two reflection-compatible conditions require no endpoint correction, while an arbitrary condition contributes at most $2nm^{3/2}$. At zero mass, the optimal-gauge energy is computed explicitly in terms of $\mathcal{H}^{-1/2} \mathcal{H}' \mathcal{H}^{-1/2}$. For a scalar hyperbolic-rotation family the massive gauge minimization reduces exactly to a one-dimensional phase functional. We prove existence of a minimizer in the principal phase sector and give an explicit trial phase that strictly and quantitatively improves the positive lift whenever the corresponding first variation is nonzero.

math.SP↗

arXiv2026

One-dimensional optimisation of indefinite-weight principal eigenvalues with asymmetric Robin parameters and a Schrödinger-type perturbation

We study the minimisation of the positive principal eigenvalue for an indefinite-weight problem with asymmetric Robin parameters. The model is motivated by diffusive logistic equations in spatially heterogeneous environments, where the weight describes allocatable favourable resources and the Robin parameters measure boundary loss. After recalling the variational setting and the bang-bang reduction, we analyse the one-dimensional optimisation problem: the optimal favourable set is an interval, and the placement problem is reduced to a branchwise criterion. The key analytical tool is a shape-derivative formula for $a\mapstoλ(a)$, which shows that interior candidates are characterised by equality of the endpoint values of the positive eigenfunction, equivalently by the coupled transfer-matrix equations $f=0$ and $g=0$. We also introduce a Schrödinger-type extension with a fixed nonnegative background potential. In the coercive case we establish the corresponding principal-eigenvalue and bang-bang results, and in one dimension with constant potential we prove a compactness-type stability result showing that minimisers for small background potential converge, along subsequences, to minimisers of the unperturbed problem. No placement classification is claimed for general positive background potential. The computations are presented as numerical illustrations generated with an adaptive root-search protocol.

math.SP↗

arXiv2026

Magnetic Dirichlet Laplacian on deformed waveguides

It is well known that the spectrum of the Dirichlet Laplacian for a two-dimensional waveguide, which is a local deformation of a straight strip, is unstable with respect to waveguide boundary deformations. This means that, when the waveguide is a straight strip, the spectrum of the Dirichlet Laplacian is purely essential. On the other hand, local boundary perturbations of the straight strip produce eigenvalues below the essential spectrum. This paper considers the Dirichlet-Laplace operator with a compactly supported magnetic field. Furthermore, we omit the condition that the boundary perturbation is local. We prove that, in this case, the spectrum of the magnetic Laplacian is stable under small deformations of the waveguide boundary.

math.SP↗

arXiv2026

Magnetic Dirichlet Laplacian on a perturbed twisted tube

It is well known that the spectrum of the Dirichlet Laplacian for a compact perturbation of a three-dimensional, periodically twisted tube is unstable with respect to domain deformations. This means that if the periodically twisted tube is unperturbed, then the spectrum of the Dirichlet Laplacian is purely essential. On the other hand, the perturbation of this domain produces eigenvalues below the essential spectrum. This paper considers the Dirichlet-Laplace operator with a magnetic field. We explicitly prove that the spectrum of the magnetic Laplacian is stable under small and local deformations of the domain.

math.SP↗

arXiv2025

Three-dimensional magnetic Schrödinger operator with the potential supported in a tube

In this paper, we study the following magnetic Schrödinger operator in $\mathbb{R}^3$: \[ H=(i \nabla +A)^2- \tilde{V}, \] where $\tilde{V}$ is non-negative potential supported over the tube built along a curve which is a local deformation of a straight one, and $B:=\mathrm{rot}(A)$ is a non-zero and local (i.e., a compact supported) magnetic field. Based on some new strategies, we first prove that the magnetic field does not change the essential spectrum of this system. Finally, in the last section of this paper, we establish the sufficient condition such that the discrete spectrum is empty.

math.SP↗

arXiv2025

The $q$-Dirac Operator on Quantum Euclidean Space

This paper provides the foundations of quantum Clifford analysis in $q$-commutative variables with symmetric difference operators. We consider a $q$-Dirac operator on the quantum Euclidean space that factorizes the $U_q(\frak{o})$-invariant Laplacian $Δ_q.$ Due to the non-commutativity of the multiplication, we need a special Clifford algebra $C\ell_{0,n}^q.$ We define $q$-monogenic functions as null solutions of the $q$-Dirac operator and $q$-spherical monogenic functions. We define an inner Fischer product and decompose the space of homogeneous polynomials of degree $k.$

math.CV↗

arXiv2025

Magnetic Dirichlet Laplacian in curved waveguides

For a two-dimensional curved waveguide, it is well known that the spectrum of the Dirichlet Laplacian is unstable. Any perturbation of the straight strip produces eigenvalues below the essential spectrum. In this paper, a magnetic field is added. We explicitly prove that the spectrum of the magnetic Laplacian is stable under small but non-local deformations of the waveguide.

math.SP↗