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We show that in-plane-magnetic-field assisted spectroscopy allows extraction of the in-plane orientation and full 3D shape of the quantum mechanical orbitals of a single electron GaAs lateral quantum dot with sub-nm precision. The method is…
A building block of noncommutative geometry is the observation that most of the geometric information of a compact riemannian spin manifold M is encoded within its Dirac operator D. Especially via Connes' distance formula one is able to…
We introduce and study {\it new} relative spectral invariants of {\it two} elliptic partial differential operators of Laplace and Dirac type on compact smooth manifolds without boundary that depend on both the eigenvalues and the…
The formalism of non-commutative geometry of A. Connes is used to construct models in particle physics. The physical space-time is taken to be a product of a continuous four-manifold by a discrete set of points. The treatment of Connes is…
We make a computational study to know what kind of isospectralities among lens spaces and lens orbifolds exist considering the Hodge--Laplace operators acting on smooth $p$-forms. Several evidenced facts are proved and some others are…
We present a new spectral scheme for analysing functions of half-integer spin-weight on the $2$-sphere and demonstrate the stability and convergence properties of our implementation. The dynamical evolution of the Dirac equation on a…
In this paper, we construct the spectral expansion for the one dimensional non-self-adjoint Dirac operator L(Q) with a complex-valued periodic matrix potential Q. To this end, we study in detail asymptotic formulas for the Bloch eigenvalues…
We propose a symmetry of the Dirac equation under the interchange of signs of eigenvalues of the Dirac's $K$ operator. We show that the only potential which obeys this requirement is the Coulomb one for both vector and scalar cases.…
We construct spectral triples on C*-algebraic extensions of unital C*-algebras by stable ideals satisfying a certain Toeplitz type property using given spectral triples on the quotient and ideal. Our construction behaves well with respect…
We look at the $L^p$ bounds on eigenfunctions for polygonal domains (or more generally Euclidean surfaces with conic singularities) by analysis of the wave operator on the flat Euclidean cone $C(\mathbb{S}^1_\rho) := \mathbb{R}_+ \times…
The geometric objects of study in this paper are K3 surfaces which admit a polarization by the unique even unimodular lattice of signature (1,17). A standard Hodge-theoretic observation about this special class of K3 surfaces is that their…
Almost commutative models provide a framework for Connes' work on the standard model of particle physics. These models are constructed as products of a the canonical spectral triple of a compact connected spin manifold with a finite…
In the paper we extend the spectral invariance of pseudodifferential operators acting on (non-weighted) classical modulation spaces to allow the Lebesgue exponents to be smaller than one. These spaces occur naturally in approximation theory…
Given a compact smooth manifold $M$ with non-empty boundary and a Morse function, a pseudo-gradient Morse-Smale vector field adapted to the boundary allows one to build a Morse complex whose homology is isomorphic to the (absolute or…
We study spectral properties of Dirichlet Laplacian on the conical layer of the opening angle $\pi-2\theta$ and thickness equal to $\pi$. We demonstrate that below the continuum threshold which is equal to one there is an infinite sequence…
The local index formula of Connes--Moscovici for the isospectral noncommutative geometry recently constructed on quantum SU(2) is discussed. The cosphere bundle and the dimension spectrum as well as the local cyclic cocycles yielding the…
This paper is concerned with spectral estimates for the first Dirichlet eigenvalue of the degenerate $p$-Laplace operator in bounded simply connected domains $\Omega \subset \mathbb C$. The proposed approach relies on the conformal analysis…
We study various noncommutative geometric aspects of the compact quantum group SU_q(2) for positive q (not equal to 1), following the suggestion of Connes and his coauthors [CL, CD] for considering the so-called true Dirac operator.…
We construct elliptic operators with scalar coefficients on the complements $(\mathbb{R}^2 \setminus S)^+$ of some Koch-type snowflakes $S$, whose Hausdorff dimensions cover the full range $(1, \ln{(4)}/\ln{(3)})$, such that the operator's…
The article deals with intrinsic metrics, Dirac operators and spectral triples induced by regular Dirichlet and resistance forms. We show, in particular, that if a local resistance form is given and the space is compact in resistance…