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The main aim of the paper is Fredholm properties of a class of bounded linear operators acting on weighted Lebesgue spaces on an infinite metric graph $\Gamma$ which is periodic with respect to the action of the group ${\mathbb Z}^n$. The…
Motivated by considerations of euclidean quantum gravity, we investigate a central question of spectral geometry, namely the question of reconstructability of compact Riemannian manifolds from the spectra of their Laplace operators. To this…
In this paper, we studied line operators in the Left-Right Symmetric Model. The gauge group of Left-Right Symmetric Electroweak Model is ${G} = SU(3) \times SU(2)_{L} \times SU(2)_{R} \times U(1)_{B - L}$. We derived the spectrum of line…
The spectral problem for self-adjoint extensions is studied using the machinery of boundary triplets. For a class of symmetric operators having Weyl functions of a special type we calculate explicitly the spectral projections in the form of…
We obtain a representation formula for the derivative of the spectral shift function $\xi(\lambda; B, \epsilon)$ related to the operators $H_0(B,\epsilon) = (D_x - By)^2 + D_y^2 + \epsilon x$ and $H(B, \epsilon) = H_0(B, \epsilon) + V(x,y),…
Let $Y$ be a $d$-dimensional random vector with unknown mean $\mu$ and covariance matrix $\Sigma$. This paper is motivated by the problem of designing an estimator of $\Sigma$ that admits tight deviation bounds in the operator norm under…
We construct an iterative procedure to compute the vertex operators of the closed superstring in the covariant formalism given a solution of IIA/IIB supergravity. The manifest supersymmetry allows us to construct vertex operators for any…
Let $D:\Omega\xrightarrow{}\Omega$ be a differential operator defined in the exterior algebra $\Omega$ of differential forms over the polynomial ring $S$ in $n$ variables. In this work we give conditions for deforming the module structure…
In a bounded domain $G$ with smooth border studied boundary value and spectral problems for operators of the rotor (vortex) and the gradient of the divergence $+\lambda\,I$ in the Sobolev spaces. For $\lambda\neq 0$ these operators are…
Differential operators are widely used in geometry processing for problem domains like spectral shape analysis, data interpolation, parametrization and mapping, and meshing. In addition to the ubiquitous cotangent Laplacian, anisotropic…
We study a spectral problem $(\mathscr{P}^{\delta})$ for a diffusion like equation in a 3D domain $\Omega$. The main originality lies in the presence of a parameter $\sigma^{\delta}$, whose sign changes on $\Omega$, in the principal part of…
Continuing previous work we develop a certain piece of functional analysis on general graphs and use it to create what Connes calls a 'spectral triple', i.e. a Hilbert space structure, a representation of a certain (function) algebra and a…
We introduce in a general setting a dynamic programming method for solving reconfiguration problems. Our method is based on contracted solution graphs, which are obtained from solution graphs by performing an appropriate series of edge…
The study of spectral properties of natural geometric elliptic partial differential operators acting on smooth sections of vector bundles over Riemannian manifolds is a central theme in global analysis, differential geometry and…
This is a survey article on a known generalization of Dirac-type operators to transverse operators called basic Dirac operators on Riemannian foliations, which are smooth foliations that have a transverse geometric structure. Construction…
We analyze the spectral properties of the volume operator of Ashtekar and Lewandowski in Loop Quantum Gravity, which is the quantum analogue of the classical volume expression for regions in three dimensional Riemannian space. Our analysis…
This paper is devoted to the construction of semiclassical spectrum and efficient (simple to implement) explicit semiclassical asymptotic eigenfunctions of the Dirac operator for relatively high-energy bound states in graphene in magnetic…
Not necessarily self-adjoint quantum graphs -- differential operators on metric graphs -- are considered. Assume in addition that the underlying metric graph possesses an automorphism (symmetry) $ \mathcal P $. If the differential operator…
The inception of spatial transcriptomics has allowed improved comprehension of tissue architectures and the disentanglement of complex underlying biological, physiological, and pathological processes through their positional contexts.…
By means of the operator extension theory, we construct an explicitly solvable model of a simple-cubic three-dimensional regimented array of quantum dots in the presence of a uniform magnetic field. The spectral properties of the model are…