Related papers: Green Operators in the Edge Calculus
Neural operators are a popular technique in scientific machine learning to learn a mathematical model of the behavior of unknown physical systems from data. Neural operators are especially useful to learn solution operators associated with…
We develop an elliptic theory based in $L^2$ of boundary value problems for general wedge differential operators of first order under only mild assumptions on the boundary spectrum. In particular, we do not require the indicial roots to be…
We present a general framework to study edge states for second order elliptic operators in a half channel. We associate an integer valued index to some bulk materials, and we prove that for any junction between two such materials, localised…
We extend to manifolds endowed with a general geometric structure, the classical notions of gradient as well as Laplace operator, and provide some of their natural properties.
We show that 2D periodic operators with local and perpendicular defects form an algebra. We provide an algorithm of finding spectrum for such operators. While the continuous spectral components can be computed by simple algebraic operations…
In some cases in two and three bulk dimensions without bulk local degrees of freedom, I look for area operators in a fixed boundary theory. In each case, I define an exact quantum error-correcting code (QECC) and show that it admits a…
This paper is a continuation of the investigation of resolvents of elliptic operators on conic manifolds from math.AP/0410178 and math.AP/0410176 to the case of manifolds with boundary and realizations of operators under boundary…
We identify Melrose's suspended algebra of pseudodifferential operators with a subalgebra of the algebra of parametric pseudodifferential operators with parameter space $\R$. For a general algebra of parametric pseudodifferential operators,…
On the one hand the algebras of linear operators here act on finite-dimensional vector spaces, and on the other hand the point of view is generally an analysts'. Also, one might think of algebras as being used to add more data to basic…
The semiconducting two-dimensional transition metal dichalcogenides MX$_{2}$ show an abundance of one-dimensional metallic edges and grain boundaries. Standard techniques for calculating edge states typically model nanoribbons, and require…
We introduce a new class of natural, explicitly defined, transversally elliptic differential operators over manifolds with compact group actions. Under certain assumptions, the symbols of these operators generate all the possible values of…
We are using the finite-gap approach for the construction of the Schr\"{o}dinger operator discretization on a quad graph. The latter is represented by a two-dimensional integer sublattice in a $d$-dimensional space. The Green's function of…
General formula for causal Green's function of linear differential operator of given degree in one variable is given according to coefficient functions of differential operator as a series of integrals. The solution also provides analytic…
Deep operator networks (DeepONets) have demonstrated their capability of approximating nonlinear operators for initial- and boundary-value problems. One attractive feature of DeepONets is their versatility since they do not rely on prior…
Green's function plays a significant role in both theoretical analysis and numerical computing of partial differential equations (PDEs). However, in most cases, Green's function is difficult to compute. The troubles arise in the following…
This is the second part of a series of four articles on weighted norm inequalities, off-diagonal estimates and elliptic operators. We consider a substitute to the notion of pointwise bounds for kernels of operators which usually is a…
Indicial operators are model operators associated to an elliptic differential operator near a corner singularity on a stratified manifold. These model operators are defined on generalized tangent cone configurations and exhibit a natural…
Centered weighted composition operators on $L^2$-spaces are characterized. The characterization is obtained without the assumption that the operator is a product of a multiplication and a composition operator. The concept of spectrally…
In this paper second order elliptic boundary value problems on bounded domains $\Omega\subset\dR^n$ with boundary conditions on $\partial\Omega$ depending nonlinearly on the spectral parameter are investigated in an operator theoretic…
We consider special elliptic operators in functional spaces on manifolds with a boundary which has some singular points. Such an operator can be represented by a sum of operators, and for a Fredholm property of an initial operator one needs…