Related papers: D-Bar Operators on Quantum Domains
In his book (II.5), Connes gives a proof of the Atiyah-Singer index theorem for closed manifolds by using deformation groupoids and appropiate actions of these on R^N. Following these ideas, we prove an index theorem for manifolds with…
We use the definition of differential operators on noncommutative rings given by V.Lunts and A.Rosenberg to find the differential operators on Azumaya algebras and the Heisenberg algebras.
For characterizing the Brownian motion in a bounded domain: $\Omega$, it is well-known that the boundary conditions of the classical diffusion equation just rely on the given information of the solution along the boundary of a domain; on…
We prove the existence of unique solutions to the Dirichlet boundary value problems for linear second-order uniformly parabolic operators in either divergence or non-divergence form with boundary blowup low-order coefficients. The domain is…
We discuss possibilities of application of Numerical Analysis methods to proving computability, in the sense of the TTE approach, of solution operators of boundary-value problems for systems of PDEs. We prove computability of the solution…
The paper presents a variational quantum algorithm to solve initial-boundary value problems described by second-order partial differential equations. The approach uses hybrid classical/quantum hardware that is well suited for quantum…
We consider a (generally, non-coercive) mixed boundary value problem in a bounded domain $D$ of ${\mathbb R}^n$ for a second order parameter-dependent elliptic differential operator $A (x,\partial, \lambda)$ with complex-valued essentially…
In this paper, Dirac operator with some integral type nonlocal boundary conditions is studied. We show that the coefficients of the problem can be uniquely determined by a dense set of nodal points. Moreover, we give an algorithm for the…
We study properties of pseudodifferential operators which arise in their use in boundary value problems. Smooth domains as well as intersections of smooth domains are considered.
This paper showed that fractional p-adic Hardy operator norms in p-adic Herz-Morrey spaces with varying exponents are bounded. Corresponding commutator operators are also estimated for p-adic variable central bounded mean oscillations…
By examining the analyticity of a sequence of topologically-proper lattice Dirac operators, we show that they tend to a nonlocal Dirac operator. This implies that a nonlocal lattice Dirac operator can have exact zero modes satisfying the…
We consider the problem of operator identification in quantum control. The free Hamiltonian and the dipole moment are searched such that a given target state is reached at a given time. A local existence result is obtained. As a by-product,…
We obtain pointwise lower bounds for heat kernels of higher order differential operators with Dirichlet boundary conditions on bounded domains in $\R^N$. The bounds exhibit explicitly the nature of the spatial decay of the heat kernel close…
We present a new solution to the index problem for hypoelliptic operators in the Heisenberg calculus on contact manifolds, by constructing the appropriate topological K-theory cocycle for such operators. Its Chern character gives a…
A new supersymmetric proof of the Atiyah-Singer index theorem is presented. The Peierls bracket quantization scheme is used to quantize the supersymmetric classical system corresponding to the index problem for the twisted Dirac operator.…
Formally symmetric differential operators on weighted Hardy-Hilbert spaces are analyzed, along with adjoint pairs of differential operators. Eigenvalue problems for such operators are rather special, but include many of the classical…
We consider a class of linear differential operators acting on vector-valued function spaces with general coupled boundary conditions. Unlike in the more usual case of so-called quantum graphs, the boundary conditions can be nonlinear.…
The present article focuses on the bounds of weighted multilinear $p$-adic Hardy operators on the product of Herz spaces and Morrey-Herz spaces. The corresponding norm on both the cases are also obtain.
We prove an Atiyah-Patodi-Singer index theorem for Dirac operators twisted by C*-vector bundles. We use it to derive a general product formula for eta-forms and to define and study new rho-invariants generalizing Lott's higher rho-form. The…
In this note we study the analytical index of pseudo-differential operators by using the notion of (infinite dimensional) operator-valued symbols (in the sense of Ruzhansky and Turunen). Our main tools will be the McKean-Singer index…