The task to construct parametrices of elliptic differential operators on a manifold with edges requires a calculus of operators with a two-component principal symbolic hierarchy, consisting of (edge-degenerate) interior and (operator-valued) edge symbols. This so-called edge-algebra can be interpreted as a generalisation of the (pseudo-differential) algebra of boundary value problems without th...

Ito’s construction of Markovian solutions to stochastic equations driven by a Lévy noise is extended to nonlinear distribution dependent integrands aiming at the effective construction of linear and nonlinear Markov semigroups and the corresponding processes with a given pseudo-differential generator. It is shown that a conditionally positive integro-differential operator (of the Lévy-Khintchin...

Quantization of BKP type equations are done through the Moyal bracket and the formalism of pseudo-differential operators. It is shown that a variant of the dressing operator can also be constructed for such quantized systems. Quantization of integrable system is one of the most important aspect of present day research on nonlinear systems. In two dimensions, a well established methodology was s...

We study the behavior of solutions to a Schrödinger equation with large, rapidly oscillating, mean zero, random potential with Gaussian distribution. We show that in high dimension d > m, where m is the order of the spatial pseudo-differential operator in the Schrödinger equation (with m = 2 for the standard Laplace operator), the solution converges in the L sense uniformly in time over finite ...

We consider gradient descent equations for energy functionals of the type S(u) = 1 2 〈u(x), A(x)u(x)〉L2 + ∫ Ω V (x, u) dx, where A is a uniformly elliptic operator of order 2, with smooth coefficients. The gradient descent equation for such a functional depends on the metric under consideration. We consider the steepest descent equation for S where the gradient is an element of the Sobolev spac...

We consider perturbed Schrödinger equation, which is an elliptic operator with unbounded coefficients. We use wavelets adapted to Schrödinger operator to deal with problems on unbounded domain. The wavelets are constructed from Hermite functions, which characterizes the space generated by the Schrödinger operator. We show that the Galerkin matrix can be preconditioned by a diagonal matrix so th...

an operator associated with the wright function is introduced in the open unit disk. differential subordination and superordination results associated with this operator are obtained by investigating appropriate classes of admissible functions. in particular, some inequalities for modified bessel functions are also obtained.

The magnetic Schrödinger operator, with Neumann boundary condition, on a smooth, bounded, and simply connected domain $$\Omega $$ of the Euclidean plane is considered in semiclassical limit. When has symmetry axis, splitting first two eigenvalues analyzed. explicit tunneling formula pure field established. analysis based pseudo-differential reduction to proof known optimal purely Agmon estimates.

The aim of this article is to study neutral type Clifford-valued high-order Hopfield neural networks with mixed delays and D operator. New criteria are established for the existence, uniqueness global exponential stability $$(\mu ,\nu )-$$ pseudo almost automorphic solutions considered model via Banach’s fixed point principle differential inequality techniques. An example given show effectivene...

When quantum fields are studied on manifolds with boundary, the corresponding one-loop quantum theory for bosonic gauge fields with linear covariant gauges needs the assignment of suitable boundary operators for elliptic differential operators of Laplace type. It is here shown that, on requiring that the full boundary operator should be a projector, at least two sets of such operators are found...