Kalyan B. Sinha

Characterization of unitary processes with independent and stationary increments

This is a continuation of the earlier work (Publ. Res. Inst. Math. Sci. 45 (2009) 745-785) to characterize unitary stationary independent increment Gaussian processes. The earlier assumption of uniform continuity is replaced by weak continuity and with technical assumptions on the domain of the generator, unitary equivalence of the process to the solution of an appropriate Hudson-Parthasarathy equation is proved.

Quantum Stochastic Calculus and Applications — A Review

It has been a little more than a decade since this subject, as it is understood today, came into being with the seminal paper of Hudson and Parthasarathy [1]. Since then the subject has seen rapid development and many of these can be found in the monographs of Parthasarathy [2] and Meyer [3]. Here I want to discuss some of the more recent developments.

Applications to Partial Differential Equations

While semigroups of operators are interesting objects of study by themselves, one of the main reasons why they are studied so extensively is due to the important role they play in the study of partial differential equations.

Vector-Valued Functions

This chapter is mostly of a preliminary nature. In the first section we collect results on measurability and integrability of vector-valued functions that will be useful throughout.

Perturbation and Convergence of Semigroups

In this chapter, the stability of various classes of semigroups under suitable sets of perturbations will be studied, viz. for general (C_0)-semigroups and contraction semigroups.

Chernoff’s Theorem and its Applications

In this chapter, a very interesting theorem, due to Chernoff [4], is proven and some of its applications, viz. the Trotter-Kato Product Formula, the Feynman-Kac Formula and the Central Limit Theorem are given.

Dissipative Operators and Holomorphic Semigroups

In this chapter we continue the study of (C_0)-semigroups concentrating on contractive and holomorphic semigroups.

Representations of q -commutation relations

Many authors have recently studied the q-deformed commutation relations (qCR) with finite or infinite degrees of freedom (see [1] – [11] ). These commutation relations also make their appearance in the study of quantum groups and their representations. Here we look at the representation theory as a simple extension of the same for the usual canonical commutation relation (CCR) [see e.g. [12]].