Research Projects

The uncertainty principle is one of the central topics of research in harmonic analysis. It states that a function and its Fourier transform cannot both be localized. There are different ways of measuring localization, and depending on many versions of the uncertainty principle are established. In this project, we begin with a recent version of the uncertainty principle where the notion of localization is introduced in terms of the uniqueness of the Fourier transform of measures supported on algebraic curves, known as Heisenberg uniqueness pair. The questions of Heisenberg uniqueness pairs are about the determining sets, which are very thin sets, where the Fourier transform of measures supported on an algebraic curve vanishes and this determines the measure uniquely. The concept of Heisenberg uniqueness pair was first introduced by Hedenmalm and Montes-Rodr\'{i}guez, and in [Ann. of Math. 173 (2011)], for measures supported on hyperbola, they consider lattice cross as determining set. We are interested in irregular lattice cross as a determining set for measures supported on the hyperbola. This will not only extend the above result, it will bring a new link with the ergodic theory. In [J. Math. Anal. Appl. 106 (1985)], Benedicks proved that if a function and its Fourier transform in the Euclidean space, both supported on sets of finite measure then the function is zero. To see an analogue of the Benedicks theorem on the Heisenberg group, note that Fourier transform on the Heisenberg group is an operator-valued function. This expects novelty when considering localization on the Fourier transform side. In [Proc. Amer. Math. Soc. 138 (2010)], Narayanan and Ratnakumar proved Benedicks theorem on the Heisenberg group by defining localization in terms of finite rank of the Fourier transform. However, the function needs to have spatial support in sets of finite measure. We are interested in Benedicks theorem for functions with arbitrary supports of finite measure on the Heisenberg group. Further, we want to prove Nazarov's uncertainty principle on the Heisenberg group. This is a quantitative interpretation of the Benedicks theorem in terms of strong annihilating inequality with an appropriate constant on the Heisenberg group. The uncertainty principle for the Fourier transform can be reformulated in terms of uncertainty principle for the solution to the Schr\"{o}dinger equation on the Euclidean space. This suggests the study of uniqueness properties for the solution to the Schr\"{o}dinger equation on the Heisenberg group. We want to obtain a strong annihilating inequality for the solution of the Schr\"{o}dinger equation with potential on the Heisenberg group. In general, this is an active area of research and there are nice questions to explore. In this project, we want to carry out several qualitative and quantitative versions of the uncertainty principle, which, we think, are significant to this area of research.