Quantum particle across Grushin singularity ^*

Approximate controllability for a 2d Grushin equation with potential having an internal singularity

This paper is dedicated to approximate controllability for Grushin equation on the rectangle (x, y) ∈ (−1, 1) × (0, 1) with an inverse square potential. This model corresponds to the heat equation for the LaplaceBeltrami operator associated to the Grushin metric on R, studied by Boscain and Laurent. The operator is both degenerate and singular on the line {x = 0}. The approximate controllabilit...

Elimination of the Standard Big Bang Singularity and Particle Horizon through Quantum Conformal Fluctuations

The standard big bang cosmology is known to suffer from two defects: (1) the space-time singularity at the beginning of the universe [ 1 ] and (2) the existence of a particle horizon which severely limits the communication between the different parts of the universe [2]. The standard models of cosmology are, however, based on the classical equations of general relativity. Earlier work [3-5] has...

Singularity resolution from polymer quantum matter

QUANTUM RING OF SINGULARITY Xp + XYq

In this paper, we will prove that the quantum ring of the quasi-homogeneous polynomial X p + XYq(p ≥ 2, q > 1) with some admissible symmetry group G defined by Fan-Jarvis-Ruan-Witten theory is isomorphic to the Milnor ring of its mirror dual polynomial X pY + Yq. We will construct an concrete isomorphism between them. The construction is a little bit different in case (p − 1, q) = 1 and case (p...

Geometry of Grushin Spaces

We compare the Grushin geometry to Euclidean geometry, through quasisymmetric parametrization, bilipschitz parametrization and bilipschitz embedding, highlighting the role of the exponents and the fractal nature of the singular hyperplanes in Grushin geometry. Consider in Rn a system of diagonal vector fields Xj = λj(x) ∂ ∂xj , j = 1, . . . , n,