scholarly journals Uncertainty principles for the Schrödinger equation on Riemannian symmetric spaces of the noncompact type

2012 ◽  
Vol 62 (3) ◽  
pp. 859-886 ◽  
Author(s):  
Angela Pasquale ◽  
Maddala Sundari
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Symmetric Spaces ◽  
Uncertainty Principles ◽  
Noncompact Type ◽  
Riemannian Symmetric Spaces

scholarly journals AN UNCERTAINTY PRINCIPLE FOR SOLUTIONS OF THE SCHRÖDINGER EQUATION ON -TYPE GROUPS

2020 ◽  
pp. 1-16
Author(s):  
AINGERU FERNÁNDEZ-BERTOLIN ◽  
PHILIPPE JAMING ◽  
SALVADOR PÉREZ-ESTEVA
Keyword(s):  
Schrödinger Equation ◽  
Uncertainty Principle ◽  
Schrodinger Equation ◽  
Schrodinger Equations ◽  
Nilpotent Lie Groups ◽  
Schrödinger Equations ◽  
Uncertainty Principles ◽  
Uniqueness Of Solutions ◽  
Free Case ◽  
Formula Type

In this paper we consider uncertainty principles for solutions of certain partial differential equations on $H$ -type groups. We first prove that, on $H$ -type groups, the heat kernel is an average of Gaussians in the central variable, so that it does not satisfy a certain reformulation of Hardy’s uncertainty principle. We then prove the analogue of Hardy’s uncertainty principle for solutions of the Schrödinger equation with potential on $H$ -type groups. This extends the free case considered by Ben Saïd et al. [‘Uniqueness of solutions to Schrödinger equations on H-type groups’, J. Aust. Math. Soc. (3)95 (2013), 297–314] and by Ludwig and Müller [‘Uniqueness of solutions to Schrödinger equations on 2-step nilpotent Lie groups’, Proc. Amer. Math. Soc.142 (2014), 2101–2118].

Theorems of Ingham and Chernoff on Riemannian symmetric spaces of noncompact type

2020 ◽  
Vol 279 (11) ◽  
pp. 108760
Author(s):  
Mithun Bhowmik ◽  
Sanjoy Pusti ◽  
Swagato K. Ray
Keyword(s):  
Symmetric Spaces ◽  
Noncompact Type ◽  
Riemannian Symmetric Spaces

HYPERSURFACES IN NONCOMPACT COMPLEX GRASSMANNIANS OF RANK TWO

2012 ◽  
Vol 23 (10) ◽  
pp. 1250103 ◽  
Author(s):  
JÜRGEN BERNDT ◽  
YOUNG JIN SUH
Keyword(s):  
Geometric Structure ◽  
Symmetric Spaces ◽  
Second Fundamental Form ◽  
Kähler Structure ◽  
Noncompact Type ◽  
The Second Fundamental Form ◽  
Special Cases ◽  
Rank One ◽  
Higher Rank ◽  
Riemannian Symmetric Spaces

Consider a Riemannian manifold N equipped with an additional geometric structure, such as a Kähler structure or a quaternionic Kähler structure, and a hypersurface M in N. The geometric structure induces a decomposition of the tangent bundle TM of M into subbundles. A natural problem is to classify all hypersurfaces in N for which the second fundamental form of M preserves these subbundles. This problem is reasonably well understood for Riemannian symmetric spaces of rank one, but not for higher rank symmetric spaces. A general treatment of this problem for higher rank symmetric spaces is out of reach at present, and therefore it is desirable to understand this problem better in a few special cases. Due to some conceptual differences between symmetric spaces of compact type and of noncompact type it appears that one needs to consider these two cases separately. In this paper we investigate this problem for the rank two symmetric space SU 2, m/S(U2Um) of noncompact type.

scholarly journals The $\Theta$-spherical transform and its inversion

2004 ◽  
Vol 95 (2) ◽  
pp. 265 ◽  
Author(s):  
Angela Pasquale
Keyword(s):  
Laplace Transform ◽  
Compact Support ◽  
Inversion Formula ◽  
Symmetric Spaces ◽  
Expansion Formula ◽  
Noncompact Type ◽  
Riemannian Symmetric Spaces

The $\Theta$-spherical transform is defined as a simultaneous generalization of the Harish-Chandra's spherical transform on Riemannian symmetric spaces of noncompact type and of the spherical Laplace transform on noncompactly causal symmetric spaces as defined by Faraut, Hilgert and Ólafsson. An extension of Ólafsson's expansion formula allows us to deduce an inversion formula for the $\Theta$-spherical transform on the space of $W_\Theta$-invariant $C^\infty$ functions with compact support.

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