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Journal of the Ramanujan Mathematical Society

Volume 33, Issue 3, September 2018  pp. 233–247.

On the extrema of the fundamental eigenvalue of a family of Schrödinger operators

Authors:  A. R. Aithal and Pratiksha M. Kadam
Author institution:Department of Mathematics, University of Mumbai, Mumbai 400 098, India

Summary:  Let D be an open regular polygon of n sides in R 2. Let ℘ 0 ⊂ D be an open regular polygon of n sides having the same center of mass and circumscribed by a circle C contained in D. We fix D and vary ℘ 0 by rotating it in C about its center of mass. Let ℘ t (t ∈ R) be the family of polygons obtained in this fashion. Let χ ℘ t denote the indicator function of the subset ℘ t of D. For any non-zero constant α ∈ R it is shown that the Fundamental Eigenvalue of the Schrödinger operators -Δ + α χ ℘ t attains its extremum when the axes of symmetry of ℘ 0 coincide with those of D.

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