# 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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