Journal of the Ramanujan Mathematical Society

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On The Extrema of the Fundamental Eigenvalue of a Family of Schrodinger Operators


Affiliations
1 Department of Mathematics, University of Mumbai, Mumbai - 400098, India
2 K. C. College, Churchgate, Mumbai - 400020, India
     

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Let D be an open regular polygon of n sides in ℝ2. 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 Schr¨odinger operators −Δ+ αχ℘t attains its extremum when the axes of symmetry of ℘0 coincide with those of D.
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  • On The Extrema of the Fundamental Eigenvalue of a Family of Schrodinger Operators

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Authors

A. R. Aithal
Department of Mathematics, University of Mumbai, Mumbai - 400098, India
Pratiksha M. Kadam
K. C. College, Churchgate, Mumbai - 400020, India

Abstract


Let D be an open regular polygon of n sides in ℝ2. 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 Schr¨odinger operators −Δ+ αχ℘t attains its extremum when the axes of symmetry of ℘0 coincide with those of D.

References