Aithal, A. R.; Kadam, Pratiksha M. On the extrema of the fundamental eigenvalue of a family of Schrödinger operators. (English) Zbl 1425.47003 J. Ramanujan Math. Soc. 33, No. 3, 233-247 (2018). Summary: Let \(D\) be an open regular polygon of \(n\) sides in \(\mathbb R^2\). Let \(\mathcal{P}_0 \subset D\) 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 \(\mathcal{P}_0\) by rotating it in \(C\) about its center of mass. Let \(\mathcal{P}_t (t \in \mathbb{R})\) be the family of polygons obtained in this fashion. Let \(\chi_{\mathcal{P}t}\) denote the indicator function of the subset \(\mathcal{P}_t\) of \(D\). For any non-zero constant \(\alpha \in \mathbb R\) it is shown that the fundamental eigenvalue of the Schrödinger operators \(-\Delta + \alpha \chi_{\mathcal{P}t}\) attains its extremum when the axes of symmetry of \(\mathcal{P}_0\) coincide with those of \(D\). MSC: 47A75 Eigenvalue problems for linear operators 47A55 Perturbation theory of linear operators 49N45 Inverse problems in optimal control 81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis PDFBibTeX XMLCite \textit{A. R. Aithal} and \textit{P. M. Kadam}, J. Ramanujan Math. Soc. 33, No. 3, 233--247 (2018; Zbl 1425.47003) Full Text: Link References: [1] Ahmad EL Soufi and Rola Kiwan, Extremal first dirichlet eigenvalue of doubly connected plane domains and dihedral symmetry, SIAM J. Math. Anal., 39 (2007) no. 4, 1112-1119. · Zbl 1156.35068 [2] A. R. Aithal and Rajesh Raut, On the extrema of Dirichlet’s first eigenvalue of a family of punctured regular polygons in two dimensional space forms, Proc. Indian Acad. Sci. (Math. Sc.), 122 (2012) no. 2, pp 257-281. · Zbl 1256.35032 [3] T. Aubin, Nonlinear analysis on manifolds-Monge-Ampere equations, SpringerVerlag (1982). · Zbl 0512.53044 [4] B. Folland Gerald, Introduction to partial differential equations, Prentice Hall of India, Second Edition (2001). · Zbl 1046.26503 [5] W. M. Gidas, B. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Commun, Math. Phys., 68 (1979) 209-243. · Zbl 0425.35020 [6] P. Grisvard, Singularities in boundary value problems, recherches en math´ematiques appliqu´ees, Masson, Paris; Springer-Verlag, Berlin, 22 (1992). · Zbl 0766.35001 [7] E. M. Harell II, P. Kroger and K. Kurata, On the placement of an obstacle or a well so as to optimize the fundamental eigenvalue, Siam J. Math. Anal., 33 no. 1, 240-259. · Zbl 0994.47015 [8] Henrot Antoine, Extremum problems for eigenvalues of elliptic operators, Birkh¨auser Verlag, Springer (2006). · Zbl 1109.35081 [9] Kreyszig Erwin, Introductory Functional Analysis with Applications, John Wiley & Sons (1978). · Zbl 0368.46014 [10] M. Reed and B. Simon, Methods of modern mathematical physics IV: analysis of operators, Academic Press (1978). · Zbl 0401.47001 [11] J. Sokolowski and J. P. Zolesio, Introduction to shape optimization-shape sensitivity analysis, Springer-Verlag (1992). · Zbl 0761.73003 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.