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Eigenvalue relationships between Laplacians of constant mean curvature hypersurfaces in $𝕊^{n + 1}$

Bingqing Ma, Guangyue Huang (2013)

Communications in Mathematics

For compact hypersurfaces with constant mean curvature in the unit sphere, we give a comparison theorem between eigenvalues of the stability operator and that of the Hodge Laplacian on 1-forms. Furthermore, we also establish a comparison theorem between eigenvalues of the stability operator and that of the rough Laplacian.

Eigenvalue stability bounds via weighted Sobolov spaces.

E.B. Davies (1993)

Mathematische Zeitschrift

Eigenvalues of conformally invariant operators on spheres

Bureš, Jarolím, Souček, Vladimír (1999)

Proceedings of the 18th Winter School "Geometry and Physics"

Eigenvalues of the Hodge Laplacian on the Heisenberg group.

Detlef Müller, Marco M. Peloso, Fulvio Ricci (2006)

Collectanea Mathematica

Eigenvalues of the laplacian of compact Riemann surfaces and minimal submanifolds

Paul C. Yang, Shing-Tung Yau (1980)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

Eigenvalues of the $p$ -Laplacian in $𝐑^{N}$ with indefinite weight

Yin Xi Huang (1995)

Commentationes Mathematicae Universitatis Carolinae

We consider the nonlinear eigenvalue problem $- {div (| \nabla u |}^{p - 2} {\nabla u) = λ g (x) | u |}^{p - 2} u$ in $𝐑^{N}$ with $p > 1$ . A condition on indefinite weight function $g$ is given so that the problem has a sequence of eigenvalues tending to infinity with decaying eigenfunctions in $W^{1, p} (𝐑^{N})$ . A nonexistence result is also given for the case $p \geq N$ .

Eight exactly solvable complex potentials in Bender-Boettcher quantum mechanics

Znojil, Miloslav (2001)

Proceedings of the 20th Winter School "Geometry and Physics"

This is a readable review of recent work on non-Hermitian bound state problems with complex potentials. A particular example is the generalization of the harmonic oscillator with the potentials: $V (x) = \frac{ω^{2}}{2} {(x - \frac{2 i β}{ω})}^{2} - \frac{ω}{2} .$ Other examples include complex generalizations of the Morse potential, the spiked radial harmonic potential, the Kratzer-Coulomb potential, the Rosen Morse oscillator and others. Instead of demanding Hermiticity $H = H^{*}$ the condition required is $H = P T H P T$ where $P$ changes the parity and $T$ transforms $i$ to $- i$ .