Abstract
The problem considered here is that of describing generically the zeros, critical points and critical values of eigenfunctions of elliptic partial differential operators. We consider operators of the form L + Ï L + \rho , where L is a fixed, second-order, selfadjoint, C â {C^\infty } linear elliptic partial differential operator on a compact manifold (without boundary) and Ï \rho is a C â {C^\infty } function. It is shown that, for almost all Ï \rho , i.e. for a residual set, the eigenvalues of L + Ï L + \rho are simple and the eigenfunctions have the following properties: (1) they are Morse functions; (2) distinct critical points have distinct critical values; (3) 0 is not a critical value.
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.