0
$\begingroup$

In Chapter 2, Section 5 of Chavel's book, regarding the Neumann eigenvalues of the Laplacian in space forms, how did Chavel determine that $T'_{l,j}$ has ($j-1$) zeros? I have consulted books on the Sturm–Liouville theory and could only confirm that $T_{l,j}$ has ($j-1$) zeros. How can one obtain the distribution of zeros of the derivative of the eigenfunctions?

Improve this question
$\endgroup$
5
  • $\begingroup$ It is best if questions are as self contained as possible, so, for example, including the definition of $T'_{l, j}$. \\ TeX note: please use $T'$ or $T^\prime$ T' or T^\prime, not $T^{'}$ T^{'}. Notice that the prime in the latter is too high and too small. I have edited accordingly. $\endgroup$
    – LSpice
    Commented Dec 10, 2023 at 19:10
  • $\begingroup$ This is almost definitional: $\lambda_{l,j}$ is chosen so that $T'_{l,j}$ has its $j^{th}$ zero at $\delta$. $\endgroup$
    – Neal
    Commented Dec 11, 2023 at 1:27
  • $\begingroup$ Compare Chavel's general description of the process to the calculation of Neumann eigenvalues in an open ball of radius $\delta$. $\endgroup$
    – Neal
    Commented Dec 11, 2023 at 1:28
  • $\begingroup$ @尼尔 Thank you for your response, but I still want to know if we don't consider it as a Bessel equation, can we still determine the number of zeros of the derivative using the general Sturm-Liouville theory? $\endgroup$
    – Ruifeng Chen
    Commented Dec 11, 2023 at 12:16
  • $\begingroup$ @LSpice Thank you for your response ! $\endgroup$
    – Ruifeng Chen
    Commented Dec 11, 2023 at 12:17

0

Reset to default

You must log in to answer this question.