Lower bound of an eigenvalue

Anton29 · June 1st, 2012, 12:30 PM

For a given real number c>0 define functions $\left(\psi_{k,c}(\cdot)\right)_{k\ge0$ }, as an eigenfunctions of the Sturm-Liouville operators L_c defined
$L_c(\psi)=(1-x^2)\frac{d^2\psi}{dx^2}-2x \frac{d\psi}{dx}-c^2x^2\psi$
( in fact, $\psi_{k,c}$ is called the prolate spheroidal wave function).

Consider a compact integral operator
$F_c(\psi)(x)=\int_{-1}^1\frac{\sin(c(x-y))}{\pi(x-y)}\psi(y)dy.$
It is known that $\psi_{k,c}$ are the eigenfunctions of the operator $F_c$ . So,
$F_c=\lambda_k\psi_{k,c},$
where $\lambda_k$ are eigenvalues of $F_c$
(Note, it is known that $\lambda_k\leq \frac{c}{2}\left(\frac{ec}{4k}\right)^{2k}$ .)

I would like to find a lower bound for $\lambda_k.$

Any references and ideas will be very helpful.

Thank you.

stainburg · June 13th, 2012, 08:35 PM

Ritz or Galerkin method?

June 1st, 2012, 12:30 PM	#1
Anton29 Member Joined: Aug 2011 Posts: 71 Thanks: 0	Lower bound of an eigenvalue For a given real number c>0 define functions $\left(\psi_{k,c}(\cdot)\right)_{k\ge0$ }, as an eigenfunctions of the Sturm-Liouville operators L_c defined $L_c(\psi)=(1-x^2)\frac{d^2\psi}{dx^2}-2x \frac{d\psi}{dx}-c^2x^2\psi$ ( in fact, $\psi_{k,c}$ is called the prolate spheroidal wave function). Consider a compact integral operator $F_c(\psi)(x)=\int_{-1}^1\frac{\sin(c(x-y))}{\pi(x-y)}\psi(y)dy.$ It is known that $\psi_{k,c}$ are the eigenfunctions of the operator $F_c$ . So, $F_c=\lambda_k\psi_{k,c},$ where $\lambda_k$ are eigenvalues of $F_c$ (Note, it is known that $\lambda_k\leq \frac{c}{2}\left(\frac{ec}{4k}\right)^{2k}$ .) I would like to find a lower bound for $\lambda_k.$ Any references and ideas will be very helpful. Thank you.
$Anton29 is offline$

June 13th, 2012, 08:35 PM	#2
stainburg Senior Member Joined: Oct 2010 From: Changchun, China Posts: 492 Thanks: 14	Re: Lower bound of an eigenvalue Ritz or Galerkin method?
$stainburg is offline$

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