# Toronto Math Forum

## APM346-2015S => APM346--Home Assignments => Web Bonus Problems => Topic started by: Victor Ivrii on February 06, 2015, 09:11:06 AM

Title: Web Bonus Problem 6
Post by: Victor Ivrii on February 06, 2015, 09:11:06 AM
Consider $LX=-X''$ on $\mathbb{R}^+:=\{x:\, x>0\}$ with boundary condition $X'(0)-\alpha X(0)=0$.

a. Find values $\alpha\in \mathbb{R}$ such that there are eigenfunctions. Find corresponding eigenvalues.

b. Find generalized eigenfunctions and the corresponding continuous spectrum.

Remark.

a. Eigenfunctions must belong to $L^2(\mathbb{R}^+)$, that means $\int_0^\infty |X(x)|^2\,dx <\infty$.

b. Generalized eigenfunctions cannot grow exponentially as $x\to +\infty$ but they do not belong $L^2(\mathbb{R}^+)$.