Spectrum of Laplacian on Half line. $\left [0, \infty \right)$

Question

I would like to calculate the spectrum of Dirichlet and Neumann Laplacian of the domain $\left [0,\infty \right)$.

To be precise, Define the Operator $T$ on $L^2\left[0,\infty\right)$ as $Tf=-f''$ and $D(T)=H^2\cap H_0^1$.

And $S$ on $L^2\left[0,\infty\right)$ as $Sf=-f''$ and $D(T)=H^2 \cap \left \{f\in H^2 | f'(0)=0 \right \}$.

Find out the Spectrum of T and S.

Here T is Dirichlet Laplacian and S is Neumann Laplacian.

Answer 1

The operators $S$ and $T$ and selfadjoint on their respective domains. These have to do with even and odd extensions. An even extension of $f \in \mathcal{D}(S)$ is in $H^{2}(\mathbb{R})$, and an odd extension of $f \in \mathcal{D}(T)$ is in $H^{2}(\mathbb{R})$. So the operators $S$ and $T$ can be expressed in terms of the Fourier cosine and sine transforms: $$ Sf = \frac{2}{\pi}\int_{0}^{\infty}s^{2}\cos(sx)\int_{0}^{\infty}f(t)\cos(st)dt\,ds,\\ Tf = \frac{2}{\pi}\int_{0}^{\infty}s^{2}\sin(sx)\int_{0}^{\infty}f(t)\sin(st)dt\,ds. $$ Both have spectrum $[0,\infty)$ with resolvents $$ (S-\lambda I)^{-1}f = \frac{2}{\pi}\int_{0}^{\infty}\frac{1}{s^{2}-\lambda}\cos(sx)\int_{0}^{\infty}f(t)\cos(st)dt\,ds,\\ (T-\lambda I)^{-1}f = \frac{2}{\pi}\int_{0}^{\infty}\frac{1}{s^{2}-\lambda}\sin(sx)\int_{0}^{\infty}f(t)\sin(st)dt\,ds. $$