tags: 224a

Lecture 18: Unbounded Spectral Theorem, Schrodinger Operators

Unbounded Spectral theorem

$$
Let

$A:D(A)\to A$ be selfadjoint. The key observation is that the characterization in the previous lecture implies

$ran(A+i)=H$, which means that

$C_A:=(A+i{)}^{-1}\in L(H)$. We further observe

$C_A$ is unitary and therefore normal. Therefore, by the spectral theorem for normal operators, there is a unitary

$U:H\to L^2(X,\mu )$ and a bounded multiplication operator

$M_h$ with

$g:X\to \mathbb{C}$ such that

$$U^{-1}{C}_{A}U=M_{h}on{L}^2(X,\mu ).$$
The change of variables

$g=h^{-1}-i$ yields

$UA{U}^{-1}=M_g$ on

$L^2(X,\mu )$, after verifying that

$h\ne 0$ a.e. and

$g$ is real.

The full proof is given in Reed and Simon VII.3.

Perturbation Theory and Schrodinger Operators

We showed that a large class of perturbations preserves selfadjointness, and used this to show that the Schrodinger operator with certain potentials (Coulomb and Yukawa) are selfadjoint. We sketched a proof that these perturbations retain some important features of the Laplacian, namely that the spectrum is bounded below and that the negative spectrum is pure point.

Details are given in Dimock 4.1-4.4.