Generalized eigenfunctions and real axis limits of the resolvent

Abstract:

Let $(\mathcal {H},( \cdot , \cdot ))$ be a Hilbert space and A, E be a selfadjoint operator and corresponding spectral measure in $\mathcal {H}\;(A = \smallint \lambda E(d\lambda ))$. It is known that for a suitable positive subspace ${\mathcal {H}_ + } \subset \mathcal {H}$ and measure $\rho$ the generalized eigenfunctions \[ {\phi _{\lambda ,f}} = \lim \limits _{h \to 0} \frac {{E(\left [ {\lambda - h,\lambda + h} \right ])f}}{{\rho (\left [ {\lambda - h,\lambda + h} \right ])}} \equiv \lim \limits _{\Delta \to \lambda } \frac {{E(\Delta )f}}{{\rho (\Delta )}}\] exist in ${\mathcal {H}_ - }$, the corresponding negative space, for $\rho$-almost every $\lambda$ and $f \in {\mathcal {H}_ + }$. It is shown that for each $\lambda$ the ${\phi _{\lambda ,f}}$ form a pre-Hilbert space ${\mathcal {H}_\lambda }$ using the natural inner product ${({\phi _f},{\phi _g})_\lambda } = {\lim _{\Delta \to \lambda }}((E(\Delta )f,g)/\rho (\Delta ))$, and that $\left \| \phi \right \| - \leq C{\left \| \phi \right \|_\lambda }$. Furthermore, if $\{ \phi (\lambda ,\alpha )\}$ is a suitably chosen basis for ${\mathcal {H}_\lambda }, - \infty < \lambda < \infty$, then one obtains the eigenfunction expansion suggested by \[ (f,g) = \int {\rho (d\lambda )\;\sum \limits _{\alpha ,\beta } {(f,\phi (\lambda ,\alpha )){\sigma _{\alpha \beta }}(\lambda )\overline {(g,\phi (\lambda ,\beta )).}} } \]. Finally it is shown that, for a suitable function $w(\varepsilon ,\lambda ),{\phi _{\lambda ,f}}$ is given by ${\lim _{\varepsilon \downarrow 0}}w(\varepsilon ,\lambda )[R(\lambda - i\varepsilon ) - R(\lambda + i\varepsilon )]f$, where $R(z) = {(z - A)^{ - 1}}$.