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Transactions of the American Mathematical Society

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Generalized eigenfunctions and real axis limits of the resolvent


Author: N. A. Derzko
Journal: Trans. Amer. Math. Soc. 174 (1972), 489-506
MSC: Primary 47A70
DOI: https://doi.org/10.1090/S0002-9947-1972-0310684-1
MathSciNet review: 0310684
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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

$\displaystyle {\phi _{\lambda ,f}} = \mathop {\lim }\limits_{h \to 0} \frac{{E(... ...hop {\lim }\limits_{\Delta \to \lambda } \frac{{E(\Delta )f}}{{\rho (\Delta )}}$

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\Vert \phi \right\Vert - \leq C{\left\Vert \phi \right\Vert _\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

$\displaystyle (f,g) = \int {\rho (d\lambda )\;\sum\limits_{\alpha ,\beta } {(f,... ...)){\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}}$.

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DOI: https://doi.org/10.1090/S0002-9947-1972-0310684-1
Article copyright: © Copyright 1972 American Mathematical Society

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