Weyl theorem

on operator perturbation

The spectrum $\sigma ( T )$ of a (possibly unbounded) self-adjoint operator $T$ is naturally divided into a point spectrum $\sigma _ { p } ( T )$ (i.e., the set of eigenvalues) and a continuous spectrum $\sigma _ { c } ( T )$, both subsets of the real axis (cf. also Spectrum of an operator). The continuous spectrum is very sensitive against perturbations. The Weyl–von Neumann theorem [a2], Sect. X.2.2, states that for any self-adjoint operator $T$ in a separable Hilbert space there exists a (compact) self-adjoint operator $S$ in the Hilbert–Schmidt class (cf. also Hilbert–Schmidt operator) such that $T + S$ has a pure point spectrum. Moreover, $S$ can be chosen arbitrarily small in the Hilbert–Schmidt norm. The set of eigenvalues obtained by perturbation of a continuous spectrum is dense in any interval covered by the continuous spectrum of the unperturbed operator. In spectral theory one introduces therefore another subset of $\sigma ( T )$ which is more stable under perturbations. The essential spectrum $\sigma _ { \text{ess} } ( T )$ is defined as $\sigma ( T ) \backslash \sigma _ { d } ( T )$, where $\sigma _ { d } ( T )$ is the set of all isolated eigenvalues with finite multiplicity. H. Weyl proved (in a special case) the following result [a4], which is now commonly known as Weyl's theorem: Let $T$ be self-adjoint and $S$ symmetric and compact. Then

\begin{equation*} \sigma _ { \text{ess} } ( T ) = \sigma _ { \text{ess} } ( T + S ). \end{equation*}

In fact, it is even sufficient to require the relative compactness of $S$, which means that the operator $S ( T + i ) ^ { - 1 }$ is compact. Another variant of the Weyl theorem states that two self-adjoint operators $A$ and $B$ have the same essential spectrum if the difference of the resolvents $( A + i ) ^ { - 1 } - ( B + i ) ^ { - 1 }$ is compact [a3] (cf. also Resolvent). The Weyl theorem is very important in mathematical quantum mechanics, where it serves to prove that $\sigma _ { ess } ( - \Delta + V ) = [ 0 , \infty )$ for a large class of two-body potentials. (A generalization to the $N$-body problem is the HVZ theorem by W. Hunziker, C. van Winter, and G. Zhislin, see [a3].) The situation is considerably more complicated for non-self-adjoint operators, where there are several possible definitions of the essential spectrum [a1]. In general, one defines the Weyl spectrum as the largest subset of the spectrum that is invariant under compact perturbations.

References

[a1]	K. Gustafson, "Necessary and sufficient conditions for Weyl's theorem" Michigan Math. J. , 19 (1972) pp. 71–81
[a2]	T. Kato, "Perturbation theory for linear operators" , Springer (1976)
[a3]	M. Reed, B. Simon, "Methods in modern mathematical physics IV. Analysis of operators" , Acad. Press (1978)
[a4]	H. Weyl, "Über quadratische Formen, deren Differenz vollstetig ist" Rend. Circ. Mat. Palermo , 27 (1909) pp. 373–392