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Let $\Omega$ be an open region or a non-compact complete manifold, $L$ be an elliptic operator with possibly non vanishing zero-order term, e.g. $-\Delta+q$. Suppose $W$ is an operator such that $W(x)\to0$ as $x\to\infty$.

Q I do not understand the following statements.

  1. $W$ defines a relatively compact perturbation of the operator of $L$.

  2. $L+W$ has the same essential spectrum as $L$, and may have only isolated eigenvalues of finite multiplicity outside the essential spectrum.

PS: I saw the above in the paper: Periodic Nonlinear Schrodinger Equations with Application to Photonic Crystals, A. Pankov(Section 5). The author(Pankov) gave the reference of, Simon's Schrodinger semigroups Theore C3.1, 3.5. But Simon gave the reference of T. Kato's Perturbation of Linear operator, but I did not follow which parts or theorems the authors used in the Book[Perturbation of Linear operator] to show the statements.

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  • $\begingroup$ What do you mean by $W(x) \to 0$ as $x\to \infty$? Is $x$ a point on your manifold or is $x$ a point in the function space? You can find a discussion of the implication 1 $\implies$ 2 in Reed-Simon's "Methods of modern mathematical physics", volume 4, section XIII.4 $\endgroup$ – Willie Wong Apr 20 '18 at 14:05
  • $\begingroup$ As $x\to\infty$, $W(x)$ tends to the zero-operator. $x$ is a point in the base space.manifold. $\endgroup$ – DLIN Apr 21 '18 at 0:37

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