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I have encountered the following functional analysis question, which I can't figure out how to prove: Prove that if $H:= - \bar{ \Delta } +V $ on $ L^2 (\mathbb{R}^n) $ , and $lim_{|x| \to \infty} V(x) = + \infty $ , then $H$ has compact resolvent.

Can someone help me figure out how to prove this exercise?

Thanks

( $H:= - \bar{ \Delta } $ stands for the closure of the laplacian)

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    On page 164 Davies says "We assume that the potentials $V$ satisfy one of the hypotheses of Theorem 8.2.1, Theorem 8.2.3 or Example 8.2.7". It's natural for us to apply the same assumptions to the exercises... By the way the subsequent proof of Lemma 8.3.3 is worded thus: "We observe that $Q(f)\ge 0$ for all $f\in C_c^\infty$ and that such $f$ form a core for $Q$, for reasons which depend upon which assumption we make on $V$. The lemma now follows by an application of the variational formula. QED" A bit on the side of a sketch..2012-07-15

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You want to show that the essential spectrum of $H$ is empty. Suppose it contains some number $\lambda\in\mathbb R$. Lemma 8.4.1 provides us, for every $\epsilon>0$, with an infinite-dimensional space $L_\epsilon $ in which $\|Hf-\lambda f\|\le \epsilon \|f\|$. Taking the inner product of $Hf-\lambda f$ with $f$, we obtain $(1)\qquad \qquad \int (|\nabla f|^2+(V-\lambda)|f|^2)\le \epsilon\int |f|^2,\qquad f\in L_\epsilon$ Now split $V-\lambda-\epsilon$ into the positive and negative parts, $V-\lambda-\epsilon=V_+-V_-$. The important thing is that $V_-$ is compactly supported. Rewriting (1) as $(2)\qquad\qquad \int (|\nabla f|^2+V_+|f|^2)\le \int V_-|f|^2,\qquad f\in L_\epsilon$ we run into trouble because $-\Delta$ has discrete spectrum on bounded domains (Theorem 6.2.3).

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    It was indeed a bit stupid to confuse the resolvent operator and the resolvent set , and to forget about the integration by parts formula. Thanks for explaining it to me! As for (3) - I've re read chapter 4 and went over your answer again, but I still can't understand why the ineqaulity you received implies you don't have discrete spectrum. Will you help me ? (Even if you'll be able to tell me more specifically what should read again, or retry to understand, it'll be very helpful! This entire subject is still very vague to me) Looking forward for your help! Thanks2012-07-16