most recent 30 from http://mathoverflow.net 2013-05-21T23:07:46Z http://mathoverflow.net/feeds/question/102464 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/102464/discrete-spectrum-of-laplacian-and-the-associated-quadratic-form Jason Mraz 2012-07-17T16:37:01Z 2012-07-17T16:37:01Z

<p>Given the following inequality: $ \int (|\nabla f|^2+V_+|f|^2)\le \int V_-|f|^2,$ , where $ V_- , V_+$ are the negative and positive parts of the potential, does this imply that the spectrum of the laplacian on the domain of integration isn't discrete? </p> <p>If so, can someone explain me the reason for that? </p> <p>I found very similar conclusion in SE: <a href="http://math.stackexchange.com/questions/170507/proving-compactness-of-resolvent-of-an-operator" rel="nofollow">http://math.stackexchange.com/questions/170507/proving-compactness-of-resolvent-of-an-operator</a> (see (2) there), but I can't understand why this is true... </p> <p>thanks all</p>