Abstract
The method of spectral deformation in configuration space, developed in the last chapter, is quite general. It has been applied to a variety of problems. Our main application is to the semiclassical theory of shape resonances. For this, we need to study the behavior of Schrödinger operators under spectral deformations. In this chapter, we first study the effect of local deformations on the Laplacian and its spectrum. We then show that the effect of adding a relatively compact potential, which is the restriction of a function analytic in some neighborhood of ℝn, does not change the essential spectrum. This shows that the hypotheses of the Aguilar-Balslev-Combes-Simon theory are satisfied and opens the way for a study of resonances.
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© 1996 Springer Science+Business Media New York
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Hislop, P.D., Sigal, I.M. (1996). Spectral Deformation of Schrödinger Operators. In: Introduction to Spectral Theory. Applied Mathematical Sciences, vol 113. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-0741-2_18
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DOI: https://doi.org/10.1007/978-1-4612-0741-2_18
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4612-6888-8
Online ISBN: 978-1-4612-0741-2
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