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28 March 2024 |
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Resonances - lost and found | Richard Froese
; Ira Herbst
; | Date: |
9 Mar 2017 | Abstract: | We consider the large $L$ limit of one dimensional Schr"odinger operators
$H_L=-d^2/dx^2 + V_1(x) + V_{2,L}(x)$ in two cases: when $V_{2,L}(x)=V_2(x-L)$
and when $V_{2,L}(x)=e^{-cL}delta(x-L)$. This is motivated by some recent work
of Herbst and Mavi where $V_{2,L}$ is replaced by a Dirichlet boundary
condition at $L$. The Hamiltonian $H_L$ converges to $H = -d^2/dx^2 + V_1(x)$
as $L o infty$ in the strong resolvent sense (and even in the norm resolvent
sense for our second case). However, most of the resonances of $H_L$ do not
converge to those of $H$. Instead, they crowd together and converge onto a
horizontal line: the real axis in our first case and the line $Im(k)=-c/2$ in
our second case. In the region below the horizontal line resonances of $H_L$
converge to the reflectionless points of $H$ and to those of $-d^2/dx^2 +
V_2(x)$. It is only in the region between the real axis and the horizontal line
(empty in our first case) that resonances of $H_L$ converge to resonances of
$H$. Although the resonances of $H$ may not be close to any resonance of $H_L$
we show that they still influence the time evolution under $H_L$ for a long
time when $L$ is large. | Source: | arXiv, 1703.3172 | Services: | Forum | Review | PDF | Favorites |
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