Geodesic
On the one-dimensional Schrödinger equation with a perturbed non-local steplike potential
Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta, no. 1 (2004), pp. 95-108 
N. I. Pletnikova. On the one-dimensional Schrödinger equation with a perturbed non-local steplike potential. Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta, no. 1 (2004), pp. 95-108. http://geodesic.mathdoc.fr/item/IIMI_2004_1_a4/
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     author = {N. I. Pletnikova},
     title = {On the one-dimensional {Schr\"odinger} equation with a perturbed non-local steplike potential},
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     pages = {95--108},
     year = {2004},
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     url = {http://geodesic.mathdoc.fr/item/IIMI_2004_1_a4/}
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Voir la notice de l'article provenant de la source Math-Net.Ru

We consider a Schrödinger operator of the form $H=-\tfrac{d^2}{dx^2}+V$ acting in $L^2(R)$ where $V=V_0\theta (x)+\varepsilon (\cdot ,\varphi _0) \varphi _0$ is non-local potential. We prove that the unique level (i.e. eigenvalue or resonance of the operator $H$) exists for all sufficiently small $\varepsilon $ and $V_0=V_0(\varepsilon)$. We investigate the asymptotic behaviour of this level. (If $V_0(\varepsilon)$ is separated from zero the levels are absent.) We study the asymptotic behaviour of eigenfunctions for $|x|\to \infty$.

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