Semiclassical spectral series of the~Schr\"odinger operator with a~delta~potential on a~straight line and on a~sphere
Teoretičeskaâ i matematičeskaâ fizika, Tome 164 (2010) no. 2, pp. 279-298
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We describe the spectral series of the Schrödinger operator $H=-(h^2/2) \Delta+V(x)+\alpha\delta(x-x_0)$, $\alpha\in\mathbb R$, with a delta potential on the real line and on the three- and two-dimensional standard spheres in the semiclassical limit as $h\to0$. We consider a smooth potential $V(x)$ such that $\lim_{|x|\to\infty}V(x)=+\infty$ in the first case and $V(x)=0$ in the last two cases. In the semiclassical limit in each case, we describe the classical trajectories corresponding to the quantum problem with a delta potential.
Keywords:
semiclassical spectrum, Schrödinger operator, delta potential, Lagrangian manifold, Maslov canonical operator.
@article{TMF_2010_164_2_a7,
author = {T. A. Filatova and A. I. Shafarevich},
title = {Semiclassical spectral series of {the~Schr\"odinger} operator with a~delta~potential on a~straight line and on a~sphere},
journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
pages = {279--298},
publisher = {mathdoc},
volume = {164},
number = {2},
year = {2010},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/TMF_2010_164_2_a7/}
}
TY - JOUR AU - T. A. Filatova AU - A. I. Shafarevich TI - Semiclassical spectral series of the~Schr\"odinger operator with a~delta~potential on a~straight line and on a~sphere JO - Teoretičeskaâ i matematičeskaâ fizika PY - 2010 SP - 279 EP - 298 VL - 164 IS - 2 PB - mathdoc UR - http://geodesic.mathdoc.fr/item/TMF_2010_164_2_a7/ LA - ru ID - TMF_2010_164_2_a7 ER -
%0 Journal Article %A T. A. Filatova %A A. I. Shafarevich %T Semiclassical spectral series of the~Schr\"odinger operator with a~delta~potential on a~straight line and on a~sphere %J Teoretičeskaâ i matematičeskaâ fizika %D 2010 %P 279-298 %V 164 %N 2 %I mathdoc %U http://geodesic.mathdoc.fr/item/TMF_2010_164_2_a7/ %G ru %F TMF_2010_164_2_a7
T. A. Filatova; A. I. Shafarevich. Semiclassical spectral series of the~Schr\"odinger operator with a~delta~potential on a~straight line and on a~sphere. Teoretičeskaâ i matematičeskaâ fizika, Tome 164 (2010) no. 2, pp. 279-298. http://geodesic.mathdoc.fr/item/TMF_2010_164_2_a7/