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 TMF, 1991, Volume 87, Number 3, Pages 323–375 (Mi tmf5495)

Splitting of the lowest energy levels of the Schrödinger equation and asymptotic behavior of the fundamental solution of the equation $hu_t=h^2\Delta u/2-V(x)u$

S. Yu. Dobrokhotov, V. N. Kolokoltsov, V. P. Maslov

Abstract: For the equation $h\partial u/\partial t=h^2\Delta u/2-V(x)u$ with positive potential $V(x)$, global exponential asymptotic behavior of the fundamental solution is obtained by the method of the tunnel canonical operator. In the case of a potential with degenerate points of global minimum, the behavior of the solutions to the Cauchy problem is investigated at times of order $t=h^{-(1+\varkappa)}$, $\varkappa>0$. The developed theory is used to obtain exponential asymptotics of the lowest eigenfunctions of the Schrödinger operator $-h^2\Delta/2-V(x)$ and to estimate the tunnel effect.

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English version:
Theoretical and Mathematical Physics, 1991, 87:3, 561–599

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Citation: S. Yu. Dobrokhotov, V. N. Kolokoltsov, V. P. Maslov, “Splitting of the lowest energy levels of the Schrödinger equation and asymptotic behavior of the fundamental solution of the equation $hu_t=h^2\Delta u/2-V(x)u$”, TMF, 87:3 (1991), 323–375; Theoret. and Math. Phys., 87:3 (1991), 561–599

Citation in format AMSBIB
\Bibitem{DobKolMas91} \by S.~Yu.~Dobrokhotov, V.~N.~Kolokoltsov, V.~P.~Maslov \paper Splitting of the lowest energy levels of the Schr\"odinger equation and asymptotic behavior of the fundamental solution of the equation $hu_t=h^2\Delta u/2-V(x)u$ \jour TMF \yr 1991 \vol 87 \issue 3 \pages 323--375 \mathnet{http://mi.mathnet.ru/tmf5495} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1129671} \zmath{https://zbmath.org/?q=an:0745.35033} \transl \jour Theoret. and Math. Phys. \yr 1991 \vol 87 \issue 3 \pages 561--599 \crossref{https://doi.org/10.1007/BF01017945} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=A1991GW78600001} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. V. P. Belavkin, V. N. Kolokoltsov, “Semiclassical asymptotics of quantum stochastic equations”, Theoret. and Math. Phys., 89:2 (1991), 1127–1138
2. B. Yu. Sternin, V. E. Shatalov, “On the Cauchy problem for differential equations in spaces of resurgent functions”, Russian Acad. Sci. Izv. Math., 40:1 (1993), 67–94
3. S. Yu. Dobrokhotov, V. N. Kolokoltsov, “Splitting amplitudes of the lowest energy levels of the Schrödinger operator with double-well potential”, Theoret. and Math. Phys., 94:3 (1993), 300–305
4. J. Brüning, S. Yu. Dobrokhotov, R. V. Nekrasov, “Splitting of lower energy levels in a quantum double well in a magnetic field and tunneling of wave packets in nanowires”, Theoret. and Math. Phys., 175:2 (2013), 620–636
5. A. Yu. Anikin, “Librations and ground-state splitting in a multidimensional double-well problem”, Theoret. and Math. Phys., 175:2 (2013), 609–619
6. Anikin A.Yu., “Asymptotic Behavior of the Maupertuis Action on a Libration and Tunneling in a Double Well”, Russ. J. Math. Phys., 20:1 (2013), 1–10
7. E. V. Vybornyi, “Tunnel splitting of the spectrum and bilocalization of eigenfunctions in an asymmetric double well”, Theoret. and Math. Phys., 178:1 (2014), 93–114
8. E. V. Vybornyi, “Energy splitting in dynamical tunneling”, Theoret. and Math. Phys., 181:2 (2014), 1418–1427
9. A. Yu. Anikin, S. Yu. Dobrokhotov, M. I. Katsnel'son, “Lower part of the spectrum for the two-dimensional Schrödinger operator periodic in one variable and application to quantum dimers”, Theoret. and Math. Phys., 188:2 (2016), 1210–1235
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