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

This article is cited in 9 scientific papers (total in 9 papers)

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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Received: 29.12.1990

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
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\paper Splitting of the lowest energy levels of the Schr\"odinger equation and asymptotic behavior of the fundamental solution of the equation $hu_t=h^2\Delta u/2-V(x)u$
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\vol 87
\issue 3
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\jour Theoret. and Math. Phys.
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\vol 87
\issue 3
\pages 561--599
\crossref{https://doi.org/10.1007/BF01017945}
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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  mathnet  crossref  mathscinet  isi
    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  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    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  mathnet  crossref  mathscinet  zmath  isi
    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  mathnet  crossref  crossref  zmath  adsnasa  isi  elib  elib
    5. A. Yu. Anikin, “Librations and ground-state splitting in a multidimensional double-well problem”, Theoret. and Math. Phys., 175:2 (2013), 609–619  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    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  crossref  isi
    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  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    8. E. V. Vybornyi, “Energy splitting in dynamical tunneling”, Theoret. and Math. Phys., 181:2 (2014), 1418–1427  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib  elib
    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  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
  • Теоретическая и математическая физика Theoretical and Mathematical Physics
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