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

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Teoreticheskaya i Matematicheskaya Fizika, 1991, Volume 87, Number 3, Pages 323–375 (Mi tmf5495)

This article is cited in 18 scientific papers (total in 18 papers)

Splitting of the lowest energy levels of the Schrödinger equation and asymptotic behavior of the fundamental solution of the equation

h u_{t} = h^{2} Δ u / 2 - V (x) u

$hu_t=h^2\Delta u/2-V(x)u$

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

Full-text PDF (5399 kB) Citations (18)

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Abstract: For the equation

h \partial u / \partial t = h^{2} Δ u / 2 - V (x) u

$h\partial u/\partial t=h^2\Delta u/2-V(x)u$ with positive potential

V (x)

$V(x)$ , 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.

Received: 29.12.1990

English version:
Theoretical and Mathematical Physics, 1991, Volume 87, Issue 3, Pages 561–599
DOI: https://doi.org/10.1007/BF01017945

Bibliographic databases:

Language: Russian

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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\pages 323--375

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\jour Theoret. and Math. Phys.

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\crossref{https://doi.org/10.1007/BF01017945}

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Linking options:

https://www.mathnet.ru/eng/tmf5495

https://www.mathnet.ru/eng/tmf/v87/i3/p323

This publication is cited in the following 18 articles:

E. V. Vybornyi, S. V. Rumyantseva, “Tunneling with oscillating effect of ground states of a quadratic operator on a hyperboloid”, Math. Notes, 116:6 (2024), 1233–1248
A. Yu. Anikin, S. Yu. Dobrokhotov, I. A. Nosikov, “Librations with large periods in tunneling: Efficient calculation and applications to trigonal dimers”, Theoret. and Math. Phys., 213:1 (2022), 1453–1476
A. Yu. Anikin, M. A. Vavilova, “Semiclassical asymptotic behavior of the lower spectral bands of the Schrödinger operator with a trigonal-symmetric periodic potential”, Theoret. and Math. Phys., 202:2 (2020), 231–242
Hiromitsu Harada, Amaury Mouchet, Akira Shudo, “Riemann surfaces of complex classical trajectories and tunnelling splitting in one-dimensional systems”, J. Phys. A: Math. Theor., 50:43 (2017), 435204
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
M. V. Karasev, E. M. Novikova, E. V. Vybornyi, “Non-Lie Top Tunneling and Quantum Bilocalization in Planar Penning Trap”, Math. Notes, 100:6 (2016), 807–819
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
E. V. Vybornyi, “Energy splitting in dynamical tunneling”, Theoret. and Math. Phys., 181:2 (2014), 1418–1427
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
A. Yu. Anikin, “Librations and ground-state splitting in a multidimensional double-well problem”, Theoret. and Math. Phys., 175:2 (2013), 609–619
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
Sergey Y. Dobrokhotov, Anatoly Y. Anikin, Nonlinear Physical Systems, 2013, 85
David Holcman, Ivan Kupka, “Singular perturbation for the first eigenfunction and blow-up analysis”, Forum Mathematicum, 18:3 (2006)
V. Sordoni, “Instantons and splitting”, Journal of Mathematical Physics, 38:2 (1997), 770
Ariel Caticha, “Construction of exactly soluble double-well potentials”, Phys. Rev. A, 51:5 (1995), 4264
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
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
V. P. Belavkin, V. N. Kolokoltsov, “Semiclassical asymptotics of quantum stochastic equations”, Theoret. and Math. Phys., 89:2 (1991), 1127–1138

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

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