Abstract
The existence of a unique solution, in terms of initial data of the hierarchy of quantum kinetic equations with delta potential, has been proven. The proof is based on nonrelativistic quantum mechanics and application of semigroup theory methods.
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References
Albertsson Martin: Analysis of the Many-Body Problem in One Dimension with Repulsive Delta-Function Interaction. Uppsala Universitet, TVE 14 025 juni Examensarbete 15 hp Juni (2014)
Bethe H.A.: On the theory of metals, I. Eigenvalues and eigenfunctions of a linear chain of atoms. Zeits. Phys. 205–226 (1931); Bethe H.A.: Selected Works of Hans A. Bethe With Commentary. World Scientific, Singapour (1996)
Bogolyubov N.N.: Lectures on Quantum Statistics, London (1970); Selected Works, 2 Naukova Dumka, Kiev (1970). [in Russian]
Bogolyubov, N.N.: (Jr): Introduction to Quantum Statitic Mechanics. Nauka, Moscow (1984)
Tracy, C.A., Harold, W.: The dynamics of one-dimentional delta-function Bose gas. J. Phys. A: Math. Theor. 41, 485204 (2008)
Izergin, A.G.: Introduction to the Bethe Ansatz Solvible Models. University di Firenze (2000)
Korepin, V.E., Bogoliubov, N.M., Izergin, A.G.: Quantum Inverse Scattering Method and Correlation Functions. Cambridge University Press, Cambridge (1993)
Lieb, E.H., Liniger, W.: 1963 Exact analysis of an interacting Bose gas. I: the general solution and the ground state. Phys. Rev. 130, 1605–1616 (1963)
Petrina, D.Y.: Mathematical Foundation of Quantum Statistical Mechanics, Continuous Systems. Kluwer Academic Publishers, Dordrecht (1995)
Petrina, D.Y., Enolsky, V.Z.: About the vibrations of one-dimensional systems. Rep. AS UkrSSR A. 8, 756–760 (1976)
Petrina D.Y., Vidybida, A.K.: Couchy problem for kinetic equation of Bogolubov, 370–378, IM AS SSSR, 136, Part 2, 370–378 (1975)
Rasulova, M.Y.: Couchy problem for kinetic equation of Bogolubov. Quantum case. Rep. AS UzSSR 2, 6–9 (1976)
Rasulova, MYu.: The soliton solution of BBGKY’S chain of quantum kinetic equations for bose systems, interacting by delta potential. Rep. Math. Phys. 40, 551–556 (1997)
Rasulova, MYu.: The soliton solution of BBGKY’S chain of quantum kinetic equations for system of particles, interacting by delta potential. Phys. A. 315, 72–78 (2002)
Schlein, B.: Derivation of effective evolution equations from many body quantum dynamics. In: Exner, P. (ed.) Sixteenth International Congress on Mathematical Physics, pp. 406–416. World Scientific, Singapore (2010)
Seringer, R., Yin, J.: The Lieb-Liniger model as a limit of dilute bosons in three dimentions. Commun. Math Phys. 284, 459 (2008). doi:10.1007/s00220-008-0521-6
Spohn, H.: Kinetic equations from Hamiltonian dynamics. Rev. Mod. Phys. 52, 569–615 (1980)
Zakharov, V.E., Shabat, A.B.: Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media. Sov. Phys. JETP 34(1), 62–69 (1972)
Zakharov, V.E., Shabat, A.B.: Interaction between solitons in a stable medium. Sov. Phys. JETP 37(5), 823–828 (1973)
Acknowledgements
Authors are grateful to Prof. H. Spohn for the discussion of results and useful comments.
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Brokate, M., Rasulova, M. (2017). The Solution of the Hierarchy of Quantum Kinetic Equations with Delta Potential. In: Manchanda, P., Lozi, R., Siddiqi, A. (eds) Industrial Mathematics and Complex Systems. Industrial and Applied Mathematics. Springer, Singapore. https://doi.org/10.1007/978-981-10-3758-0_10
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