Self-adjoint approximations of the degenerate Schrödinger operator

Research Articles

Abstract

The problem of construction a quantum mechanical evolution for the Schrödinger equation with a degenerate Hamiltonian which is a symmetric operator that does not have selfadjoint extensions is considered. Self-adjoint regularization of the Hamiltonian does not lead to a preserving probability limiting evolution for vectors from the Hilbert space but it is used to construct a limiting evolution of states on a C*-algebra of compact operators and on an abelian subalgebra of operators in the Hilbert space. The limiting evolution of the states on the abelian algebra can be presented by the Kraus decomposition with two terms. Both of these terms are corresponded to the unitary and shift components of Wold’s decomposition of isometric semigroup generated by the degenerate Hamiltonian. Properties of the limiting evolution of the states on the C*-algebras are investigated and it is shown that pure states could evolve into mixed states.

Key words

the degenerate Schrödinger operator C*-algebra pure and mixed states 

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References

  1. 1.
    G. Fichera, “On a unified theory of boundary value problems for elliptic-parabolic equations of second order,” Boundary Problems in Differential Equations, 97–120 (Univ. of Wisconsin Press, Madison, 1960).MATHGoogle Scholar
  2. 2.
    O. A. Oleinik and E. V. Radkevich, “Second order equations with nonnegative characteristic form,” Itogi Nauki, Ser. Mat.,Mat. Anal. 1969, 7–252 (1971) [in Russian].MathSciNetMATHGoogle Scholar
  3. 3.
    A. D. Ventcel’ and M. I. Freidlin, “Some problems concerning stability under small random perturbations,” Theory Probab. Appl. 17 (2), 269–283 (1973).CrossRefMATHGoogle Scholar
  4. 4.
    I. V. Volovich and V. Zh. Sakbaev, “Universal boundary value problem for equations ofmathematical physics,” Proc. Steklov Inst. Math. 285 (1), 56–80 (2014).MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    V. Zh. Sakbaev and O.G. Smolyanov, “Diffusion and quantum dynamics of particles with position-dependent mass,” DokladyMath. 86 (1), 460–463 (2012).MathSciNetMATHGoogle Scholar
  6. 6.
    V. Zh. Sakbaev, “Averaging of quantum dynamical semigroups,” Theor. Math. Phys. 164 (3), 1215–1221 (2010).CrossRefMATHGoogle Scholar
  7. 7.
    V. Zh. Sakbaev, “On dynamics of quantum states generated by the Cauchy problem for the Schroedinger equation with degeneration on the half-line, J. Math. Sci. 151 (1), 2741–2753 (2008).Google Scholar
  8. 8.
    V. Zh. Sakbaev, “On the Cauchy problem for linear differential equation with degeneration and the averaging of its regularization,” Sovrem. Mat. Fundam.Napravl. 43, 3–174 (2012).Google Scholar
  9. 9.
    M. Gadella, S. Kuru and J. Negro, “Self-adjoint Hamiltonians with a mass jump: General matching conditions,” Phys. Lett. A 362, 265–268 (2007).MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    M. Gadella and O. G. Smolyanov, “Feynman formulas for particles with position-dependent mass,” Doklady Math. 77 (1), 120–123 (2007).MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    V. L. Chernyshev, A. A. Tolchennikov and A. I. Shafarevich, “Behavior of quasi-particles on hybrid spaces. Relations to the geometry of geodesics and to the problems of analytic number theory,” Regul. Chaotic Dyn. 21 (5), 531–537 (2016).MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Di Perna and P. Lions, “Ordinary differential equation, transport theory and Sobolev spaces,” Invent. Math. V 98), 511–547 (1989).MathSciNetCrossRefGoogle Scholar
  13. 13.
    M. Ohya and I. V. Volovich, Mathematical Foundations of Quantum Information and Computation and its Applications to Nano- and Bio-Systems (Springer, Dordrecht, 2011).CrossRefMATHGoogle Scholar
  14. 14.
    L. Accardi, Yu. G. Lu and I. V. Volovich, Quantum Theory and its Stochastic Limit (Springer, 2002).CrossRefMATHGoogle Scholar
  15. 15.
    A. S. Trushechkin and I. V. Volovich, “Perturbative treatment of inter-site couplings in the local description of open quantum networks,” EPL 113 (3), 30005 (2016).CrossRefGoogle Scholar
  16. 16.
    I. Ya. Aref’eva, I. V. Volovich and S. V. Kozyrev, “Stochastic limit method and interference in quantum manyparticle systems,” Theor. Math. Phys. 183 (3), 782–799 (2015).MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    I. V. Volovich and S. V. Kozyrev, “Manipulation of states of a degenerate quantum system,” Proc. Steklov Inst.Math. 294, 241–251 (2016).CrossRefGoogle Scholar
  18. 18.
    O. Bratteli and D. W. Robinson, Operator Algebras and Quantum Statistical Mechanics (Springer-Verlag, 1979).CrossRefMATHGoogle Scholar
  19. 19.
    M. Takesaki, “On the conjugate space of operator algebra,” TohikuMath. J. 10, 194–203 (1958).MathSciNetMATHGoogle Scholar
  20. 20.
    W. I. M. Wils, “Stone-Cech compactification and representations of operator algebras,” http://hdl.handle.net/2066/107571.Google Scholar
  21. 21.
    V. Zh. Sakbaev, “On the variational description of the trajectories of averaging quantum dynamical maps,” p-Adic Numbers Ultrametric Anal. Appl. 4 (2), 120–134 (2012).MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    G. F. Dell’ Antonio, “On the limits of sequences of normal states,” Comm. Pure Appl. Math. 20, 413–429 (1967).MathSciNetCrossRefGoogle Scholar
  23. 23.
    B. Sz.-Nagy and C. Foias, Analyse harmonique des operateurs de l’espace de Hilbert (1967).MATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2017

Authors and Affiliations

  1. 1.Steklov Mathematical InstituteRASMoscowRussia

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