The main directions of the research

The aim of the project is a dynamical description of basic phenomena of quantum and statistical physics: long time convergence to the quantum stationary states and thermodynamic equilibrium, wave-particle duality, thermal and electric conductivity of solid state, thermoelectronic emission, quantum scattering and renormalization, formation of nanostructures, and other.

The mathematical tools of the research area are rooted in Complex, Harmonic, and Functional Analysis, Spectral Theory, and Scattering theory for Partial Differential Equations. The physical background is provided by the methods of Quantum Mechanics and Quantum Electrodynamics: the Feynman diagrams, the Feynman functional integral, and renormalization.

This interconnection of several disciplines allows to engage top specialists from adjacent fields, who represent the main body of the group. Their collaboration started in the focused research group of A.I.Komech at the Faculty of Mathematics of the University of Vienna in 2002-2005 and continued in IITP RAN in 2005-2008, BIRS focused research group in May 2007, and Oberwolfach miniworkshop in February 2008.

Dynamical problems of quantum

and statistical physics

Global Attraction to Quantum Stationary

States (QSS)

The concept of QSS has been introduced in 1913 by Nils Bohr for a phenomenological treatment of the Rydberg-Ritz combinational principle for the atomic spectra. In 1926, Schr\"odinger has identified the QSS with the eigenfunctions of the Schr\"odinger operator. However, an inherent dynamical meaning of the QSS is unknown until now.

Possible dynamical treatment of the QSS as global attractor of the corresponding dynamical equations has been suggested first by A. Komech in [40]. Corresponding mathematical results for model systems were obtained in [B2,2,3,4,9,11,25,37,39, 42]. However, for the coupled nonlinear Maxwell-Schroedinger and Maxwell-Dirac equations, the global attraction to QSS is not proved yet.

Wave-Particle Duality

First elementary particle (the electron) has been discovered in 1897 by J.J.Thomson who observed the deflection of the cathode rays in weak electrostatic and magnetic fields.

On the other hand, later theoretical development by L. de Broglie (1923), E. Schroedinger (1926), suggested the wave nature of the electrons. The diffraction of the electrons has been observed experimentally first by Davisson and Germer (1927), and confirmed recently by novel experiments (A. Tonomura, J. Endo, T. Matsuda, T. Kawasaki, American J.Phys. 57 (1989), Issue 2, 117-120).

However a dynamical treatment of the wave-particle duality is missing up to now.

Possible dynamical treatment of the wave-particle duality as global decay to the solitons has been suggested first by A. Komech in [40]. Then the stability of elementary particles EP should correspond to asymptotic stability of the solitons, and the particle like deflection of the EP in the weak external fields should correspond to the adiabatic effective dynamics of the solitons in slowly varying external potentials.

First mathematical results for model systems were obtained

i) in [20,21,26,27,30,36,41] on the global decay to the solitons;

ii) in [1,5,12,13] on asymptotic stability of the solitons;

iii) in [38] on adiabatic effective dynamics of the solitons in slowly varying external potentials.

However, for the coupled nonlinear Maxwell-Schroedinger and Maxwell-Dirac equations, the global decay, asymptotic stability and adiabatic effective dynamics are not proved yet.

Electron Beams

We are interested in the Quantum Physics description of processes in klystron, synchrotron, and electron microscope.

Photoeffect

Solid State Physics

Problems of Thermodynamics: black body emission law, thermionic emission, thermoelectric effect, Ohm law.

Nanostructures

Books and Lecture notes

B1. A. Komech, Lectures on Elliptic Partial Differential Equations (Method of Pseudodifferential Operators), undergraduate course given at Vienna University during October-December 2006.

B2. A. Komech, On Global Attractors of Hamilton Nonlinear Wave Equations, Lecture Notes of the Max Planck Institute for Mathematics in the Sciences, LN 24/2005, Leipzig, 2005. http://www.mis.mpg.de/preprints/ln/lecturenote-2405-abstr.html

B3. A. Komech, Lectures on Quantum Mechanics (nonlinear PDE point of view), Lecture Notes of the Max Planck Institute for Mathematics in the Sciences, LN 25/2005, Leipzig, 2005. http://www.mis.mpg.de/preprints/ln/lecturenote-2505-abstr.html

B4. A.I. Komech, A.A. Komech, Book of Practical PDEs, 2006.

B5. A. Komech, Practical Solution of Equations of Mathematical Physics, 2006 (revised edition) [Russian].

Papers

  1. A.I. Komech, E. Kopylova, D. Stuart, On asymptotic stability of solitary waves for Sch\"odinger equation coupled to nonlinear oscillator, II, submitted to \em SIAM J. Math. Analysis, 2008. arXiv:0807.1878
  2. A.I. Komech, A.A. Komech, On global attraction to solitary waves for the Klein-Gordon field coupled to several nonlinear oscillators, submitted to Acta Mathematica, 2007.arXiv:math/0702660
  3. A.I. Komech, A.A. Komech, On global attraction to solitary waves with mean field interaction Klein-Gordon equation, accepted in Annales l'IHP ANL , 2008. arXiv:math/0708.1131
  4. A.I. Komech, A.A. Komech, Global Attraction to Solitary Waves in Models Based on the Klein-Gordon Equation , SIGMA, Symmetry Integrability Geom. Methods Appl. 4 (2008), Paper 010, 23 pages, electronic only. http://www.emis.de/journals/SIGMA/2008/ arXiv:math/0711.0041
  5. V. Buslaev, A. Komech, E. Kopylova, D. Stuart, On asymptotic stability of solitary waves in nonlinear Schrödinger equation, Comm. Partial Diff. Eqns 33 (2008), no. 4, 669-705. arXiv:math-ph/0702013
  6. A. Komech, E. Kopylova, B. Vainberg, On dispersive properties of discrete 2D Schr\"odinger and Klein-Gordon equations , J. Funct. Anal. 254 (2008), no. 8, 2227-2254.
  7. A.I. Komech, A.A. Komech, Global well-posedness for the Schrodinger equation coupled to a nonlinear oscillator, Russ. J. Math. Phys. 14 (2007), no. 2, 164-173. math.AP/0608780
  8. A.I. Komech, A.E. Merzon, Relation between Cauchy data in the scattering by wedge , Russ. J. Math. Phys. 14 (2007), no. 3, 279-303.
  9. A.I. Komech, A.A. Komech, Global attractor for a nonlinear oscillator coupled to the Klein-Gordon field, Arch. Rat. Mech. Anal.185 (2007), 105-142. arXiv:math.AP/0609013
  10. A. Komech, E. Kopylova, M. Kunze, Dispersive estimates for 1D discrete Schrö dinger and Klein-Gordon equations, Applicable Analysis 85 (2006), no. 12, 1487-1508.
  11. A.I. Komech, A.A. Komech, On global attraction to solitary waves for the Klein-Gordon equation coupled to nonlinear oscillator, C. R., Math., Acad. Sci. Paris 343 (2006), no. 2, 111-114.
  12. V.Imaikin, A. Komech, B. Vainberg, On scattering of solitons for the Klein-Gordon equation coupled to a particle, Comm. Math. Phys. 268 (2006), no. 2, 321-367. arXiv:math.AP/0609205
  13. A. Komech, E.A. Kopylova, Scattering of solitons for Schrödinger equation coupled to a particle, Russian J. Math. Phys. 50 (2006), no. 2, 158-187. arXiv:math.AP/0609649
  14. M. Freidlin, A. Komech, On metastable regimes in stochastic Lamb system, Journal of Mathematical Physics 47 (2006), 043301-1 -- 043301-12.
  15. A. Komech, A.E.Merzon, Limiting amplitude principle in the diffraction by wedges, Mathematical Methods in Applied Sciences 29 (2006), 1147-1185.
  16. T. Dudnikova, A. Komech, Two-temperature problem for the Klein-Gordon equation, J. Theory Probability and Appl. 50 (2005), no. 4, 675-710. [Russian]. English translation: On two-temperature problem for the Klein-Gordon equation, Theory Prob. Appl. 50 (2006), no. 4, 582-611.))
  17. A. Komech, E.Kopylova, N.Mauser, On convergence to equilibrium distribution for Schrödinger equation, Markov Processes and Related Fields 11 (2005), no. 1, 81-110.
  18. T. Dudnikova, A. Komech, On the convergence to a statistical equilibrium in the crystal coupled to a scalar field, Russ. J. Math. Phys. 12 (2005), no. 3, 301-325.
  19. A. Komech, N.J. Mauser, A.E. Merzon, On Sommerfeld representation and uniqueness in diffraction by wedges, Mathematical Methods in Applied Sciences 28 (2005), no. 2, 147-183.
  20. A. Komech, N.J. Mauser, A. Vinnichenko, On attraction to solitons in relativistic nonlinear wave equations, Russ. J. Math. Phys. 11 (2004), no. 3, 289-307.
  21. V.Imaikin, A. Komech, N.J. Mauser, Soliton-type asymptotics for the coupled Maxwell-Lorentz equations, Ann. Inst. Poincaré, Phys. Theor. 5 (2004), 1117-1135.
  22. V.Imaikin, A. Komech, H.Spohn, Rotating charge coupled to the Maxwell field: scattering theory and adiabatic limit, Monatshefte fuer Mathematik 142 (2004), no. 1-2, 143-156.
  23. T.Dudnikova, A. Komech, N.Mauser, On two-temperature problem for harmonic crystals , Journal of Statistical Physics 114 (2004), no. 3/4, 1035-1083.
  24. A. Komech, E.Kopylova, N.Mauser, On convergence to equilibrium distribution for wave equation in even dimensions, Ergodic Theory and Dynamical Systems 24 (2004), 1-30.
  25. A. Komech, On attractor of a singular nonlinear U(1)-invariant Klein-Gordon equation , p. 599-611 in: Proceedings of the 3rd ISAAC Congress, Freie Universitat Berlin, Berlin, 2003.
  26. V.Imaikin, A. Komech, H.Spohn, Scattering theory for a particle coupled to a scalar field, Journal of Discrete and Continuous Dynamical Systems 10 (2003), no. 1&2, 387-396.
  27. V.Imaikin, A. Komech, P.Markowich, Scattering of solitons of the Klein-Gordon equation coupled to a classical particle, Journal of Mathematical Physics 44 (2003), no. 3, 1202-1217.
  28. T.Dudnikova, A. Komech, H.Spohn, On convergence to statistical equilibrium for harmonic crystals, Journal of Mathematical Physics 44 (2003), no. 6, 2596-2620.
  29. T.Dudnikova, A. Komech, N.Mauser, On the convergence to a statistical equilibrium for the Dirac equation, Russian Journal of Math. Phys. 10 (2003), no. 4, 399-410.
  30. A. Bensoussan, C. Iliine, A. Komech, Breathers for a relativistic nonlinear wave equation, Arch. Rat. Mech. Anal. 165 (2002), 317-345.
  31. T.V. Dudnikova, A.I. Komech, E.A. Kopylova, Yu.M. Suhov, On convergence to equilibrium distribution, I. Klein-Gordon equation with mixing, Comm. Math. Phys. 225 (2002), no. 1, 1-32.
  32. T.V. Dudnikova, A.I. Komech, N.E. Ratanov, Yu.M. Suhov, On convergence to equilibrium distribution, II. Wave equation with mixing, Journal of Statistical Physics 108 (2002), no. 4, 1219-1253.
  33. T. Dudnikova, A. Komech, H. Spohn, On a two-temperature problem for wave equation with mixing, Markov Processes and Related Fields 8 (2002), no. 1, 43-80.
  34. A.Merzon, A. Komech, P.Zhevandrov, A method of complex characteristics for elliptic problems in angles and its applications, Translations. Series 2. American Mathematical Society. 206, American Mathematical Society (AMS), Providence, RI, 2002.
  35. T. Dudnikova, A. Komech, H. Spohn, Energy-momentum relation for solitary waves of relativistic wave equation, Russian Journal Math. Phys. 9 (2002), no. 2, 153-160.
  36. V.Imaikin, A. Komech, H.Spohn, Soliton-like asymptotics and scattering for a particle coupled to Maxwell field, Russian Journal of Mathematical Physics 9 (2002), no. 4, 428-436.
  37. A. Komech, On transitions to stationary states in one-dimensional nonlinear wave equations, Arch. Rat. Mech. Anal. 149 (1999), no. 3, 213-228.
  38. A. Komech, M. Kunze, H. Spohn, Effective Dynamics for a mechanical particle coupled to a wave field, Comm. Math. Phys. 203 (1999), 1-19.
  39. A. Komech, P. Joly, O. Vacus, On transitions to stationary states in a Maxwell-Landau- Lifschitz-Gilbert system, SIAM J. Math. Anal. 31 (1999), no. 2, 346-374.
  40. A. Komech, On transitions to stationary states in Hamiltonian nonlinear wave equations, Phys. Letters A 241 (1998), 311-322.
  41. A. Komech, H. Spohn, Soliton-like asymptotics for a classical particle interacting with a scalar wave field, Nonlinear Analysis 33 (1998), no. 1, 13-24.
  42. A. Komech, On stabilization of string-nonlinear oscillator interaction, J. Math. Anal. Appl. 196 (1995), 384-409.