Wigner Measure and Semiclassical Limits of Nonlinear Schrodinger Equations

The classical WKB method Wigner measure The limit from the one-dimensional Schrodinger-Poisson to Vlasov-Poisson equations Semiclassical limit of Schrodinger-Poisson equations Semiclassical limit of the cubic Schrodinger equation in an exterior domain Incompressible and compressible limits of coupled systems of nonlinear Schrodinger equations High-frequency limit of the Helmholtz equation Global solutions to (3.14) Denseness of polynomials Global existence of a solution to (5.1) Global smooth solution to (6.1) Bibliography.

[1]  E. Stein,et al.  Introduction to Fourier Analysis on Euclidean Spaces. , 1971 .

[2]  C. D. Levermore,et al.  Singular limits of dispersive waves , 1994 .

[3]  Ping Zhang,et al.  The limit from the Schrödinger‐Poisson to the Vlasov‐Poisson equations with general data in one dimension , 2002 .

[4]  J. Cooper.  SINGULAR INTEGRALS AND DIFFERENTIABILITY PROPERTIES OF FUNCTIONS , 1973 .

[5]  N. Mauser,et al.  The Selfconsistent Pauli Equation , 2001 .

[6]  L. Tartar.  H-measures, a new approach for studying homogenisation, oscillations and concentration effects in partial differential equations , 1990, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[7]  Xue Ping Wang.  MICROLOCAL ESTIMATES OF THE STATIONARY SCHR¨ ODINGER EQUATION IN SEMI-CLASSICAL LIMIT , 2007 .

[8]  Luis Vega,et al.  Morrey–Campanato Estimates for Helmholtz Equations , 1999 .

[9]  M. Puel.  CONVERGENCE OF THE SCHRÖDINGER–POISSON SYSTEM TO THE INCOMPRESSIBLE EULER EQUATIONS , 2002 .

[10]  Víctor M Pérez-García,et al.  Feshbach resonance induced shock waves in Bose-Einstein condensates. , 2004, Physical review letters.

[11]  E. Mourre.  Operateurs conjugués et propriétés de propagation , 1983 .

[12]  Reinhard Illner,et al.  Global existence, uniqueness and asymptotic behaviour of solutions of the Wigner–Poisson and Schrödinger‐Poisson systems , 1994 .

[13]  L. Hörmander.  The analysis of linear partial differential operators , 1990 .

[14]  J. Mayer,et al.  On the Quantum Correction for Thermodynamic Equilibrium , 1947 .

[15]  A. I. Vol'pert,et al.  Analysis in classes of discontinuous functions and equations of mathematical physics , 1985 .

[16]  Ping Zhang,et al.  Wigner Measure and the Semiclassical Limit of Schrödinger-Poisson Equations , 2002, SIAM J. Math. Anal..

[17]  A. I. Vol'pert.  THE SPACES BV AND QUASILINEAR EQUATIONS , 1967 .

[18]  Ping Zhang,et al.  Incompressible and Compressible Limits of Coupled Systems of Nonlinear Schrödinger Equations , 2006 .

[19]  S. Schochet.  The compressible Euler equations in a bounded domain: Existence of solutions and the incompressible limit , 1986 .

[20]  Norbert J. Mauser,et al.  THE CLASSICAL LIMIT OF A SELF-CONSISTENT QUANTUM-VLASOV EQUATION IN 3D , 1993 .

[21]  Ping Zhang,et al.  On the weak solutions to a shallow water equation , 2000 .

[22]  Jack Schaeffer,et al.  Global existence of smooth solutions to the vlasov poisson system in three dimensions , 1991 .

[23]  André Martinez,et al.  An Introduction to Semiclassical and Microlocal Analysis , 2002 .

[24]  I. Segal.  Non-Linear Semi-Groups , 1963 .

[25]  On smooth solutions to the initial-boundary value problem for the nonlinear Schro¨dinger equation in two space dimensions , 1989 .

[26]  Pierre-Louis Lions,et al.  Propagation of moments and regularity for the 3-dimensional Vlasov-Poisson system , 1991 .

[27]  Tosis Kato,et al.  Schrödinger operators with singular potentials , 1972 .

[28]  E. Mourre.  Absence of singular continuous spectrum for certain self-adjoint operators , 1981 .