Computation of the Semiclassical Limit of the Schrödinger Equation with Phase Shift by a Level Set Method

In this paper, we show how the level set method, developed in [Cheng, Liu and Osher, (2003). Comm. Math. Sci. 1(3), 593–621; Jin, Liu, Osher and Tsai, (2005). J. comp. Phys. 205, 222–241; Jin and Osher, (2003). Comm. Math. Sci. 1(3), 575–591] for the numerical computation of the semiclassical limit of the Schrödinger equation, can be amended to include the phase shift using the Keller-Maslov index. This gives a more accurate approximation of the physical observables for multivalued solutions in the semiclassical limit. Numerical examples in one and two spaces dimensions demonstrate the improved accuracy of our approach away from caustics.

[1]  S. Osher,et al.  A level set-based Eulerian approach for anisotropic wave propagation , 2003 .

[2]  Y. Egorov,et al.  Fourier Integral Operators , 1994 .

[3]  S. Osher,et al.  COMPUTATIONAL HIGH-FREQUENCY WAVE PROPAGATION USING THE LEVEL SET METHOD, WITH APPLICATIONS TO THE SEMI-CLASSICAL LIMIT OF SCHRÖDINGER EQUATIONS∗ , 2003 .

[4]  Peter A. Markowich,et al.  A Wigner-Measure Analysis of the Dufort-Frankel Scheme for the Schrödinger Equation , 2002, SIAM J. Numer. Anal..

[5]  P. Souganidis,et al.  Convergence of MUSCL and filtered schemes for scalar conservation laws and Hamilton-Jacobi equations , 1995 .

[6]  B. Engquist,et al.  Multi-phase computations in geometrical optics , 1996 .

[7]  S. Osher,et al.  Regular Article: A PDE-Based Fast Local Level Set Method , 1999 .

[8]  Chohong Min,et al.  Simplicial isosurfacing in arbitrary dimension and codimension , 2003 .

[9]  Shi Jin,et al.  Multi-phase computations of the semiclassical limit of the Schrödinger equation and related problems: Whitham vs Wigner , 2003 .

[10]  Stanley Osher,et al.  Geometric Optics in a Phase-Space-Based Level Set and Eulerian Framework , 2002 .

[11]  Y. Brenier,et al.  A kinetic formulation for multi-branch entropy solutions of scalar conservation laws , 1998 .

[12]  S. Osher,et al.  A LEVEL SET METHOD FOR THE COMPUTATION OF MULTIVALUED SOLUTIONS TO QUASI-LINEAR HYPERBOLIC PDES AND HAMILTON-JACOBI EQUATIONS , 2003 .

[13]  P. Markowich,et al.  On time-splitting spectral approximations for the Schrödinger equation in the semiclassical regime , 2002 .

[14]  Laurent Gosse,et al.  TWO MOMENT SYSTEMS FOR COMPUTING MULTIPHASE SEMICLASSICAL LIMITS OF THE SCHRÖDINGER EQUATION , 2003 .

[15]  J. Benamou,et al.  A geometrical optics-based numerical method for high frequency electromagnetic fields computations near fold caustics--Part I , 2003 .

[16]  S. Osher,et al.  A PDE-Based Fast Local Level Set Method 1 , 1998 .

[17]  L. Gosse.  Using K-Branch Entropy Solutions for Multivalued Geometric Optics Computations , 2002 .

[18]  Laurent Gosse,et al.  Numerical aspects of nonlinear Schrodinger equations in the presence of caustics , 2007 .

[19]  Stanley Osher,et al.  A level set method for the computation of multi-valued solutions to quasi-linear hyperbolic PDE's and Hamilton-Jacobi equations , 2003 .

[20]  Peter A. Markowich,et al.  Numerical approximation of quadratic observables of Schrödinger-type equations in the semi-classical limit , 1999, Numerische Mathematik.

[21]  T. Paul,et al.  Sur les mesures de Wigner , 1993 .

[22]  G. M.,et al.  Partial Differential Equations I , 2023, Applied Mathematical Sciences.

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

[24]  P. Markowich,et al.  Quantum hydrodynamics, Wigner transforms, the classical limit , 1997 .

[25]  L. Hörmander.  Fourier integral operators. I , 1995 .

[26]  P. Markowich,et al.  Homogenization limits and Wigner transforms , 1997 .

[27]  Stanley Osher,et al.  Computing multivalued physical observables for the semiclassical limit of the Schrödinger equation , 2005 .

[28]  Jean-David Benamou,et al.  A geometric optics method for high-frequency electromagnetic fields computations near fold caustics: part II. the energy , 2004 .

[29]  Stanley Osher,et al.  Numerical solution of the high frequency asymptotic expansion for the scalar wave equation , 1995 .

[30]  S. Orszag,et al.  Advanced Mathematical Methods For Scientists And Engineers , 1979 .

[31]  William E. Schiesser,et al.  Linear and nonlinear waves , 2009, Scholarpedia.