PARALLEL METHODS FOR THE SCHROEDINGER EQUATION

WE WILL DEVELOP WAVELET-BASED DOMAIN DECOMPOSITION PROCEDURES FOR THE SOLUTION OF THE INITIAL VALUE PROBLEM FOR THE SCHROEDINGER EQUATION ON MULTIDIMENSIONAL SPATIAL DOMAINS. ACCURATE AND PRACTICAL NUMERICAL CALCULATIONS WOULD BE OF GREAT USE IN THE IMPLEMENTATION AND EXTENSION OF THE BORN-OPPENHEIMER PROCEDURE. BEING BOTH COMPACTLY SUPPORTED AND ORTHOGONAL, WAVELETS COMBINE THE ADVANTAGES OF FINITE ELEMENT, SPLINE AND FOURIER SPECTRAL METHODS. THE SYMMETRIES OF OPERATORS AND OMAINS ARE PRESERVED BY OUR METHOD. FOR DOMAINS WITH A HIGH LEVEL OF SYMMETRY, THIS ALLOWS THE ACCURATE NUMERICAL RESOLUTION OF DISCRETELY ORTHOGONAL SPACES THAT BLOCK DIAGONALIZE THE OPERATOR IN THESE DOMAINS. THESE RESULTS HAVE INTERESTING APPLICATIONS TO DOMAIN DECOMPOSITION, SINCE THEY ALLOW THE DEFINITION OF HIGHLY ACCURATE WAVELET ELEMENTS THAT SHOULD BE SIMPLE TO MATCH ACROSS ELEMENT BOUNDARIES. THE COUPLING OF SOLUTION AND BOUNDARY CONDITION IMPLIED BY OUR METHOD WILL LEAD TO A SUBSTANTIAL IMPROVEMENT IN SPEED AND ACCURACY THAT WILL SCALE WITH INCREASES IN THE SIZE AND COMPLEXITY OF A PROBLEM.