RESEARCH ON FUNCTIONAL ANALYTICAL FOUNDATION OF QUANTUM MECHANICS RELATING TO CANONICAL COMMUTATION RELATIONS

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(1) I DEALT WITH AN OPERATOR COMPOSED OF A DIFFERENTIAL OPERATOR AND A SINGULAR TERM. I CONSTRUCTED A TRANSFORM, WHICH CONVERTS THE OPERATOR INTO A MULTIPLICATION OPERATOR. THIS TRANSFORM IS A GENERALIZED FOURIER TRANSFORM. I DERIVED A THEOREM OF PLANCHEREL TYPE RELATED TO THIS TRANSFORM. I APPLIED THIS TRANSFORM TO SOME CAUCHY PROBLEMS FOR PARTIAL DIFFERENTIAL EQUATIONS WITH SINGULAR COEFFICIENTS AND OBTAINED THE SOLUTIONS EXPLICITLY. USING THIS TRANSFORM I ALSO DEFINED A SPACE OF SOBOLEV TYPE AND SHOWED AN EMBEDDING THEOREM OF SOBOLEV TYPE CONCERNING THIS SPACE. IT IS WELL KNOWN THAT BY THE SOBOLEV EMBEDDING THEOREM WE OBTAIN INFORMATION CONCERNING SMOOTHNESS. ON THE OTHER HAND, THANKS TO OUR EMBEDDING THEOREM, WE CAN OBTAIN INFORMATION CONCERNING NOT ONLY SMOOTHNESS BUT ALSO SINGULARITY OF EACH ELEMENT IN THIS SPACE. THEN I APPLIED THIS TRANSFORM TO THE STUDY OF THE MOTION OF A QUANTUM MECHANICAL PARTICLE UNDER THE INVERSE SQUARE POTENTIAL. I FOUND OUT AN EASIER CONDITION UNDER WHICH THERE EXISTS THE MOTION OF SUCH A PARTICLE.(2) I DEALT WITH THE DYNAMICAL VARIABLES IN QUANTUM MECHANICS ON THE 1-SPHERE BASED ON DIRAC FORMALISM. I SHOWED SELFADJOINTNESS OF THE DYNAMICAL VARIABLES SUCH AS THE MOMONTUM OPERATORS AND THE HAMILTONIAN. AS A RESULT, I FOUND THAT THE DYNAMICAL SYSTEM IS ACTUALLY A QUANTUM MECHANICAL SYSYTEM. IT IS OF INTEREST THAT THE CONTINUOUS SPECTRUM OF EACH MOMENTUM OPERATOR COINCIDES WITH THE SET OF ALL REAL NUMBERS. I APPLIED THESE RESULTS TO THE CAUCHY PROBLEM FOR THE SCHRODINGER EQUATION FOR A FREE QUANTUM MECHANICAL PARTICLE MOVING ON THE 1-SPHERE.