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

(1)我处理了一个由微分算子和奇异项组成的算子。我构造了一个转换，它将运算符转换为乘法运算符。该变换是一种广义傅里叶变换。我得到了一个与这种变换有关的Plancherel型定理。我把这个变换应用于一些奇异系数偏微分方程柯西问题，得到了显式的解。利用这个变换，我还定义了一个Sobolev型空间，并证明了关于该空间的一个Sobolev型嵌入定理。众所周知，通过Sobolev嵌入定理，我们得到了关于光滑性的信息。另一方面，由于我们的嵌入定理，我们不仅可以得到关于该空间中每个元素的光滑性的信息，而且可以得到关于奇异性的信息。然后，我将这个变换应用于研究反平方势下量子力学粒子的运动。我找到了这样的粒子存在运动的一个更容易的条件。(2)基于狄拉克形式，我处理了1-球上量子力学中的动力学变量。我证明了动力学变量，如MOMONTUM算符和哈密顿量的自洽性。结果，我发现这个动力系统实际上是一个量子力学系统。有趣的是，每个动量算符的连续谱与所有实数的集合重合。我将这些结果应用于1-球上运动的自由量子力学粒子的薛定谔方程的柯西问题。

(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.