Bargmann型变换和修正的谐振子
Bargmann-type transforms and modified harmonic oscillators
摘要：
本文研究了实数上的完备规范正交系。这些系统由Bargmann型变换确定，Bargmann型变换是具有复值二次相位函数的傅里叶积分算子。每个系统由二阶椭圆微分算子的本征函数组成，如谐振子的哈密顿量。我们还研究了一类称为非交换谐振子的二阶微分算子系统的交换情形。利用对角化技术，我们计算了非对易谐振子在对易情况下的本征值和本征函数。最后，我们研究了相平面上椭圆的函数族。我们证明了该族是实直线上的完备正交系。
英文摘要：
We study some complete orthonormal systems on the real-line. These systems are determined by Bargmann-type transforms, which are Fourier integral operators with complex-valued quadratic phase functions. Each system consists of eigenfunctions for a second-order elliptic differential operator like the Hamiltonian of the harmonic oscillator. We also study the commutative case of a certain class of systems of second-order differential operators called the non-commutative harmonic oscillators. By using the diagonalization technique, we compute the eigenvalues and eigenfunctions for the commutative case of the non-commutative harmonic oscillators. Finally, we study a family of functions associated with an ellipse in the phase plane. We show that the family is a complete orthogonal system on the real-line.
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