Differential commuting operator and closed-form eigenfunctions for linear canonical transforms

作者：Pei Soo Chang^*; Liu Chun Lin

来源：Journal of the Optical Society of America A-Optics Image Science and Vision, 2013, 30(10): 2096-2110.

DOI：10.1364/JOSAA.30.002096

The linear canonical transform (LCT) with a, b, c, d parameter plays an important role in quantum mechanics, optics, and signal processing. The eigenfunctions of the LCT are also important because they describe the self-imaging phenomenon in optical systems. However, the existing solutions for the eigenfunctions of the LCT are divided into many cases and they lack a systematic way to solve these eigenfunctions. In this paper, we find a linear, second-order, self-adjoint differential commuting operator that commutes with the LCT operator. Hence, the commuting operator and the LCT share the same eigenfunctions with different eigenvalues. The commuting operator is very general and simple when it is compared to the existing multiple-parameter differential equations. Then, the eigenfunctions can be derived systematically. The eigenvalues of the commuting operator have closed-form relationships with the eigenvalues of the LCT. We also simplify the eigenfunctions for vertical bar a + d vertical bar > 2 and a + d = +/- 2, b not equal 0 into the more compact closed form instead of the integral form. For vertical bar a + d vertical bar > 2, the eigenfunctions are related to the parabolic cylinder functions.

• 出版日期2013-10