高次幂压缩态、雅可比矩阵和汉堡矩问题
Higher power squeezed states, Jacobi matrices, and the Hamburger moment problem
摘要：
k次幂（振幅）压缩态定义为单模湮灭算符的k次幂实部和虚部在Schroedinger-Robertson测不准关系中相等的归一化态。等价地，它们是Bogolubov变换的“k次幂湮灭算子”的归一化本征态（对于所有可能的复本征值）的集合。本征值方程用数字表示，导出了展开系数的三项递推关系，在k = 1，2的情况下，可以显式求解。解本质上分别是Hermite和Pollaczek多项式。k = 1给出了普通的压缩态，即位移压缩真空。对于k等于或大于3，其中没有明确的解决方案已被发现，递归关系的对称运营商给出的实部的k：th权力的湮灭运营商定义了一个雅可比矩阵对应于一个经典的汉堡矩问题，这是未定的。这意味着该算子有无穷多个自伴扩张，且都具有不相交的离散谱。然而，相应的压缩态是明确定义的。
英文摘要：
k:th power (amplitude-)squeezed states are defined as the normalized states giving equality in the Schroedinger-Robertson uncertainty relation for the real and imaginary parts of the k:th power of the one-mode annihilation operator. Equivalently they are the set of normalized eigenstates (for all possible complex eigenvalues) of the Bogolubov transformed "k:th power annihilation operators". Expressed in the number representation the eigenvalue equation leads to a three term recursion relation for the expansion coefficients, which can be explicitly solved in the cases k = 1, 2. The solutions are essentially Hermite and Pollaczek polynomials, respectively. k = 1 gives the ordinary squeezed states, i.e. displaced squeezed vacua. For k equal to or larger than three, where no explicit solution has been found, the recursion relation for the symmetric operator given by the real part of the k:th power of the annihilation operator defines a Jacobi matrix corresponding to a classical Hamburger moment problem, which is undetermined. This implies that the operator has an infinity of self-adjoint extensions, all with disjoint discrete spectra. The corresponding squeezed states are well-defined, however.
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