Higher power squeezed states, Jacobi matrices, and the Hamburger moment problem

TL;DR

This paper explores higher power squeezed states, their mathematical properties via Jacobi matrices, and their connection to the Hamburger moment problem, revealing new insights into their spectral structure and state definitions.

Contribution

It introduces a novel analysis of higher power squeezed states using Jacobi matrices and links to the Hamburger moment problem, extending understanding beyond known cases.

Findings

Explicit solutions for k=1,2 using Hermite and Pollaczek polynomials.

For k ≥ 3, the recursion relation defines an undetermined Hamburger moment problem.

Higher power squeezed states are well-defined despite the spectral complexity.

Abstract

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.