Semiclassical dynamics of higher-order squeezing: self-adjoint extensions and coherent returns

Abstract

Higher-order squeezing Hamiltonians have several self-adjoint realizations, while their classical trajectories reach infinity in finite time. We relate these extension problems through boundary amplitudes and the semiclassical propagation of coherent states. A discrete Fourier transformation gives the sink-to-source ray matrix of each quantum realization. After the first passage through infinity, the full Husimi probability measure converges to point masses on outgoing classical branches, with weights equal to the squared moduli of the matrix entries. The proof combines finite-order approximate tail solutions satisfying the exact extension boundary condition with full-norm Gaussian matching and a square-summable residual estimate. For positive energy, we determine the limit after any fixed number of passages: amplitudes evolve by powers of the unitary ray matrix, not by iterations of its squared-modulus matrix. We also realize these limits by concentrated complex amplitudes for an energy-preserving Liouville extension, distinguishing deterministic classical gluings from coherent boundary mixing.

For higher-order squeezing Hamiltonians $\widehat H_{k,\hbar}=\lambda\hbar^{k/2}(a^k+(a^\dagger)^k)$ with $k>2$, quantum evolution requires a self-adjoint extension, while classical trajectories escape to infinity in finite time. The paper asks how the extension determines semiclassical continuation beyond escape, and whether repeated passages admit a stochastic description. Boundary amplitudes parameterize the quantum extensions by $U\in U(k)$. Discrete Fourier transforms convert this parameter into an exact sink-to-source ray matrix, $S_{\rm q}=F_-UF_+^*$. For a coherent state with nonzero classical energy escaping along ray $j$, the full Husimi probability measure subsequently converges to point masses on outgoing classical branches, weighted by $|(S_{\rm q})_{lj}|^2$. Deterministic continuations correspond precisely to monomial ray matrices. At positive energy, after any fixed number $\nu$ of passages, the weights are $|(S_{\rm q}^{\nu}e_j)_l|^2$, generally not those obtained by iterating the one-pass probability matrix. A Fourier-matrix example splits uniformly on the first passage and recombines into one branch on the second. The propagation proof combines compact-orbit Gaussian approximations with finite-order discrete WKB tail packets satisfying the exact extension boundary condition; square-summable residual bounds enable full-norm Duhamel estimates. An energy-preserving Liouville evolution of complex amplitudes reproduces these concentration limits. The results exclude zero-energy initial centers and observations exactly at escape times; repeated-passage quantum limits are established only at positive energy and for fixed passage counts. They do not establish convergence on a full hybrid observable algebra or unitary equivalence of quantum and Liouville dynamics. Numerical illustrations retain substantial radial-cutoff dependence and provide no full spatial-error bound.