论文标题
具有三角洲潜力的半古典限制
The semi-classical limit with a delta-prime potential
论文作者
论文摘要
我们考虑量子进化$ e^{ - i \ frac {t} {\ hbar}h_β}ψ_² （q，p）\ in \ mathbb {r}^{2} $，其中$h_β$表示正式的哈密顿$ - \ frac {\ hbar^{2}}}} {2m} {2m} {2m} {2m} \， β\，δ'_{0} $，带有$δ'_{0} $ dirac的dirac Delta分布的衍生物为$ x = 0 $ and $β$ a真实参数。我们表明，在半古典限制中，可以近似这样的量子进化（W.R.T. $ l^{2}（\ Mathbb {r}）$ - norm- n norm，对于任何$ t \ in \ Mathbb {r} $ a collision the Collision time by $ e^{ e^{it l_ {b}} ϕ^{\ hbar} _ {x} $，其中$ a_ {t} = \ frac {p^{2} t} {2m} {2m} $，$ vary，$ dem，$ ϕ_ {x}}}}} $ l_ {b} $是将限制到$ \ MATHCAL {C}^{\ infty} _ {c}（{\ Mathscr M} _ {0}）$，$ {\ MATHSCR M} _ {0} _ {0} _ {0}： \ Mathbb {r}^{2} \，| \，Q \ neq 0 \} $（$ -i $ times）是免费经典动力学的生成器。虽然此处使用的操作员$ l_ {b} $类似于我们以前的工作中出现的[C。 Cacciapuoti, D. Fermi, A. Posilicano, The semi-classical limit with a delta potential, Annali di Matematica Pura e Applicata (2020)] regarding the semi-classical limit with a delta potential, in the present case the approximation gives a smaller error: it is of order $\hbar^{7/2-λ}$, $0 < λ< 1/2$,事实证明，它是$ \ hbar^{3/2-λ} $，$ 0 <λ<3/2 $，用于增量电位。我们还为波和散射算子提供了相似的近似结果。
We consider the quantum evolution $e^{-i\frac{t}{\hbar}H_β} ψ_ξ^{\hbar}$ of a Gaussian coherent state $ψ_ξ^{\hbar}\in L^{2}(\mathbb{R})$ localized close to the classical state $ξ\equiv (q,p) \in \mathbb{R}^{2}$, where $H_β$ denotes a self-adjoint realization of the formal Hamiltonian $-\frac{\hbar^{2}}{2m}\,\frac{d^{2}\,}{dx^{2}} + β\,δ'_{0}$, with $δ'_{0}$ the derivative of Dirac's delta distribution at $x = 0$ and $β$ a real parameter. We show that in the semi-classical limit such a quantum evolution can be approximated (w.r.t. the $L^{2}(\mathbb{R})$-norm, uniformly for any $t \in \mathbb{R}$ away from the collision time) by $e^{\frac{i}{\hbar} A_{t}} e^{it L_{B}} ϕ^{\hbar}_{x}$, where $A_{t} = \frac{p^{2}t}{2m}$, $ϕ_{x}^{\hbar}(ξ) := ψ^{\hbar}_ξ(x)$ and $L_{B}$ is a suitable self-adjoint extension of the restriction to $\mathcal{C}^{\infty}_{c}({\mathscr M}_{0})$, ${\mathscr M}_{0} := \{(q,p) \in \mathbb{R}^{2}\,|\,q \neq 0\}$, of ($-i$ times) the generator of the free classical dynamics. While the operator $L_{B}$ here utilized is similar to the one appearing in our previous work [C. Cacciapuoti, D. Fermi, A. Posilicano, The semi-classical limit with a delta potential, Annali di Matematica Pura e Applicata (2020)] regarding the semi-classical limit with a delta potential, in the present case the approximation gives a smaller error: it is of order $\hbar^{7/2-λ}$, $0 < λ< 1/2$, whereas it turns out to be of order $\hbar^{3/2-λ}$, $0 < λ< 3/2$, for the delta potential. We also provide similar approximation results for both the wave and scattering operators.
