TFPT scientific source excerpt
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000001 | 
000002 | Because $U_6$ has eigenphases $e^{in\pi/3}$, the regularized transport spectrum is
000003 | \begin{equation}
000004 | \label{eq:transport-spectrum}
000005 | \lambda_{y,n}^{-1}
000006 | =
000007 | y^2+\delta_{\mathrm{ph}}^2-2y\delta_{\mathrm{ph}}\cos\!\left(\frac{n\pi}{3}\right)+\varepsilon^2,
000008 | \qquad
000009 | n=0,\dots,5,
000010 | \end{equation}
000011 | for the eigenvalues $\lambda_{y,n}$ of $\mathcal Y_y^{(\varepsilon)}$. In particular,
000012 | \begin{lemma}[Finite admissible hexagon gap]
000013 | \label{lem:finite-admissible-hexagon-gap}
000014 | On the admissible cusp set
000015 | \[
000016 | y\in\left\{1,\frac13,\frac23\right\},
000017 | \]
000018 | the smallest singular value of the unregulated hexagon denominator is attained at $y=\frac23$ and $n=0$. Equivalently,
000019 | \[
000020 | \epsilon_f
000021 | :=
000022 | \min_{\substack{y\in\{1,\frac13,\frac23\}\\ n=0,\dots,5}}
000023 | \left|y-\delta_{\mathrm{ph}}e^{in\pi/3}\right|
000024 | =
000025 | \frac23-\delta_{\mathrm{ph}}.
000026 | \]
000027 | Hence the finite admissible hexagon has a strictly positive spectral gap $\epsilon_f$.
000028 | \end{lemma}
000029 | 
000030 | \begin{proof}
000031 | For $\varepsilon=0$, the eigenvalues in \Cref{eq:transport-spectrum} reduce to
000032 | $|y-\delta_{\mathrm{ph}}e^{in\pi/3}|^{-2}$. By \Cref{thm:transport-pole-algebraic},
000033 | $\delta_{\mathrm{ph}}$ is the unique admissible root in $(1/3,2/3)$. Hence the positive-real cusp
000034 | $y=2/3$ with $n=0$ is the nearest admissible point, while every other cusp or nonzero phase has
000035 | strictly larger distance. Therefore
000036 | \[
000037 | \epsilon_f=\frac23-\delta_{\mathrm{ph}},
000038 | \]
000039 | which is positive because $\delta_{\mathrm{ph}}<2/3$.
000040 | \end{proof}
000041 | 
000042 | \begin{definition}[Gap-stable admissibility]
000043 | \label{def:gap-stable-admissibility}
000044 | Let
000045 | \[
000046 | D_0:=D_{2/3}.
000047 | \]
000048 | We say that the admissible transport sector is gap-stable if the physical transport operator has the form
000049 | \[
000050 | D_f=D_0+V_{\mathrm{adm}},
000051 | \qquad
000052 | [P_{\mathrm{str}},V_{\mathrm{adm}}]=0,
000053 | \qquad
000054 | \|V_{\mathrm{adm}}\|<\frac{\epsilon_f}{2}.
000055 | \]
000056 | \end{definition}
000057 | 
000058 | \begin{proposition}[Structural gap lift]
000059 | \label{prop:gap-lift-full-operator}
000060 | If admissibility is gap-stable in the sense of \Cref{def:gap-stable-admissibility}, then for every
000061 | \[
000062 | \psi\in\mathrm{Ran}(P_{\mathrm{str}})
000063 | \]
000064 | one has
000065 | \[
000066 | \|D_f\psi\|
000067 | \ge
000068 | \frac{\epsilon_f}{2}\|\psi\|.
000069 | \]
000070 | Consequently
000071 | \begin{equation}
000072 | \label{eq:gap-lift-full-operator}
000073 | P_{\mathrm{str}} D_f^\dagger D_f P_{\mathrm{str}}
000074 | \ge
000075 | \frac{\epsilon_f^2}{4}\,P_{\mathrm{str}}
000076 | >
000077 | 0.
000078 | \end{equation}
000079 | In particular, the final admissible restriction to $\mathrm{Ran}(\Padm)$ is strictly positive.
000080 | \end{proposition}
000081 | 
000082 | \begin{proof}
000083 | Since $\psi\in\mathrm{Ran}(P_{\mathrm{str}})$ and $[P_{\mathrm{str}},V_{\mathrm{adm}}]=0$,
000084 | \[
000085 | \|D_f\psi\|
000086 | \ge
000087 | \|D_0\psi\|-\|V_{\mathrm{adm}}\psi\|
000088 | \ge
000089 | \bigl(\epsilon_f-\|V_{\mathrm{adm}}\|\bigr)\|\psi\|
000090 | \ge
000091 | \frac{\epsilon_f}{2}\|\psi\|.
000092 | \]
000093 | Squaring gives \Cref{eq:gap-lift-full-operator}.
000094 | \end{proof}
000095 | 
000096 | \begin{equation}
000097 | \label{eq:mobius-quotient}
000098 | \frac{\lambda_{y,0}}{\lambda_{y,3}}
000099 | =
000100 | \frac{(y+\delta_{\mathrm{ph}})^2+\varepsilon^2}{(y-\delta_{\mathrm{ph}})^2+\varepsilon^2}.