AI 中文总结
研究超临界状态下正双曲线分支和移位格点十字的海森堡唯一性，解决任意移位问题的无限维条款，通过将零化条件简化为图方程，结合相关估计和刚性得出结论，还给出\(L^1\)预零化子范式及相关算子性质。
AI 中文摘要
我们研究超临界状态\(q = \alpha\gamma>1\)下正双曲线分支和移位格点十字的海森堡唯一性。我们解决了Giri和Manna提出的任意移位问题的无限维条款：对于双臂上的任意移位，归一化预零化子是无限维的。更准确地说，每个\(v\in BV((1,q))\)都有一个全局\(BV\)预零化扩展；除非两个扭曲相位都平凡，否则扩展是唯一的，在这种情况下其模糊性是一维的。证明将零化条件简化为扭曲的Perron-Frobenius算子的图方程，并结合了相位均匀的Lasota-Yorke估计和外围谱刚性。我们还根据相关格林级数的最大收敛域给出了整个\(L^1\)预零化子的精确算子理论范式。用\(Q\)表示扭曲乘积，\(A\)表示强迫算子，我们表明\(Q\)在\(L^1((0,1))\)上的谱是闭单位圆盘，\(\Ran(I - Q)\)不是闭的，并且在\(q>1\)的可数个代数值集之外，对于每个\(N\geq1\)，算子\(\sum_{j = 0}^{N - 1}Q^jA:L^1((1,q))\to L^1((0,1))\)的范数为\(2N\)。
英文摘要
We study Heisenberg uniqueness for the positive hyperbola branch and shifted lattice crosses in the supercritical regime $q=αγ>1$. We resolve the infinite-dimensionality clause of the arbitrary-shift problem posed by Giri and Manna: for arbitrary shifts on both arms, the normalized pre-annihilator is infinite-dimensional. More precisely, every $v\in BV((1,q))$ has a global $BV$ pre-annihilating extension; the extension is unique unless both twisting phases are trivial, in which case its ambiguity is one-dimensional. The proof reduces the annihilation conditions to a graph equation for a twisted Perron--Frobenius operator and combines a phase-uniform Lasota--Yorke estimate with peripheral spectral rigidity. We also give an exact operator-theoretic normal form for the entire $L^1$ pre-annihilator in terms of the maximal convergence domain of the associated Green series. Writing $Q$ for the twisted product and $A$ for the forcing operator, we show that $Q$ has the closed unit disk as its spectrum on $L^1((0,1))$, that $\Ran(I-Q)$ is not closed, and that, outside a countable set of algebraic values of $q>1$, the operator $\sum_{j=0}^{N-1}Q^jA:L^1((1,q))\to L^1((0,1))$ has norm $2N$ for every $N\ge1$.
Comments30 pages, no figure
