AI 中文总结
研究自伴算子\(A\)的二进预解式表示，通过将底层算子视为服从Landen加倍递归的单参数尺度族等方法，得到相关级数及特殊值，推导方程基本解的二进表示，还能从二进表示重建\(A\)的谱数据，且二进样本唯一确定\(A\)。
AI 中文摘要
对于希尔伯特空间上的自伴算子\(A\)，Castillo、Costin和Costin的二进预解式表示将预解式\(R_A(i\lambda)=(A - i\lambda)^{-1}\)表示为酉群\(U_t = e^{-itA}\)在二进时间\(\{2^{-k}\}_{k\geq0}\)处采样的级数。我们从结构上并通过几个谱应用来发展这种表示。首先将底层算子展示为服从Landen加倍递归的单参数尺度族，其伸缩恢复该表示，并记录其读数为一系列Zak变换和二进滤波器组。作为Zak变换读数的一个实例，我们得到\(\mathbb{Z}^d\)的格点格林函数的二进贝塞尔级数及特殊值。对于\(\mathbb{R}^n\)上的拉普拉斯算子\(A = -\Delta\)，将\(R_A(i\lambda)\varphi\)展开为与自由薛定谔传播子的卷积级数，并推导\(\mathbb{R}^3\)中拉普拉斯和泊松方程以及一维热方程基本解的显式二进表示。最后，从二进表示重建\(A\)的谱数据，包括\(\sigma(A)\)上的谱测度等。还表明二进样本唯一确定\(A\)。
英文摘要
For a self-adjoint operator $A$ on a Hilbert space, the dyadic resolvent representation of Castillo, Costin and Costin expresses the resolvent $R_A(iλ)=(A-iλ)^{-1}$ as a series in the unitary group $U_t=e^{-itA}$ sampled at the dyadic times $\{2^{-k}\}_{k\ge 0}$. We develop this representation structurally and through several spectral applications. We first exhibit the underlying operators as a one-parameter scale family obeying a Landen doubling recursion whose telescoping recovers the representation, and record its readings as a series of Zak transforms and as a dyadic filter bank. As a worked instance of the Zak-transform reading we obtain dyadic-Bessel series for the lattice Green functions of $\mathbb{Z}^d$, with closed-form special values: the lemniscatic constant $Γ(1/4)$ in two dimensions and Watson's integral in three. For the Laplacian $A=-Δ$ on $\mathbb{R}^n$ we expand $R_A(iλ)φ$, for $φ\in L^1(\mathbb{R}^n)\cap L^2(\mathbb{R}^n)$, as a series of convolutions with the free Schrödinger propagator, and derive explicit dyadic representations of the fundamental solutions of the Laplace and Poisson equations in $\mathbb{R}^3$ and of the one-dimensional heat equation. Finally, we reconstruct spectral data of $A$ from the dyadic representation: the spectral measures on $σ(A)$, including, through the limiting absorption principle, the absolutely continuous spectral density of $-Δ+V$, together with the density of states and the spectral zeta function, the last reducing to the Riemann zeta function for $-Δ$ on the circle and yielding the functional determinant of $-Δ+m^2$ there. We close by showing that the dyadic samples determine $A$ uniquely, an exact anti-aliasing of the propagator, so that all of this spectral data is a function of the dyadic samples alone.
