For complex scalar waves, a convenient measure of the local oscillations and (‘faster than Fourier’) superoscillations is the phase gradient vector: the local wavevector, or weak value of the momentum operator. This vanishes for standing waves, described by real functions ψ ( r ); for such waves, an alternative descriptor of oscillations is the local weak value of the square of one of the momentum components, i.e. ##IMG## [http://ej.iop.org/images/1751-8121/53/22/225201/aab8b3bieqn1.gif] {${K}_{2}\left(\mathbf{r}\right)=-{\partial }^{2}\psi \left(\mathbf{r}\right)/\partial {x}^{2}/\psi \left(\mathbf{r}\right)$} , here called the ‘weak curvature’. Superoscillations correspond to places where K 2 lies outside the interval 0 ⩽ K 2 ⩽ 1. Two illustrations are given. First is an explicit family of real waves in dimension d = 2, with arbitrarily strong superoscillations; this could represent Neumann sta...

中文翻译：

---

单色驻波的超振荡

对于复杂的标量波，可以方便地测量局部振荡和（比傅立叶更快）的超振荡是相位梯度矢量：局部波矢量或动量算符的弱值。这对于驻波消失了，由实函数ψ（r）描述；对于此类波，振荡的替代描述子是动量分量之一的平方的局部弱值，即## IMG ## [http://ej.iop.org/images/1751-8121/53/22 /225201/aab8b3bieqn1.gif] {$ {K} _ {2} \ left（\ mathbf {r} \ right）=-{\ partial} ^ {2} \ psi \ left（\ mathbf {r} \ right） / \ partial {x} ^ {2} / \ psi \ left（\ mathbf {r} \ right）$}，这里称为“弱曲率”。超振动对应于K 2位于区间0⩽K 2⩽1之外的地方。给出了两个说明。首先是维数= 2的显式实波族 具有任意强烈的超振荡；这可能代表了诺伊曼站...