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In mathematical physics, the gradient operator with nonconstant coefficients encompasses various models, including Fourier's law for heat propagation and Fick's first law, that relates the diffusive flux to the gradient of the concentration. Specifically, consider $n\geq 3$ orthogonal unit vectors $e_1,\dots,e_n\in\mathbb{R}^n$, and let $\Omega\subseteq\mathbb{R}^n$ be some (in general unbounded) Lipschitz domain. This paper investigates the spectral properties of the gradient operator $T=\sum_{i=1}^ne_ia_i(x)\frac{\partial}{\partial x_i}$ with nonconstant positive coefficients $a_i:\overline{\Omega}\to(0,\infty)$. Under certain regularity and growth conditions on the $a_i$, we identify bisectorial or strip-type regions that belong to the $S$-resolvent set of $T$. Moreover, we obtain suitable estimates of the associated resolvent operator. Our focus lies in the spectral theory on the $S$-spectrum, designed to study the operators acting in Clifford modules $V$ over the Clifford algebra $\mathbb{R}_n$, with vector operators being a specific crucial subclass. The spectral properties related to the $S$-spectrum of $T$ are linked to the inversion of the operator $Q_s(T):=T^2-2s_0T+|s|^2$, where $s\in\mathbb{R}^{n+1}$ is a paravector, i.e., it is of the form $s=s_0+s_1e_1+\dots+s_ne_n$. This spectral problem is substantially different from the complex one, since it allows to associate general boundary conditions to $Q_s(T)$, i.e., to the squared operator $T^2$.

The path-integral technique in quantum mechanics provides an intuitive framework for comprehending particle propagation and scattering. Calculating the propagator for the Aharonov-Bohm potential fits into the range of potentials in multiply-connected spaces, with the propagator represented through a series expansion. In this paper, we analyze the Schr\"odinger evolution of superoscillations, showing the supershift properties of the solution to the Schr\"odinger equation for this potential. Our proof is based on the continuity of particular infinite order differential operators acting on spaces of entire functions.

In contrast to the classical complex spectral theory, where the spectrum is related to the invertibility of $\lambda-A:D(A)\subseteq X_\mathbb{C}\rightarrow X_\mathbb{C}$, in the noncommutative quaternionic $S$-spectral theory one uses the invertibility of the second order polynomial $Q_s(T):=T^2-2\text{Re}(s)T+|s|^2:D(T^2)\subseteq X\rightarrow X$ to define the $S$-spectrum, where $X$ is a quaternionic Banach space. In this paper we will consider quaternionic operators $T$, for which at least one ray $\{te^{i\omega}\;|\;t>0\}$, $\omega\in[0,\pi]$, $i\in\mathbb{S}$ is contained in the $S$-resolvent set, and the inverse operator $Q_s^{-1}(T)$ admits certain decay properties on this ray. Utilizing the $K$-interpolation method, we then demonstrate that the domain $D(T^k)$ of the $k$-th power of $T$ is an intermediate space between $D(T^n)$ and $D(T^m)$, whenever $n