Correctness of the definition of the Laplace operator with delta-like potentials

Universal properties of the isotropic Laplace operator on homogeneous trees

Let $P$ be the isotropic nearest neighbor transition operator on a homogeneous tree. We consider the $λ$-eigenfunctions of $P$ for $λ$ outside its $\ell^2$ spectrum, i.e., the eigenfunctions with eigenvalue $γ=λ- 1$ of the Laplace operator $Delta=P- \mathbb I$, and also the $λ-$polyharmonic functions, that is, the union of the kernels of $(Delta-γ\mathbb I)^n$ for $n\geqslant 0$. We prove that, on a suitable Banach space generated by the $λ-$polyharmonic functions, the operator $e^{Delta-γ\mathbb I}$ is hypercyclic, although $Delta-γ\mathbb I$ is not.

A novel approach to fractional calculus: utilizing fractional integrals and derivatives of the Dirac delta function

While the definition of a fractional integral may be codified by Riemann and Liouville, an agreed-upon fractional derivative has eluded discovery for many years. This is likely a result of integral definitions including numerous constants of integration in their results. An elimination of constants of integration opens the door to an operator that reconciles all known fractional derivatives and shows surprising results in areas unobserved before, including the appearance of the Riemann Zeta Function and fractional Laplace and Fourier Transforms. A new class of functions, known as Zero Function…

二维 Dirac delta Schrödinger 势的唯一连续 L$^2(\mathbb{R}^2)$ 束缚态解的精确点谱与特征向量

The Exact Point Spectrum and Eigenvector of the Unique Continuous L$^2(\mathbb{R}^2)$ Bound State Solution to the Dirac Delta Schrodinger Potential in Two Dimensions

本文针对二维 Dirac delta Schrödinger 势，构造特殊算子定义域并证明其连续 L$^2$ 束缚态唯一，从而无需正则化或重整化即精确确定唯一点谱。

作为二维薛定谔方程奇异势的狄拉克δ函数

The Dirac Delta as a Singular Potential for the 2D Schrodinger Equation

本文在分布广义量子理论框架下，将二维薛定谔方程中的狄拉克δ函数作为奇异势进行严格数学处理，不依赖重整化或自伴扩张，得到与文献一致的束缚态能量，并提出C-谱概念。