We give error bounds of the Trotter-Kato product formula of relativistic Schrödinger operators $\sqrt{(i \partial_x - a(x))^2 + 1} - 1 + V(x)$, where $B(x) = \nabla \times a(x)$ is bounded and $V(x)$ behaves like $|x|^{\rho}$, $\rho > 0$.

In this paper we study the essential self-adjointness of Dirac operators with a variable mass term $m(x)$ and an electric potential $V(x)$. We are mainly interested in the local singularities of $m(x)$ and $V(x)$. We can treat singularities of $m(x)$ and $V(x)$ which are stronger than those of Coulomb potentials.

Let $\mathbf{N}$ denote the set of natural numbers $\{1, 2, 3, \ldots\}$. $n$ being an odd natural number, we consider the Diophantine equation as mentioned in the title and solve it completely for $n \leq 15$, i.e. find all $(x,y) \in \mathbf{N}^2$ satisfying this equation.

This is a résumé of the author's recent work on certain analogies between primes and links. The purpose of this article is to introduce a new invariant, called Milnor invariant, in algebraic number theory, based on an analogy between the structure of a certain Galois group over the rational number field and that of the group of a link in three dimensional Euclidean space. It then turns out that the Legendre, Rédei symbols are interpreted as our link invariants. We expect that this is a tip of an arithmetical theory after the model of link theory which may give a new insight in algebraic number theory. The details will be published elsewhere.

We compute explicitly the Selberg trace formula for principal congruence subgroups $\Gamma$ of $PGL(2, \mathbf{F}_q[t])$, which is the modular group in positive characteristic cases. It is known that $\Gamma \backslash X$ is an infinite Ramanujan diagram, where $X$ is the $q + 1$-regular tres. We express the Selberg zeta function for $\Gamma$ as the determinant of the adjacency operator which is composed of both discrete and continuous spectra. They are rational functions in $q^{-s}$. We also discuss the limit distribution of eigenvalues of $\Gamma \backslash X$ as the level tends to infinity.