Narrow Results

We consider the spectral problem associated with the Klein–Gordon equation for unbounded electric potentials such that the spectrum is contained in two disjoint real intervals related to positive and negative energies, respectively. If the two inner boundary points are eigenvalues, we show that these extremal eigenvalues are simple and possess strictly positive eigenfunctions. Examples of electric potentials satisfying these assumptions are given.

In this article, we extend Pólya's legendary inequality for the Dirichlet Laplacian to the fractional Laplacian. Pólya's argument is revealed to be a powerful tool for proving such extensions on tiling domains. As in the Dirichlet Laplacian case, Pólya's inequality for the fractional Laplacian on any bounded domain is still an open problem. Moreover, we also investigate the equivalence of several related inequalites for bounded domains by using the convexity, the Lieb–Aizenman procedure (the Riesz iteration), and some transforms such as the Laplace transform, the Legendre transform, and the Weyl fractional transform.

This paper deals with the massive three-dimensional Dirac operator coupled with a Lorentz scalar shell interaction supported on a compact smooth surface. The rigorous definition of the operator involves suitable transmission conditions along the surface. After showing the self-adjointness of the resulting operator, we switch to the investigation of its spectral properties, in particular, to the existence and non-existence of eigenvalues. In the case of an attractive coupling, we study the eigenvalue asymptotics as the mass becomes large and show that the behavior of the individual eigenvalues and their total number are governed by an effective Schrödinger operator on the boundary with an external Yang–Mills potential and a curvature-induced potential.

We study the spectral analysis and the scattering theory for time evolution operators of position-dependent quantum walks. Our main purpose of this paper is the construction of generalized eigenfunctions of the time evolution operator. Roughly speaking, the generalized eigenfunctions are not square summable but belong to ${\ell }^{\infty }$-space on $Z$. Moreover, we derive a characterization of the set of generalized eigenfunctions in view of the time-harmonic scattering theory. Thus we show that the S-matrix associated with the quantum walk appears in the singularity expansion of generalized eigenfunctions.

In this paper, some properties of resonances for multi-dimensional quantum walks are studied. Resonances for quantum walks are defined as eigenvalues of complex translated time evolution operators in the pseudo momentum space. For some typical cases, we show some results of existence or nonexistence of resonances. One is a perturbation of an elastic scattering of a quantum walk which is an analogue of classical mechanics. Another one is a shape resonance model which is a perturbation of a quantum walk with a non-penetrable barrier.