arXiv subjects
Publications and source records attributed to Young-Ran Lee.
We consider a polyharmonic operator $H=(-Δ)^l+V(x)$ in dimension two with $l\geq 6$, $l$ being an integer, and a limit-periodic potential $V(x)$. We prove that the spectrum contains a semiaxis of absolutely continuous spectrum.
We study the decay of eigenfunctions of the non self-adjoint matrix operator $\calH = (\begin{smallmatrix} -Δ+μ+U & W \W & Δ-μ-U \end{smallmatrix})$, for $μ>0$, corresponding to eigenvalues in the strip $-μ<\re E <μ$.
Let G(x,y) and G_D(x,y) be the Green functions of rotationally invariant symmetric α-stable process in R^d and in an open set D respectively, where 0<α< 2. The inequality G_D(x,y)G_D(y,z)/G_D(x,z) \le c(G(x,y)+G(y,z)) is a very useful tool in studying (local) Schrodinger operators. When the above inequality is true with a constant c=c(D)>0, then we say that the 3G theorem holds in D. In this paper, we establish a generalized version of 3G theorem when D is a bounded κ-fat open set, which includes a bounded John domain. The 3G we consider is of the form G_D(x,y)G_D(z,w)/G_D(x,w), where y may be different from z. When y=z, we recover the usual 3G. The 3G form G_D(x,y)G_D(z,w)/G_D(x,w) appears in non-local Schrodinger operator theory. Using our generalized 3G theorem, we give a concrete class of functions belonging to the non-local Kato class, introduced by Chen and Song, on κ-fat open sets. As an application, we discuss relativistic α-stable processes (relativistic Hamiltonian when α=1) in κ-fat open sets. We identify the Martin boundary and the minimal Martin boundary with the Euclidean boundary for relativistic α-stable processes in κ-fat open sets. Furthermore, we show that relative Fatou type theorem is true for relativistic stable processes in κ-fat open sets. The main results of this paper hold for a large class of symmetric Markov processes, as are illustrated in the last section of this paper. We also discuss the generalized 3G theorem for a large class of symmetric stable Levy processes.
We consider a polyharmonic operator $H=(-Δ)^l+V(x)$ in dimension two with $l\geq 6$ and a limit-periodic potential $V(x)$. We prove that the spectrum of $H$ contains a semiaxis and there is a family of generalized eigenfunctions at every point of this semiaxis with the following properties. First, the eigenfunctions are close to plane waves $e^{i< \vec k,\vec x>}$ at the high energy region. Second, the isoenergetic curves in the space of momenta $\vec k$ corresponding to these eigenfunctions have a form of slightly distorted circles with holes (Cantor type structure).