Related papers: Weyl-Titchmarsh M-Function Asymptotics for Matrix-…
We explore the connections between singular Weyl-Titchmarsh theory and the single and double commutation methods. In particular, we compute the singular Weyl function of the commuted operators in terms of the original operator. We apply the…
We investigate the connections between Weyl-Titchmarsh-Kodaira theory for one-dimensional Schr\"odinger operators and the theory of $n$-entire operators. As our main result we find a necessary and sufficient condition for a one-dimensional…
We study Toeplitz operators on the Bargmann space, with Toeplitz symbols that are exponentials of inhomogeneous quadratic polynomials. It is shown that the boundedness of such operators is implied by the boundedness of the corresponding…
Sharp spectral asymptotics for multidimensional Schroedinger operators with the strong magnetic field are derived under rather weak smoothness conditions. I assume that magnetic intensity matrix has constant defect r>0 at each point. In…
We prove a matrix inequality for convex functions of a Hermitian matrix on a bipartite space. As an application we reprove and extend some theorems about eigenvalue asymptotics of Schr\"odinger operators with homogeneous potentials. The…
We prove a universal bound for the number of negative eigenvalues of Schr\"odinger operators with Neumann boundary conditions on bounded H\"older domains, under suitable assumptions on the H\"older exponent and the external potential. Our…
In three-dimensional case, we consider two classical operators: Schrodinger operator and an operator in the divergence form. For slowly-decaying oscillating potentials, we establish spatial asymptotics of the Green's function. The main term…
We will discuss the asymptotic behaviour of the eigenvalues of Schr\"{o}dinger operator with a matrix potential defined by Neumann boundary condition in $L_2^m(F)$, where $F$ is $d$-dimensional rectangle and the potential is a $m \times m$…
We provide the necessary and sufficient conditions for a generalized Nevanlinna function $Q$ ($Q\in N_{\kappa }\left( \mathcal{H} \right)$) to be a Weyl function (also known as a Weyl-Titchmarch function). We also investigate an important…
We explain in which sense Schr\"odinger operators with complex potentials appear to violate locality (or Weyl's asymptotics), and we pose three open problems related to this phenomenon.
We consider asymptotic behavior of $e^{-itH}f$ for $N$-body Schr\"odinger operator $H=H_0+\sum_{1\le i<j\le N}V_{ij}(x)$ with long- and short-range pair potentials $V_{ij}(x)=V_{ij}^L(x)+V_{ij}^S(x)$ $(x\in {\mathbb R}^\nu)$ such that…
The matrix Schroedinger equation with a selfadjoint matrix potential is considered on the half line with the general selfadjoint boundary condition at the origin. When the matrix potential is integrable, the high-energy asymptotics are…
We study the Weyl representation of metaplectic operators associated to a symplectic matrix having no non-trivial fixed point, and justify a formula suggested in earlier work of Mehlig and Wilkinson. We give precise calculations of the…
We study spectra of alloy-type random Schr\"odinger operators on metric graphs. For finite edge subsets of general graphs we prove a Wegner estimate which is linear in the volume (i.e. the number of edges) and the length of the considered…
We obtain a complete asymptotic expansion of the integrated density of states of operators of the form H =(-\Delta)^w +B in R^d. Here w >0, and B belongs to a wide class of almost-periodic self-adjoint pseudo-differential operators of order…
We study the theory of scattering for the Maxwell-Schr"odinger system in space dimension 3, in the Coulomb gauge. We prove the existence of modified wave operators for that system with no size restriction on the Schr"odinger and Maxwell…
The main goal of this paper is to characterise all the possible Ces\`aro and $L$-asymptotic limits of powerbounded, complex matrices. The investigation of the $L$-asymptotic limit of a powerbounded operator goes back to Sz.-Nagy and it…
We provide a precise description of the bottom of the spectrum in the semiclassical limit of a harmonic-type Schr\"odinger operator with an inverse square potential. By exploiting the connection between the eigenfunctions of these operators…
In this article we obtain eigenvalue asymptotics for 2D and 3D-Schroedinger, Schroedinger-Pauli and Dirac operators in the situations in which the role of the magnetic field is important. These operators are essentially different and there…
We explore the sparsity of Weyl-Titchmarsh $m$-functions of discrete Schr\"odinger operators. Due to this, the set of their $m$-functions cannot be dense on the set of those for Jacobi operators. All this reveals why an inverse spectral…