Spectral Theory
For a very general class of unbounded self-adjoint operator function we prove upper bounds for eigenvalues which lie within arbitrary gaps of the essential spectrum. These upper bounds are given by triple variations. Furthermore, we find…
The self-adjoint and $m$-sectorial extensions of coercive Sturm-Liouville operators are characterised, under minimal smoothness conditions on the coefficients of the differential expression.
The commuting graph of a non-commutative ring $R$ with center $Z(R)$ is a simple undirected graph whose vertex set is $R\setminus Z(R)$ and two vertices $x, y$ are adjacent if and only if $xy = yx$. In this paper, we compute the spectrum…
We obtain a simple formula for the first-order trace of a regular differential operator on a segment perturbated by a multiplication operator. The main analytic ingredient of the proof is an improvement of the Tamarkin equiconvergence…
We obtain a new simple formula for the regularized traces of singular ordinary differential operators.
We give a mathematically rigorous construction of a magnetic Schr\"odinger operator corresponding to a field with flux through finitely many holes of the Sierpinski Gasket. The operator is shown to have discrete spectrum accumulating at…
It is shown that transfer functions, which play a crucial role in M.G. Krein's study of inverse spectral problems, are a proper tool to formulate local spectral uniqueness conditions.
The eigenvalues of the normalized Laplacian of a graph provide information on its topological and structural characteristics and also on some relevant dynamical aspects, specifically in relation to random walks. In this paper we determine…
We investigate the effect of non-symmetric relatively bounded perturbations on the spectrum of self-adjoint operators. In particular, we establish stability theorems for one or infinitely many spectral gaps along with corresponding…
We study the asymptotic properties of the conormal cycle of nodal sets associated to a random superposition of eigenfunctions of the Laplacian on a smooth compact Riemannian manifold without boundary. In the case where the dimension is odd,…
We solve two inverse spectral problems for star graphs of Stieltjes strings with Dirichlet and Neumann boundary conditions, respectively, at a selected vertex called root. The root is either the central vertex or, in the more challenging…
The Anderson Hamiltonian $H_0=-\Delta+V(x,\omega)$ is considered, where $V$ is a random potential of Bernoulli type. The operator $H_0$ is perturbed by a non-random, continuous potential $-w(x) \leq 0$, decaying at infinity. It will be…
In this note we present some abstract ideas how one can construct spaces from building blocks according to a graph. The coupling is expressed via boundary pairs, and can be applied to very different spaces such as discrete graphs, quantum…
In this paper, we show that every pseudo B-Fredholm operator is a pseudo Fredholm operator. Afterwards, we prove that the pseudo B-Weyl spectrum is empty if and only if the pseudo B-Fredholm spectrum is empty. Also, we study a symmetric…
We construct the biharmonic heat kernel for a suitable self-adjoint extension of the bi-Laplacian on a manifold with incomplete edge singularities. We employ a microlocal description of the biharmonic heat kernel to establish mapping…
The Bochner Classification Theorem (1929) characterizes the polynomial sequences $\p_{n}\}_{n=0}^{\infty}$, with $\text{deg}\,p_{n}=n$ that simultaneously form a complete set of eigenstates for a second-order differential operator and are…
\AA. Pleijel (1956) has proved that in the case of the Laplacian with Dirichlet condition, the equality in the Courant nodal theorem (Courant sharp situation) can only be true for a finite number of eigenvalues when the dimension is $\geq…
We study the (generalized) semi-Weyl commutation relations $$ U_gAU_g^*=g(A) \quad \text{ on }\quad \Dom(A), $$ where $A$ is a densely defined operator and $G\ni g\mapsto U_g$ is a unitary representation of the subgroup $G$ of the affine…
We study the spectrum of the operator $D^*D$, where the operator $D$, introduced in \cite{KMR}, is a forward derivative on the $p$-adic tree, a weighted rooted tree associated to $\mathbb Z_p$ via Michon's correspondence. We show that the…
We discuss several geometric conditions guaranteeing the finiteness or the infiniteness of the discrete spectrum for Robin Laplacians on conical domains.