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We investigate the discrete spectrum of the Hamiltonian describing a quantum particle living in the two-dimensional curved strip. We impose the Dirichlet and Neumann boundary conditions on opposite sides of the strip. The existence of the…

The longstanding open problem of approximating all singular vertex couplings in a quantum graph is solved. We present a construction in which the edges are decoupled; an each pair of their endpoints is joined by an edge carrying a $\delta$…

We consider boundary conditions at the vertex of a star graph which make Schroedinger operators on the graph self-adjoint, in particular, the two-parameter family of such conditions invariant with respect to permutations of graph edges. It…

On a finite connected metric graph, we establish upper bounds for the eigenvalues of the Laplacian. These bounds depend on the length, the Betti number, and the number of pendant vertices. For trees, these estimates are sharp. We also…

On a compact metric graph, we consider the spectrum of the Laplacian defined with a mix of standard and Dirichlet vertex conditions. A Cheeger-type lower bound on the gap $\lambda_2 - \lambda_1$ is established, with a constant that depends…

We study asymptotic behaviors of solutions $f$ to the Dirichlet problem for minimal graphs in the hyperbolic space with singular asymptotic boundaries under the assumption that the boundaries are piecewise regular with positive curvatures.…

We discuss approximations of the vertex coupling on a star-shaped quantum graph of $n$ edges in the singular case when the wave functions are not continuous at the vertex and no edge-permutation symmetry is present. It is shown that the…

In this work, we analyze the Dirichlet Laplacian $-\Delta_{\Omega}^D$ in an unbounded waveguide $\Omega \subset \mathbb R^3$, where the cross-section is translated in a constant direction and rotated along a spatial line. We focus on the…

We consider the spectrum of the Laplace operator on 3D rod structures, with a small cross section depending on a small parameter $\varepsilon$. The boundary conditions are of Dirichlet type on the basis of this structure and Neumann on the…

We investigate quantum graphs with infinitely many vertices and edges without the common restriction on the geometry of the underlying metric graph that there is a positive lower bound on the lengths of its edges. Our central result is a…

Let $K$ be a p.c.f. self-similar set equipped with a strongly recurrent Dirichlet form. Under a homogeneity assumption, for an open set $\Omega\subset K$ whose boundary $\partial \Omega$ is a graph-directed self-similar set, we prove that…

The paper concerns scattering of plane waves by a bounded obstacle with complex valued impedance boundary conditions. We study the spectrum of the Neumann-to-Dirichlet operator for small wave numbers and long wave asymptotic behavior of the…

It is known that, for an oriented hypergraph with (vertex) coloring number $\chi$ and smallest and largest normalized Laplacian eigenvalues $\lambda_1$ and $\lambda_N$, respectively, the inequality $\chi\geq…

We prove lower bounds for the Dirichlet Laplacian on possibly unbounded domains in terms of natural geometric conditions. This is used to derive uncertainty principles for low energy functions of general elliptic second order divergence…

In this note we give a new upper bound for the Laplacian eigenvalues of an unweighted graph. Let $G$ be a simple graph on $n$ vertices. Let $d_{m}(G)$ and $\lambda_{m+1}(G)$ be the $m$-th smallest degree of $G$ and the $m+1$-th smallest…

The Dirichlet Laplacian between two parallel hypersurfaces in Euclidean spaces of any dimension in the presence of a magnetic field is considered in the limit when the distance between the hypersurfaces tends to zero. We show that the…

We consider the Laplacian in a strip $\mathbb{R}\times (0,d)$ with the boundary condition which is Dirichlet except at the segment of a length $2a$ of one of the boundaries where it is switched to Neumann. This operator is known to have a…

We study expansions near the boundary of solutions to the Dirichlet problem for minimal graphs in the hyperbolic space and prove the local convergence of such expansions if the boundary is locally analytic. As a consequence, we prove a…

Let $(G_\epsilon)_{\epsilon>0}$ be a family of '$\epsilon$-thin' Riemannian manifolds modeled on a finite metric graph $G$, for example, the $\epsilon$-neighborhood of an embedding of $G$ in some Euclidean space with straight edges. We…

We discuss optimal lower bounds for eigenvalues of Laplacians on weighted graphs. These bounds are formulated in terms of the geometry and, more specifically, the inradius of subsets of the graph. In particular, we study the first non-zero…