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The Kepler problem is a dynamical system that is well defined not only on the Euclidean plane but also on the sphere and on the Hyperbolic plane. First, the theory of central potentials on spaces of constant curvature is studied. All the…

The problem of determining the domain of the closure of the Laplace-Beltrami operator on a 2D almost-Riemannian manifold is considered. Using tools from theory of Lie groupoids natural domains of perturbations of the Laplace-Beltrami…

We study the one-dimensional Laplace operator with point interactions on the real line identified with two copies of the half-line $[0,\infty)$. All possible boundary conditions that define generators of $C_0$-semigroups on…

We present a systematic perturbative construction of the most general metric operator (and positive-definite inner product) for quasi-Hermitian Hamiltonians of the standard form, H= p^2/2 + v(x), in one dimension. We show that this problem…

Analyzing the point spectrum, i.e. bound state energy eigenvalue, of the Dirac delta function in two and three dimensions is notoriously difficult without recourse to regularization or renormalization, typically both. The reason for this in…

We consider a convolution-type operator on vector bundles over metric-measure spaces. This operator extends the analogous convolution Laplacian on functions in our earlier work to vector bundles, and is a natural extension of the graph…

Finite rank perturbations of a semi-bounded self-adjoint operator A are studied in the scale of Hilbert spaces associated with A. A concept of quasi-boundary value space is used to describe self-adjoint operator realizations of regular and…

We study the two-dimensional magnetic Laplacian when the magnetic field is allowed to be complex-valued. Under the assumption that the imaginary part of the magnetic potential is relatively form-bounded with respect to the real part of the…

Given a sample from a probability measure with support on a submanifold in Euclidean space one can construct a neighborhood graph which can be seen as an approximation of the submanifold. The graph Laplacian of such a graph is used in…

Let $X=G/H$ be a symmetric space for a real simple Lie group $G$, equipped with a $G$-invariant complex structure. Then, $X$ is a pseudo-Hermitian manifold, and in this geometric setting, higher Laplacians $L_m$ are defined for each…

It is natural to investigate if the quantization of an integrable or superintegrable classical Hamiltonian systems is still integrable or superintegrable. We study here this problem in the case of natural Hamiltonians with constants of…

The discrete Laplacian on Euclidean triangulated surfaces is a well-established notion. We introduce discrete Laplacians on spherical and hyperbolic triangulated surfaces. On the one hand, our definitions are close to the Euclidean one in…

The problem of self-adjoint extensions of Dirac-type operators in manifolds with boundaries is analysed. The boundaries might be regular or non-regular. The latter situation includes point-like interactions, also called delta-like…

Eigenfunctions of the Laplace-Beltrami operator on a hyperboloid are studied in the spirit of the treatment of the spherical harmonics by Stein and Weiss. As a special case, a simple self-contained proof of Laplace's integral for a Legendre…

The Dirichlet eigenvalues of the Laplace-Beltrami operator are larger on a flat disc than on any other surface of revoltuion immersed in Euclidean space with the same boundary.

We introduce a technique to obtain pointwise upper and lower bounds for the Green's function of elliptic operators whose principal part is the Laplacian and that include a drift term diverging near the boundary like a power of the inverse…

We develop the first steps towards an analysis of geometry on the quantum spacetime proposed in [1]. The homogeneous elements of the universal differential algebra are naturally identified with operators living in tensor powers of Quantum…

A bosonic Laplacian is a conformally invariant second order differential operator acting on smooth functions defined on domains in Euclidean space and taking values in higher order irreducible representations of the special orthogonal…

We establish an entanglement principle for fractional powers of the Laplace-Beltrami operator on hyperbolic space $\mathbb H^n$, $n\ge 2$. More precisely, we prove that if finitely many distinct noninteger powers of $-\Delta_{\mathbb H^n}$,…

2nd-order conformal superintegrable systems in $n$ dimensions are Laplace equations on a manifold with an added scalar potential and $2n - 1$ independent 2nd order conformal symmetry operators. They encode all the information about…