The aim of this review is to provide an overview of a recent work concerning``leaky''quantum
graphs described by Hamiltonians given formally by the expression $-\Delta-\alpha\delta (x …

The spectral theory of linear operators plays a key role in the mathematical formulation of
quantum theory. This textbook provides a concise and comprehensible introduction to the …

The scattering properties of regularizing finite-range potentials constructed in the form of
squeezed rectangles, which approximate the first and second derivatives of the Dirac delta …

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 …

We rigorously define the self-adjoint Hamiltonian of the harmonic oscillator perturbed by an
attractive δ'-interaction, of strength β, centred at 0 (the bottom of the confining parabolic …

We consider the one-dimensional Hamiltonian with a V-shaped potential H 0= 1 2− d 2 dx
2+| x|, decorated with a point impurity of either δ-type, or local δ′-type or even nonlocal δ …

We show that formal Schr\" odinger operators with singular potentials from the space W^{-1}
_ {2, unif}(R) can be naturally defined to give selfadjoint and bounded below operators …

We determine and study the ground states of a focusing Schrödinger equation in dimension
one with a power nonlinearity| ψ| 2 μ ψ and a strong inhomogeneity represented by a …

In one-dimensional quantum gases there is a well known “duality” between hard core
bosons and noninteracting fermions. However, at the field theory level, no exact duality …

We discuss approximations of vertex couplings of quantum graphs using families of thin
branched manifolds. We show that if a Neumann-type Laplacian on such manifolds is …