Ondrej Turek

Approximation of a general singular vertex coupling in quantum graphs

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

On the spectrum of a bent chain graph

delta

APPROXIMATIONS OF SINGULAR VERTEX COUPLINGS IN QUANTUM GRAPHS

potential and a vector potential coupled to the loose edges by a

Fulop–Tsutsui interactions on quantum graphs

delta

Spectral Filtering in Quantum Y-Junction

coupling. It is shown that if the lengths of the connecting edges shrink to zero and the potentials are properly scaled, the limit can yield any prescribed singular vertex coupling, and moreover, that such an approximation converges in the norm-resolvent sense.

Tripartite connection condition for a quantum graph vertex

We study Schrodinger operators on an infinite quantum graph of a chain form which consists of identical rings connected at the touching points by δ-couplings with a parameter . If the graph is straight, i.e. periodic with respect to ring shifts, its Hamiltonian has a band spectrum with all the gaps open whenever α ≠ 0. We consider a bending deformation of the chain consisting of changing one position at a single ring and show that it gives rise to eigenvalues in the open spectral gaps. We analyze dependence of these eigenvalues on the coupling α and the bending angle as well as resonances of the system coming from the bending. We also discuss the behaviour of the eigenvalues and resonances at the edges of the spectral bands.

Hermitian unitary matrices with modular permutation symmetry

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 Cheon–Shigehara technique using δ interactions with nonlinearly scaled couplings yields a 2n-parameter family of boundary conditions in the sense of norm resolvent topology. Moreover, using graphs with additional edges, one can approximate the

Equiangular tight frames and unistochastic matrices

{n+1choose 2}

Spectrum of a Dilated Honeycomb Network

-parameter family of all time-reversal invariant couplings.

High-energy asymptotics of the spectrum of a periodic square lattice quantum graph

We examine scale invariant Fulop–Tsutsui couplings in a quantum vertex of a general degree n. We demonstrate that essentially same scattering amplitudes as for the free coupling can be achieved for two (n−1)-parameter Fulop–Tsutsui subfamilies if n is odd, and for three (n−1)-parameter Fulop–Tsutsui subfamilies if n is even. We also work up an approximation scheme for a general Fulop–Tsutsui vertex, using only n δ function potentials.