Narrow Results

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 -parameter family of all time-reversal invariant couplings.

We study the stability problem for a non-relativistic quantum system in dimension three composed by N ≥ 2 identical fermions, with unit mass, interacting with a different particle, with mass m, via a zero-range interaction of strength α ∈ ℝ. We construct the corresponding renormalized quadratic (or energy) form and the so-called Skornyakov–Ter–Martirosyan symmetric extension H_α, which is the natural candidate as Hamiltonian of the system. We find a value of the mass m*(N) such that for m > m*(N) the form is closed and bounded from below. As a consequence, defines a unique self-adjoint and bounded from below extension of H_α and therefore the system is stable. On the other hand, we also show that the form is unbounded from below for m < m*(2). In analogy with the well-known bosonic case, this suggests that the system is unstable for m < m*(2) and the so-called Thomas effect occurs.

Restricting ourselves to a simple rectangular approximation but using properly a two-scale regularization procedure, additional resonant tunneling properties of the one-dimensional Schrödinger operator with a delta derivative potential are established, which appear to be lost in the zero-range limit. These "intrinsic" properties are complementary to the main already proved result that different regularizations of Dirac's delta function produce different limiting self-adjoint operators. In particular, for a given regularizing sequence, a one-parameter family of connection condition matrices describing bound states is constructed. It is proposed to consider the convergence of transfer matrices when the potential strength constant is involved into the regularization process, resulting in an extension of resonance sets for the transmission across a δ′-barrier.

We investigate the propagation of gravitational waves in linearized Chern–Simons (CS) modified gravity by considering two nondynamical models for the coupling field $𝜃$: (i) a domain wall and (ii) a surface layer of $𝜃$, motivated by their relevance in condensed matter physics. We demonstrate that the metric and its first derivative become discontinuous for a domain wall of $𝜃$, and we determine the boundary conditions by realizing that the additional contribution to the wave equation corresponds to one of the self-adjoint extensions of the D'Alembert operator. Nevertheless, such discontinuous metric satisfies the area matching conditions introduced by Barrett. On the other hand, the propagation through a surface layer of $𝜃$ behaves similarly to the propagation of electromagnetic waves in CS extended electrodynamics. In both cases, we calculate the corresponding reflection and transmission amplitudes. As a consequence of the distributional character of the additional terms in the equations that describe wave propagation, the results obtained for the domain wall are not reproduced when the thickness of the surface layer goes to zero, as one could naively expect.

We study the bound state problem for $N$ attractive point Dirac $\delta$-interactions in two- and three-dimensional Riemannian manifolds. We give a sufficient condition for the Hamiltonian to have $N$ bound states and give an explicit criterion for it in hyperbolic manifolds ${ℍ}^2$ and ${ℍ}^3$. Furthermore, we study the same spectral problem for a relativistic extension of the model on ${ℝ}^2$ and ${ℍ}^2$.

We discuss the stability problem for a system of N identical fermions with unit mass interacting with a different particle of mass m via zero-range interactions in dimension three. We find a stability parameter m*(N) > 0, increasing with N, such that the Hamiltonian of the system is self-adjoint and bounded from below for m > m*(N).