Isospectrality and non-locality of generalized Dirac combs

TL;DR

The paper studies a one-dimensional quantum system with a periodic array of generalized point interactions, described by the four-parameter self-adjoint extension family U(2) (encompassing $\delta$- and $\delta'$-like cases) and organized as a generalized Dirac comb. Using a Bloch-Floquet decomposition, it derives a spectral function $F_{U,k}(\epsilon)$ that determines the band structure and exposes rich isospectrality structures under unitary and anti-unitary maps, some of which act nonlocally in the bulk. It identifies two continuous families of boundary symmetries (vertical and oblique inner automorphisms) and discrete anti-automorphisms, and shows how these lift to bulk unitary/antiunitary equivalences between Hamiltonians $H_U$ and $H_{\phi_k(U)}$, including both local and nonlocal realizations. The work highlights that non-confining Dirac combs are spectrally degenerate with a $U(1)$ family of isospectral partners, while symmetric Robin (confining) Dirac combs are spectrally unique, offering a clear delineation between hearable and non-hearable spectral data with potential applications to band engineering and topological perspectives.

Abstract

We consider a generalization of Dirac's comb model, describing a non-relativistic particle moving in a periodic array of generalized point interactions. The latter represent the most general point interactions rendering the kinetic-energy operator self-adjoint, and form a four-parameters family that includes the $δ$-potential and the $δ'$-potential as particular cases. We study the parameter dependence of the spectral properties of this system, finding a rich isospectrality structure. We systematically classify a large class of isospectral relations, determining which Hamiltonians are spectrally unique, and which are instead related by a unitary or anti-unitary transformation.

Figures (7)

• Figure 1: Schematic representation of a generalized Dirac comb. At each lattice point $n\in\mathbb{Z}$ there is a generalized point interaction imposing the $\mathrm{U}(2)$ boundary conditions $(I-U)\varPsi_n'=\mathrm{i}\mkern1mu (I+U)\varPsi_n$.
• Figure 2: The physical system described by the fiber Hamiltonian $h_U(k)$ is a ring with two generalized point interactions at antipodal points. In this figure, the left and right point interactions implement respectively the Bloch condition $\psi^{(j)}(1/2)=\mathrm{e}^{\mathrm{i}\mkern1mu k}\psi^{(j)}(-1/2)$ with $j=0,1$ and the $\mathrm{U}(2)$ condition $(I-U)\varPsi_0'=\mathrm{i}\mkern1mu (I+U)\varPsi_0$.
• Figure 3: The first two energy bands for the $\updelta$-potentials $\boldsymbol{g}=(g_1,0,0,0)$ with $g_1=-1$ (left) and $g_1=-2$ (right). The positive energies are shown in red and the negative ones in blue.
• Figure 4: The first two energy bands for the singular metric $\frac{\mathrm{d}}{\mathrm{d}x}\updelta(x)\frac{ \mathrm{d} }{\mathrm{d}x}$ interactions $\boldsymbol{g}=(0,0,0,g_4)$ with $g_4=0.4$ (left) and $g_4=1$ (right). The positive energies are shown in red and the negative ones in blue.
• Figure 5: Pictorial representation of a vertical unitary transformation. Notice how, ignoring the Bloch pseudo-periodicity, the $k$-independent original domain (in blue) is mapped to a $k$-depending domain (in red), and the corresponding bulk Hamiltonian can no longer be interpreted as a (local) generalized Dirac comb.