One-dimensional semirelativistic Hamiltonian with multiple Dirac delta potentials

In this paper, we consider the one-dimensional semirelativistic Schrödinger equation for a particle interacting with $N$ Dirac delta potentials. Using the heat kernel techniques, we establish a resolvent formula in terms of an $N\times N$ matrix, called the principal matrix. This matrix essentially includes all the information about the spectrum of the problem. We study the bound state spectrum by working out the eigenvalues of the principal matrix. With the help of the Feynman-Hellmann theorem, we analyze how the bound state energies change with respect to the parameters in the model. We also prove that there are at most $N$ bound states and explicitly derive the bound state wave function. The bound state problem for the two-center case is particularly investigated. We show that the ground state energy is bounded below, and there exists a self-adjoint Hamiltonian associated with the resolvent formula. Moreover, we prove that the ground state is nondegenerate. The scattering problem for $N$ centers is analyzed by exactly solving the semirelativistic Lippmann-Schwinger equation. The reflection and the transmission coefficients are numerically and asymptotically computed for the two-center case. We observe the so-called threshold anomaly for two symmetrically located centers. The semirelativistic version of the Kronig-Penney model is shortly discussed, and the band gap structure of the spectrum is illustrated. The bound state and scattering problems in the massless case are also discussed. Furthermore, the reflection and the transmission coefficients for the two delta potentials in this particular case are analytically found. Finally, we solve the renormalization group equations and compute the beta function nonperturbatively.

Figure 1

Eigenvalues ${\omega }_1$ and ${\omega }_2$ as a function of $E/m$ for different values of $E_B/m$ (assuming that delta centers are twin, i.e., $E_B^1=E_B^2$ for simplicity) and $2ma=1$. Here $a_1=-a$ and $a_2=a$.

Figure 2

Eigenvalues ${\omega }_1$ and ${\omega }_2$ as a function of $2ma$ for the values $E_B/m=1/2$ and $E/m=1/2$.

Figure 3

Eigenvalues ${\omega }_1$ and ${\omega }_2$ as a function of $E/m$ for $2ma=5$ and $E_B/m=1/2$.

Figure 4

The determinant of the principal matrix as a function of $E/m$ for different values of $2ma$. Here $E_B/m=1/2$.

Figure 5

The reflection and transmission coefficients of two symmetric twin Dirac delta centers as a function of $k/m$ for the values $E_B/m=1/2$, $2ma=1$.

Figure 6

The transmission coefficient as a function of $k/m$ plotted near its first peak $k/m=4$ for $E_B/m=1/2$ and $2ma=1$.

Figure 7

The reflection coefficient as a function of $k/m$ in different scales for $2ma=0.775$ and $E_B/m=1/2$.

Figure 8

The reflection coefficient as a function of $2ma$ in different scales for different values of $k/m$. Here we choose $E_B/m=1/2$.

Figure 9

The reflection coefficient as a function of $2ma$ for $E_B/m=1/10$.

Figure 10

The reflection coefficient $R_{\text{approx}}$ in the asymptotic approximation and the reflection coefficient $R_{\text{numerical}}$ obtained numerically for the particular values of $E_B^1/m=E_B^2/m=1/2$ and $2ma=1$. Notice that they slightly differ only near the region when $k/m$ is zero for a fixed value of $2ma$. This is expected since the asymptotic approximation becomes better and better as $2ka$ takes larger values.

Figure 11

Phase shift $\delta (k)$ as a function of $k/m$ for three different values of $E_B/m$ and for $2ma=1$.

Figure 12

The transmission coefficient as a function of $k/m$ for different values $N=1$, 2, 4, 8, respectively. Here we choose that all $E_B^i$’s are the same, $E_B/m=1/10$, and $m|a_i-a_j|=2$.

Figure 13

The flow of the eigenvalues of the principal matrix as a function of $aE$ in the massless case for different values of $a{E}_B$.

Figure 14

The reflection coefficient as a function of $ka$ for a pair of symmetrically located centers in the massless case for $a|E_B|=1/2$.

Figure 15

The reflection coefficient as a function of $a|E_B|$ for a pair of symmetrically located centers in the massless case ($ka=0.01$).