Resonant-state expansion applied to one-dimensional quantum systems

The resonant-state expansion, a rigorous perturbation theory recently developed in electrodynamics, is applied to nonrelativistic quantum-mechanical systems in one dimension. The method is used here for finding the resonant states in various potentials approximated by combinations of Dirac $\delta$ functions. The resonant-state expansion is first verified for a triple-quantum-well system, showing convergence to the available analytic solution as the number of resonant states in the basis increases. The method is then applied to multiple-quantum-well and barrier structures, including finite periodic systems. Results are compared with the eigenstates in triple quantum wells and infinite periodic potentials, revealing the nature of the resonant states in the studied systems.

Figure 1

(a) Eigen wave numbers of the resonant states for a double symmetric quantum well with $\gamma =3/a$ (open circles) and a triple symmetric quantum well with $b=0$ and $\gamma =\beta =3/a$, calculated using the QM-RSE (red crosses) and the analytic secular equation (13) (blue squares, shown for even RSs only). The wave numbers of even (black circles) and odd (green circles) unperturbed RSs for a double-quantum-well structure are calculated via Eq. (9). The insets show a zoom-in of a particular area and a sketch of the perturbed potential (black lines) and the perturbation used (red line). (b) Relative error of the QM-RSE values of the RS wave numbers as a function of the real or imaginary part of the wave number, for different basis sizes $M$ as given. The dashed line shows a power-law dependence as labeled.

Figure 2

As Fig. 1, but for $b=a/3$.

Figure 3

As Fig. 2, but for $b=a/2$ (a), $b=3a/5$ (b), and $b=4a/5$ (c).

Figure 4

As Fig. 1, but for a finite periodic potential with $N=4$ quantum wells of depth $\gamma =10/a$. The perturbation used in the QM-RSE is given by Eq. (19). The resonant states of a triple-well structure with $\beta =\gamma =10/a$ and $b=a/3$ are shown additionally (blue stars). The relative error in (b) is calculated using the QM-RSE values for $M=4480$ replacing the exact solution.

Figure 5

As Fig. 4 but for $N=5$, 6, and 11. Additionally, the resonant states for a triple-well structure with $\beta =\gamma =10/a$ and $b=a/2, 3a/5$, and $4a/5$ are shown in (a), (b), and (c), respectively.

Figure 6

(Top) Solution of the Kronig-Penney model Eq. (23). (Bottom) Resonant-state wave numbers of a finite periodic quantum lattice with $N=20$ calculated using the QM-RSE, for $\gamma =10/a$ (wells) and $\gamma =-10/a$ (barriers).

Figure 7

Resonant-state wave numbers of a finite periodic lattice of $N=20$ quantum barriers, calculated using the QM-RSE for different barrier strengths $\gamma$ as given.