Figure 1

Difference in spectrum between the DVR Hamiltonian (24) and the HO energies $(n+1/2)\hslash \omega$. The three (blue) curves with pluses have fixed UV scale (lattice constant $a=1$, $k_c=\pi /a$) with $L=\in \{30,40,50\}$ and $\omega =2\pi /L$ from left to right. The (red) curves with dots have fixed $L=30$ but varying lattice constant $a\in \{1/2,1/3\}$ demonstrating the UV convergence. The sizes of the DVR basis sets are $L{k}_c/\pi =30$, 40, and 50 (blue pluses) and 60 and 90 (red circles), respectively. For the blue pluses, the corresponding number of HO wave functions suggested in Refs. [1, 2] [see also Eqs. (4)] would be $N=L{k}_c/4=L\pi /4a\approx 24$, 31, 39; and 47 and 71 for the red dots, respectively. Notice that the size of the DVR basis set can be reduced by a factor of 2 to $L{k}_c/2\pi =$ 15, 20, 25 (blue) and 30, 45 (red) respectively, by imposing Dirichlet boundary conditions; however, in that case, states not localized to a single cell will not be reproduced.Reuse & Permissions

Figure 2

Exponential convergence of the periodic DVR basis for the energy of the bound states of the analytically solvable Scarf II potential $V\left(x\right)=[a+b\sinh x]/{\cosh }^{2}x$ (with $\hslash =m=1$). For $a=7/2$ and $b=-11/2$, the potential has three bound states: $E_n=-{(3-n)}^2/2$ (shown in black, blue, and green from left to right, respectively). The left plot demonstrates the IR convergence for increasing $L$ with fixed $k_c$; the right plot, the UV convergence for increasing $k_c$ for fixed $L$. The various values for $k_c\in \{5,10,15,20\}$ (left) and $L\in \{5,15,25,35\}$ (right) correspond to dotted, dot-dashed, dashed, and solid lines with increasing convergence, respectively.Reuse & Permissions

Figure 3

Binding energy of a three-particle “triton” and four-particle “$\alpha$” ground state using various multiples (specified by numerical factors in the figure) of the potentials $V_{\text{PT}}\left(r\right)\propto -{\mathrm{sech}}^2(2r/r_0)$ and $V_{2\text{G}}\left(r\right)\propto \exp (-r^2/r_0^2)-4\exp (-4r^2/r_0^2)$ with $r_0=3\mathrm{fm}$ (see [24] for explicit normalizations). Upper and lower bounds are obtained from Dirichlet and periodic boundary conditions, respectively. The deeply bound four-body state with $1.5V_{\text{PT}}$ is not converged and has comparable UV and IR errors (each “band” has fixed lattice spacing). The other results are UV converged: the different lattice spacings lying on the same curves describing the dependence on the box size. The inset shows the radial profile of the two potentials.Reuse & Permissions