Universal properties of infrared oscillator basis extrapolations

Recent work has shown that a finite harmonic oscillator basis in nuclear many-body calculations effectively imposes a hard-wall boundary condition in coordinate space, motivating infrared extrapolation formulas for the energy and other observables. Here we further refine these formulas by studying two-body models and the deuteron. We accurately determine the box size as a function of the model space parameters, and compute scattering phase shifts in the harmonic oscillator basis. We show that the energy shift can be well approximated in terms of the asymptotic normalization coefficient and the bound-state momentum, discuss higher-order corrections for weakly bound systems, and illustrate this universal property using unitarily equivalent calculations of the deuteron.

Figure 1

(a) The exact radial wave function (dashed) for a square well Eq. (3) with depth $V_0=4$ (and $\hslash =\mu =R=1$) is compared to the wave function obtained from an HO basis truncated at $N=4$ with $\hslash \Omega =6$ (solid). The spatial extent of the wave function obtained from the HO basis truncation is dictated by the square of HO wave function for the highest radial quantum number (dot-dashed). (b) The wave functions obtained from imposing a Dirichlet boundary condition at $L_0$, $L_0^{\prime }$, and $L_2$ are compared to the wave function in truncated HO basis.

Figure 2

Ground-state energies versus (a) $L_0$, (b) $L_0^{\prime }$, and (c) $L_2$ for a Gaussian potential well Eq. (5) with $V_0=5$ (and $\hslash =\mu =R=1$). The crosses are the energies from HO basis truncation. The energies obtained by numerically solving the Schrödinger equation with a Dirichlet boundary condition at $L$ lie on the solid line. The horizontal dotted lines mark the exact energy, $E_{\infty }=-1.27$.

Figure 3

Ground-state energies versus (a) $L_0$, (b) $L_0^{\prime }$, and (c) $L_2$ for a square well potential well Eq. (3) with $V_0=4$ (and $\hslash =\mu =R=1$). The crosses are the energies from HO basis truncation. The energies obtained by numerically solving the Schrödinger equation with a Dirichlet boundary condition at $L$ lie on the solid line. The horizontal dotted lines mark the exact energy, $E_{\infty }=-1.51$.

Figure 4

Ground-state energies versus (a) $L_0$, (b) $L_0^{\prime }$, and (c) $L_2$ for the Entem-Machleidt 500 MeV N${}^3$LO potential [7]. The horizontal dotted lines mark the exact energy, $E_{\infty }=-2.2246\text{MeV}$.

Figure 5

Deuteron radius squared versus (a) $L_0$ and (b) $L_2$ for the Entem-Machleidt 500 MeV N${}^3$LO potential [7]. The horizontal dotted lines mark the exact radius squared, $r_{\infty }^2=3.9006{\mathrm{fm}}^2$. The insets show a magnification of data at smaller lengths $L_n$.

Figure 6

The staircase function of the $s$ states of the operator $p^2$ in a finite oscillator basis with $N=32$ (black) compared to its semiclassical estimate (smooth red curve). $M\left(k\right)$ denotes the number of states of the operator $p^2$ with eigenvalues $p^2\le {\hslash }^2{k}^2$.

Figure 7

The ${}^1{S}_0$ phase shifts (in degrees) of the N${}^3$LO chiral interaction (solid line) compared to the phase shifts computed directly in the harmonic oscillator basis (circles).

Figure 8

The ${}^3{P}_1$ phase shifts (in degrees) of the N${}^3$LO chiral interaction (solid line) compared to the phase shifts computed directly in the harmonic oscillator basis (circles).

Figure 9

The IR energy correction $\Delta E_L$ versus $L_2$ for a Gaussian potential well Eq. (5) with $V_0=5$ (and $\hslash =\mu =R=1$) using a wide range of $N$ and $\hslash \Omega$. The energies are fitted with (a) exponential, (b) Gaussian, and (c) power-law dependence on $L_2$.

Figure 10

The IR energy correction $\Delta E_L$ versus $L_2$ for the deuteron calculated with the chiral EFT potential from Ref. [7] using a wide range of $N$ and $\hslash \Omega$. The energies are fitted with (a) exponential, (b) Gaussian, and (c) power-law dependence on $L_2$.

Figure 11

Testing the linear energy approximation Eq. (24) for (a) deep ($V_0=10$) and (b) shallow ($V_0=2$) Gaussian potential well Eq. (5) ($\hslash =\mu =R=1$). The solid lines are the exact solutions $u_L\left(r\right)$ for energies $-3.5$ and $-0.020$, respectively, whose zero crossings determine the corresponding values for $L$.

Figure 12

Energy versus $L_2$ for a quartic potential well Eq. (6) for a wide range of $N$ and $\hslash \Omega$ (circles) ($\hslash =\mu =R=1$). The solid line is a fit to Eq. (1) with $A$, $k_{\infty }$, and $E_{\infty }$ as fit parameters while the dashed and dot-dashed lines are predictions from Eqs. (42) and (44). The horizontal line is the exact energy, $E_{\infty }=-1.0115$. The inset illustrates the calculation of the asymptotic normalization coefficient (ANC) from the (normalized) wave function.

Figure 13

Energy versus $L_2$ for moderate-depth (a) square well Eq. (3) and for (b) Gaussian potential well Eq. (5) ($\hslash =\mu =R=1$) for a wide range of $N$ and $\hslash \Omega$ (circles). The solid line is a fit to Eq. (1) with $A$, $k_{\infty }$, and $E_{\infty }$ as fit parameters while the dashed and dot-dashed lines are predictions from Eqs. (42) and (44). The horizontal dotted lines are the exact energies; square well: $E_{\infty }=-1.5088$, Gaussian well: $E_{\infty }=-1.2717$.

Figure 14

Energy versus $L_2$ for the deeply bound ground state of a Gaussian potential for a wide range of $N$ and $\hslash \Omega$ (circles) ($\hslash =\mu =R=1$). These are compared to the predictions of Eq. (42) (dashed) and Eq. (44) (dot-dashed). The solid line is a fit to Eq. (1) with $A$, $k_{\infty }$, and $E_{\infty }$ as fit parameters. The horizontal dotted line is the exact energy, $E_{\infty }=-4.2806$.

Figure 15

Energy versus $L_2$ for the deeply bound ground state of an exponential potential well for a wide range of $N$ and $\hslash \Omega$ (circles) ($\hslash =\mu =R=1$). These are compared to the predictions of Eq. (42) (dashed) and Eq. (44) (dot-dashed). The solid line is a fit to Eq. (1) with $A$, $k_{\infty }$, and $E_{\infty }$ as fit parameters. The horizontal dotted line is the exact energy, $E_{\infty }=-3.3121$.

Figure 16

Energy versus $L_2$ for the first excited states of deep (a) Gaussian Eq. (5) and (b) quartic Eq. (6) potential wells for a wide range of $N$ and $\hslash \Omega$ (circles) ($\hslash =\mu =R=1$). The solid line is a fit to Eq. (1) with $A$, $k_{\infty }$, and $E_{\infty }$ as fit parameters while the dashed and dot-dashed lines are predictions from Eqs. (42) and (44). The horizontal dotted lines are the exact energies for the first excited states; Gaussian well: $E_{\infty }=-1.2147$, quartic well: $E_{\infty }=-1.8236$.

Figure 17

(a) Ground-state energy versus $L_2$ for model Gaussian potential. (b) Energy versus $L$ for the square well. The energies for the square well are from solving the Schrödinger equation exactly with a Dirichlet boundary condition on wave functions at $r=L$. The dashed and dot-dashed lines are predictions from Eqs. (42) and (44). The depths of these model potentials are chosen so that the scaled energies (with $\hslash =\mu =R=1$) are the same as the deuteron binding energy.

Figure 18

Deuteron energy versus $L_2$ for the potential of Ref. [7]. To eliminate the UV contamination we only keep points for which $\hslash \Omega >49$. The dashed and dot-dashed lines are predictions from Eqs. (42) and (44). The horizontal dotted line is the deuteron binding energy.

Figure 19

Deuteron energy versus $L_2$ for the potential of Ref. [7] evolved by the SRG to four different resolutions (specified by $\lambda$). To eliminate the UV contamination we only keep points for which $\hslash \Omega >40$. The dashed and dot-dashed lines are predictions from Eqs. (42) and (44). The horizontal dotted line is the deuteron binding energy.

Figure 20

The same SRG-evolved potentials as in Fig. 19 are used to generate energies, but with $N$ fixed at (a) 8 and (b) 12 and no restriction on $\hslash \Omega$. Thus UV corrections are not negligible everywhere. The dashed and dot-dashed lines are predictions from Eqs. (42) and (44). The horizontal dotted line is the deuteron binding energy.

Figure 21

Triton energy versus $L_2$ (here calculated with the deuteron-neutron reduced mass) for the two- and three-nucleon potential in Ref. [27] unitarily evolved by the SRG to four different resolutions (specified by $\lambda$) with the same binding energy [27, 28]. Only larger $\hslash \Omega$ points are plotted to minimize the UV contamination. The horizontal dotted line is the exact triton binding energy for this interaction.