Bootstrapping Shape Invariance: Numerical Bootstrap as a Detector of Solvable Systems

TL;DR

The paper demonstrates that the numerical bootstrap method can diagnose solvability in one-dimensional quantum systems by producing tightly constrained or exact energy spectra. For shape-invariant (SI) systems, the bootstrap matrix built from suitably chosen operators yields exact energy eigenvalues and reveals the underlying annihilation-operator structure, connecting positivity bounds directly to SI algebra. Through detailed analyses of the harmonic oscillator and Morse potentials (and related SI families like Rosen–Morse and hyperbolic Scarf), the authors show how rigid boundaries emerge and how annihilation data can be read from zero modes, thereby explaining solvability. The approach offers a practical, scalable detector of solvable systems, with potential extensions to Krein–Adler-transformed models and beyond, including many-body and quasi-exactly solvable settings.

Abstract

Determining the solvability of a given quantum mechanical system is generally challenging. We discuss that the numerical bootstrap method can help us to solve this question in one-dimensional quantum mechanics. We show that the bootstrap method can derive exact energy eigenvalues in systems with shape invariance, which is a sufficient condition for solvability and which many solvable systems satisfy. The information of the annihilation operators is also obtained naturally, and thus the bootstrap method tells us why the system is solvable. We numerically demonstrate this explicitly for shape invariant potentials: harmonic oscillators, Morse potentials, Rosen-Morse potentials and hyperbolic Scarf potentials. Therefore, the numerical bootstrap method can determine the solvability of a given unknown system if it satisfies shape invariance.

Figures (12)

• Figure 1: Numerical bootstrap result for the anharmonic oscillator \ref{['H-AHO']}. We used the bootstrap matrix ${\mathcal{M}}_{xp}^{(AHO)}$\ref{['bootstrap-XP2']} and $K_x$ and $K_p$ defined in Eq. \ref{['operators-XP']} determine the size of the bootstrap matrix. We use the numerical linear programming to find the upper and lower bounds of $\Braket{x}$ at each fixed value of $E$. The colored regions show the allowed regions in the $(E,\Braket{x})$ plane that satisfy the condition ${\mathcal{M}}_{xp}^{(AHO)} \succeq 0$. The results for $(K_x,K_p)=(4,3)$ are almost points, and they are highlighted with red circles. The regions for $(K_x,K_p)=(3,2)$ and $(K_x,K_p)=(4,3)$ are difficult to see in this figure, and see Table \ref{['Table-AHO-XP']} for the allowed values of $E$. As $K_x$ and $K_p$ increase, the allowed regions shrink, and converge to the results at $(K_x,K_p)=(4,3)$. Here, $\Braket{x}$ converges to 0, as expected from the parity.
• Figure 2: Numerical bootstrap result for the anharmonic oscillator \ref{['H-AHO']}. We use the same analysis as the one in Fig. \ref{['fig-x4-x']}, but we solve the linear programming with respect to $\Braket{x^2}$ instead of $\Braket{x}$. The obtained allowed regions for $E$ is consistent with Fig. \ref{['fig-x4-x']}, and they are summarized in Table \ref{['Table-AHO-XP']}. Again, the regions for $(K_x,K_p)=(3,2)$ and $(K_x,K_p)=(4,3)$ are difficult to see in this figure, and see Table \ref{['Table-AHO-XP']} for the details.
• Figure 3: The Morse potential \ref{['eq-Morse-H']} at $(h,\mu)=(13/4,1)$. The red dashed lines represent the energy eigenvalues \ref{['eq-Morse-E2']} of the bound states (four bound states appear at $h=13/4=3.25$). The bottom of the potential is set to be $-h-1/4$ (at $\mu e^x=h+1/2$) such that the ground state energy $E_0$ is zero. The continuous spectrum appears in $E\geq h^2$.
• Figure 4: Bootstrap analysis of the Morse potential \ref{['eq-Morse-H']} at $(h,\mu)=(13/4,1)$. We use the operators $\{ e^{mx} p^n \}$ ($m=0,1,\cdots,K_x$ and $n=0,1,\cdots,K_p$) in Eq. \ref{['operators-XP-Morse']} to construct the bootstrap matrix ${\mathcal{M}}^{(\text{Morse})}$\ref{['eq-bootstrap-matrix-Morse-XP']}, and solve the linear programming discussed in Sec. \ref{['sec-SDP']}. At $h=13/4=3.25$, $[h]' =3$ and four bound states appear. The small red circles represent the exact energy eigenstates \ref{['eq-Morse-E2']} (the continuous spectrum $E>h^2$ is omitted). The inside of the curves and the dots represent the allowed regions where the energy eigenstates can exist. At $(K_x,K_p)=(3,3)$, the numerical bootstrap method reproduces the known analytic results including the continuous spectrum. Since some allowed points are not visible because they overlap, see Table \ref{['Table-Morse-XP']} for the details. Note that the allowed points in the numerical linear programming are slightly smeared due to numerical error. See Appendix \ref{['app-exact-numerical']} for more details.
• Figure 5: The Tolerance dependence of the allowed regions of the Morse potential and the anharmonic oscillator obtained by using the numerical linear programming. These are the enlarged view of Figs. \ref{['fig-x4']} and \ref{['fig-Morse-XP']}. The curves represent the boundaries of the isolated allowed regions obtained by changing the Tolerance with a fixed size of the bootstrap matrix. In the anharmonic oscillator case, the curve is almost insensitive to the Tolerance, and the difference is almost invisible in the figure. On the other hand, in the Morse potential case, it strongly depends on the Tolerance, and converges to the exact result (the details are summarized in Table. \ref{['Table-Morse-Tol']}). Also, the boundary is jagged and noisy due to numerical error. Thus, the isolated allowed regions of the Morse potential and the anharmonic oscillator are significantly different.

Theorems & Definitions (5)

• Definition 1
• Definition 2
• Definition 3
• Definition 4
• Definition 5