Quantum Mechanics Bootstrap

• Quantum-mechanics bootstrap is a method that infers spectral data using positivity constraints, operator identities, and self-consistency conditions instead of solving the Schrödinger equation.
• It employs semidefinite optimization and moment matrices to derive rigorous energy bounds and capture non-perturbative features in diverse quantum systems.
• The framework has broad applications including 1D confining Hamiltonians, periodic and non-Hermitian systems, and low-energy nuclear physics, demonstrating precise spectral predictions.

Quantum-mechanics bootstrap is a constraint-based program for quantum systems in which spectral data, moments, and, in some extensions, dynamical observables are inferred from positivity, operator identities, and self-consistency conditions rather than from direct solution of the Schrödinger equation. The bootstrap philosophy dates back to the 1960s, but its recent formulation in quantum mechanics was shaped by analogies with conformal bootstrap and relativistic scattering-amplitude bootstrap, and it has since been applied to confining one-dimensional Hamiltonians, periodic systems, scattering on the half-line, low-energy nuclear Hamiltonians, non-Hermitian models, Calabi–Yau quantum systems, time evolution, and three-dimensional central potentials (Bai, 2022, Berenstein et al., 2022, Berenstein et al., 2023, Khan et al., 2024, Lawrence et al., 2024, Lawrence et al., 9 Dec 2025).

1. Foundational framework

The basic quantum-mechanical input is the positivity statement that for any state $|\Psi\rangle$ and any operator $\mathcal O$,

$$\langle \Psi|\mathcal O^\dagger \mathcal O|\Psi\rangle \ge 0.$$

Choosing a finite operator set $\{\mathcal O_i\}$ and writing $\mathcal O=\sum_i \alpha_i \mathcal O_i$ turns this into positivity of a bootstrap, Gram, or moment matrix,

$$\mathcal M_{ij}=\langle \mathcal O_i^\dagger \mathcal O_j\rangle, \qquad \mathcal M\succeq 0.$$

In one-dimensional polynomial problems this often reduces to a Hankel matrix $M^{(K)}_{ij}=x_{i+j}$, where $x_n=\langle x^n\rangle$ and $x_0=1$; in finite-dimensional Hamiltonian truncations, such as the deuteron in a harmonic-oscillator basis, it becomes a matrix of expectation values of projectors $|i\rangle\langle j|$ (Berenstein et al., 2022, Bai, 2022).

The second ingredient is the eigenstate condition. For an energy eigenstate $\mathcal O$0 with $\mathcal O$1, one imposes relations such as

$\mathcal O$2

or their appropriate non-Hermitian generalizations. These identities generate recursions among moments and mixed correlators. Depending on the problem, the operator basis may be monomials in $\mathcal O$3 and $\mathcal O$4, exponentials $\mathcal O$5, circle operators $\mathcal O$6, or finite-basis projectors $\mathcal O$7 (Berenstein et al., 2022, Du et al., 2021, Aikawa et al., 2021).

In this formulation, the bootstrap can also be regarded as a generalization of uncertainty relations. For fixed-energy states, the same positivity machinery that yields the Heisenberg inequality yields state-independent bounds on observables such as $\mathcal O$8, $\mathcal O$9, and $$\langle \Psi|\mathcal O^\dagger \mathcal O|\Psi\rangle \ge 0.$$0. For the harmonic oscillator, for example, the universal bound

$$\langle \Psi|\mathcal O^\dagger \mathcal O|\Psi\rangle \ge 0.$$1

follows from this broader positivity structure (Morita, 2022).

2. Linearization and semidefinite optimization

A central technical step is to fix the trial energy $$\langle \Psi|\mathcal O^\dagger \mathcal O|\Psi\rangle \ge 0.$$2. In polynomial one-dimensional systems this linearizes the recursion relations among the moments. For a potential

$$\langle \Psi|\mathcal O^\dagger \mathcal O|\Psi\rangle \ge 0.$$3

the bootstrap reduces higher moments to a finite set of independent moments $$\langle \Psi|\mathcal O^\dagger \mathcal O|\Psi\rangle \ge 0.$$4, and the truncated Hankel matrix can be written as

$$\langle \Psi|\mathcal O^\dagger \mathcal O|\Psi\rangle \ge 0.$$5

The spectral question then becomes a semidefinite feasibility problem: does there exist a set of low moments such that $$\langle \Psi|\mathcal O^\dagger \mathcal O|\Psi\rangle \ge 0.$$6? A useful equivalent formulation introduces a slack variable $$\langle \Psi|\mathcal O^\dagger \mathcal O|\Psi\rangle \ge 0.$$7 and maximizes the minimal eigenvalue through

$$\langle \Psi|\mathcal O^\dagger \mathcal O|\Psi\rangle \ge 0.$$8

Positive optimal $$\langle \Psi|\mathcal O^\dagger \mathcal O|\Psi\rangle \ge 0.$$9 means that the chosen $\{\mathcal O_i\}$0 is allowed at truncation depth $\{\mathcal O_i\}$1; negative optimal $\{\mathcal O_i\}$2 quantifies failure of positivity (Berenstein et al., 2022).

The same logic extends beyond the simplest Hankel setup. In half-line problems one must impose positivity of both $\{\mathcal O_i\}$3 and $\{\mathcal O_i\}$4, conveniently encoded by a block-diagonal semidefinite constraint. In finite harmonic-oscillator spaces, as in the deuteron problem, one solves linear expectation-value constraints from normalization and the discrete eigenvalue equation and then checks positivity of the resulting finite bootstrap matrix (Berenstein et al., 2023, Bai, 2022).

A compact summary of recurrent formulations is given below.

Setting	Bootstrap data	Representative papers
1D polynomial Schrödinger operators	$\{\mathcal O_i\}$5, Hankel matrix $\{\mathcal O_i\}$6	(Berenstein et al., 2022)
Finite-basis nuclear Hamiltonians	$\{\mathcal O_i\}$7, projector bootstrap matrix	(Bai, 2022)
Circle and periodic systems	$\{\mathcal O_i\}$8 or mixed position-momentum moments	(Aikawa et al., 2021, Blacker et al., 2022)
Half-line scattering	$\{\mathcal O_i\}$9, $\mathcal O=\sum_i \alpha_i \mathcal O_i$0 with boundary anomalies	(Berenstein et al., 2023)

This optimization viewpoint also clarifies why the method often produces bounds rather than exact values at finite truncation. The admissible set is convex, and increasing the operator basis or matrix size monotonically tightens the feasible region.

3. Benchmark systems and exact solvability

The earliest systematic tests focused on analytically tractable systems and simple confining Hamiltonians. For the hydrogen atom and the harmonic oscillator, the bootstrap resolves many energy levels and the spectra converge exponentially fast as the matrix size increases (Berenstein et al., 2021). For the quartic anharmonic oscillator and the double-well potential, the method captures non-perturbative structure; in particular, the double-well study found that the bootstrap correctly captures non-perturbative aspects, while supersymmetric partner potentials reproduce the expected paired spectra, and the singlet sector of the $\mathcal O=\sum_i \alpha_i \mathcal O_i$1 vector model agrees with large-$\mathcal O=\sum_i \alpha_i \mathcal O_i$2 saddle-point analysis (Bhattacharya et al., 2021).

A distinct line of work concerns exactly solvable systems. For shape-invariant one-dimensional Hamiltonians, the bootstrap can derive exact energy eigenvalues analytically, and the information of the annihilation operators is also obtained naturally. This was demonstrated numerically for harmonic oscillators, Morse potentials, Rosen–Morse potentials, and hyperbolic Scarf potentials, leading to the claim that the numerical bootstrap can determine the solvability of a given unknown system if it satisfies shape invariance (Aikawa et al., 11 Apr 2025).

Exactness can also hinge on operator ordering. In the interval model with Hamiltonian

$\mathcal O=\sum_i \alpha_i \mathcal O_i$3

defined on $\mathcal O=\sum_i \alpha_i \mathcal O_i$4, one ordering of the bootstrap constraints restricts the energy only into bands, whereas an alternative ordering makes a finite number of constraints sufficient to fix the low-lying energy levels exactly. Since the exact spectrum is

$\mathcal O=\sum_i \alpha_i \mathcal O_i$5

this model provides a concrete example in which finite bootstrap data reproduces exact results rather than asymptotic bounds (Sword et al., 2024).

For generic polynomial confining potentials, exactness is replaced by rapid numerical convergence. In the eighth-degree test potential

$\mathcal O=\sum_i \alpha_i \mathcal O_i$6

the interval widths for the first five excited states shrink exponentially with truncation depth,

$\mathcal O=\sum_i \alpha_i \mathcal O_i$7

and the bootstrap values at $\mathcal O=\sum_i \alpha_i \mathcal O_i$8 match finite-element benchmarks to the quoted precision (Berenstein et al., 2022).

4. Periodicity, boundary conditions, and scattering

Periodic and compact-configuration systems require operator bases adapted to translation or winding structure. In the Kronig–Penney problem, the bootstrap was applied to the Schrödinger equation with periodic potentials, with an operator basis involving position and momenta. In this setting the method efficiently computes band gaps of the energy spectrum but has trouble effectively constraining the minimum energy; more complex constraints involving higher powers of momenta were proposed as a possible remedy. The same study also proposed an approach for analytically constructing the dispersion relation associated with the Bloch momentum of the system (Blacker et al., 2022).

A related compact system is quantum mechanics on a circle with a $\mathcal O=\sum_i \alpha_i \mathcal O_i$9-term, implemented as a charged particle on $$\mathcal M_{ij}=\langle \mathcal O_i^\dagger \mathcal O_j\rangle, \qquad \mathcal M\succeq 0.$$0 in a constant gauge potential. Here the bootstrap correctly reproduces correlations among observables for energy eigenstates for any $$\mathcal M_{ij}=\langle \mathcal O_i^\dagger \mathcal O_j\rangle, \qquad \mathcal M\succeq 0.$$1, but it is hard to determine physical quantities as functions of $$\mathcal M_{ij}=\langle \mathcal O_i^\dagger \mathcal O_j\rangle, \qquad \mathcal M\succeq 0.$$2, such as $$\mathcal M_{ij}=\langle \mathcal O_i^\dagger \mathcal O_j\rangle, \qquad \mathcal M\succeq 0.$$3, except at $$\mathcal M_{ij}=\langle \mathcal O_i^\dagger \mathcal O_j\rangle, \qquad \mathcal M\succeq 0.$$4 and $$\mathcal M_{ij}=\langle \mathcal O_i^\dagger \mathcal O_j\rangle, \qquad \mathcal M\succeq 0.$$5. This identifies a structural difficulty associated with gauge choice and periodicity rather than a failure of positivity itself (Aikawa et al., 2021).

Scattering on the half-line requires a further modification because boundary conditions become part of the bootstrap data. For

$$\mathcal M_{ij}=\langle \mathcal O_i^\dagger \mathcal O_j\rangle, \qquad \mathcal M\succeq 0.$$6

on $$\mathcal M_{ij}=\langle \mathcal O_i^\dagger \mathcal O_j\rangle, \qquad \mathcal M\succeq 0.$$7, the basic operator identity becomes

$$\mathcal M_{ij}=\langle \mathcal O_i^\dagger \mathcal O_j\rangle, \qquad \mathcal M\succeq 0.$$8

where the second term is an anomaly reflecting the fact that $$\mathcal M_{ij}=\langle \mathcal O_i^\dagger \mathcal O_j\rangle, \qquad \mathcal M\succeq 0.$$9 may not preserve the domain of the self-adjoint Hamiltonian. For Robin boundary conditions

$M^{(K)}_{ij}=x_{i+j}$0

the bootstrap imposes positivity of both $M^{(K)}_{ij}=x_{i+j}$1 and $M^{(K)}_{ij}=x_{i+j}$2. Varying the Robin parameter $M^{(K)}_{ij}=x_{i+j}$3 traces the discrete half-line spectrum $M^{(K)}_{ij}=x_{i+j}$4, and for a purely reflecting one-dimensional problem the reflection coefficient is recovered from

$M^{(K)}_{ij}=x_{i+j}$5

This framework was applied to half-harmonic walls, a metastable quartic potential, and the exponential potential of Liouville theory, with WKB used to connect the bootstrap data to the asymptotic scattering phase (Berenstein et al., 2023).

5. Generalizations to realistic, geometric, non-Hermitian, and dynamical settings

One prominent extension is low-energy nuclear physics. “Bootstrapping the deuteron” formulates the deuteron in the $M^{(K)}_{ij}=x_{i+j}$6 channel within pionless effective field theory in a harmonic-oscillator basis and reports the first bootstrap results in low-energy nuclear physics. The Hamiltonian is a finite real symmetric matrix, the operator basis is built from projectors $M^{(K)}_{ij}=x_{i+j}$7, and positivity of the corresponding bootstrap matrix, together with linear self-consistency conditions, reproduces the deuteron ground-state energy and excited levels in excellent agreement with exact diagonalization (Bai, 2022).

Another extension concerns mirror-curve quantum mechanics from local toric Calabi–Yau geometries. For the local $M^{(K)}_{ij}=x_{i+j}$8 and local $M^{(K)}_{ij}=x_{i+j}$9 Hamiltonians, the bootstrap is built from exponentials $x_n=\langle x^n\rangle$0, and an important improvement is to use a larger set of two-dimensional operators instead of one-dimensional ones. The same improved strategy also gives better numerical accuracies for the two-body non-relativistic Toda system and the quartic anharmonic oscillator (Du et al., 2021).

The bootstrap has also been generalized beyond Hermitian stationary spectra. For generic complex polynomial potentials,

$x_n=\langle x^n\rangle$1

one replaces the Hermitian constraints by

$x_n=\langle x^n\rangle$2

$x_n=\langle x^n\rangle$3

$x_n=\langle x^n\rangle$4

and then bootstraps complex spectra directly. This formulation handles generic non-Hermitian systems, includes PT-symmetric Hamiltonians as a special case, and captures PT-symmetric phase transitions such as the onset of complex eigenvalues in $x_n=\langle x^n\rangle$5 (Khan et al., 2024).

Time evolution admits a different but closely related bootstrap. Instead of a static moment matrix, one considers a time-dependent matrix

$x_n=\langle x^n\rangle$6

subject to positivity, operator identities, known initial data, and Heisenberg equations. This yields a hierarchy of rigorous bounds on observables at later times, systematically generalizing Mandelstam–Tamm-like relations. For any fixed hierarchy level the bounds are tightest at short times and loosen over time, while for fixed time the evidence indicates that increasing the hierarchy can make them arbitrarily tight; the computational effort scales polynomially with the number of degrees of freedom at fixed level (Lawrence et al., 2024).

Three-dimensional central potentials provide a further major generalization. The bootstrap has now been applied to Coulomb, Yukawa, Gaussian, Cornell, and conformal quantum mechanics, including non-algebraic potentials such as Yukawa and Gaussian. In this setting the method is particularly effective for ground-state lower bounds: the Cornell critical coupling is determined to better than one part in $x_n=\langle x^n\rangle$7, and lower bounds on energies are occasionally accurate to one part in greater than $x_n=\langle x^n\rangle$8. The same study also analyzes when meaningful upper bounds can and cannot be obtained (Lawrence et al., 9 Dec 2025).

Domain	Characteristic feature	Representative papers
Low-energy nuclear physics	Finite-basis projector bootstrap for pionless EFT deuteron	(Bai, 2022)
Mirror-curve / Calabi–Yau QM	Exponential operator basis $x_n=\langle x^n\rangle$9 and 2D bootstrap matrices	(Du et al., 2021)
Non-Hermitian QM	Complex spectra and PT phase transitions	(Khan et al., 2024)
Time evolution	Hierarchy of rigorous bounds on $x_0=1$0	(Lawrence et al., 2024)
3D central potentials	High-precision ground-state bounds for Coulomb, Yukawa, Gaussian, Cornell	(Lawrence et al., 9 Dec 2025)

6. Limitations, misconceptions, and open directions

A recurring limitation is that the strongest results are usually lower bounds on low-lying states, especially the ground state. In three-dimensional central potentials, meaningful upper bounds are obtained only under specific circumstances, whereas lower bounds are routinely much sharper (Lawrence et al., 9 Dec 2025). In periodic problems the method efficiently constrains band gaps but may leave the minimum energy weakly constrained unless higher-momentum constraints are added (Blacker et al., 2022).

Another limitation concerns global data that are not naturally encoded by moment positivity. For systems with a $x_0=1$1-term, the bootstrap reproduces correlations among observables in energy eigenstates but has difficulty reconstructing $x_0=1$2 except at $x_0=1$3 and $x_0=1$4 (Aikawa et al., 2021). For identical particles, the method faces a more basic obstruction: it has difficulty distinguishing bosons from fermions at the level of expectation-value constraints, and its predictive power in multi-particle systems is therefore limited in the derivation of observables including energy eigenstates. In thermal applications, temperature and entropy cannot be handled, although some quantities in micro-canonical ensembles of integrable systems governed by generalized Gibbs ensembles can still be derived (Morita, 2022).

Boundary conditions and operator orderings are also structural, not merely technical, inputs. Half-line and interval problems require anomaly terms or domain-sensitive identities, and in the interval model $x_0=1$5 one operator ordering gives only bands while another produces exact low-lying eigenvalues from finite constraints (Berenstein et al., 2023, Sword et al., 2024). This suggests that bootstrap performance depends strongly on how the operator algebra is represented in the positivity problem.

The current literature nevertheless delineates a coherent research program. Bootstrap methods in quantum mechanics now cover discrete spectra, band structure, reflection phases, realistic finite-basis nuclear systems, generic complex spectra, exact solvability via shape invariance, and rigorous bounds on real-time evolution (Aikawa et al., 11 Apr 2025). What remains open is not the existence of useful bootstrap constraints, but the extent to which basis choice, boundary data, and truncation hierarchies can be systematized so that exactness, when present, can be detected reliably and high-precision bounds can be made routine across broader classes of quantum systems.