Francesco-Zeno-Costanzo / Bootstrap_MQ

an example of boostrap tecnique applied to a simple system in the framework of quantum mechanics: anharmonic oscillators and double-well

Bootstrap_MQ

an example of boostrap tecnique applied to a simple system in the framework of quantum mechanics: anharmonic oscillators and double-well.

Starting from an hamiltonian like:

$$ \mathcal{H} = P^2 + V(x) $$

Now choosing a basis of the energy eigenstates any operator $\mathcal{O}$ should obey the identity:

$$ \langle [ H, \mathcal{O}] \rangle = 0 \quad \text{and} \quad \langle \mathcal{H}\mathcal{O}\rangle = E\langle \mathcal{O}\rangle $$

From an accurate choos of $\mathcal{O}$ we can obtein the following relation:

$$ 4nE \langle x^{n - 1} \rangle - 4n \langle x^{n-1} V(x) \rangle + n(n-1)(n-2)\langle x^{n - 3} \rangle - 4\langle x^{n} V^{'}(x) \rangle = 0 $$

If we define a matrix: $M_{ij} = \langle x^{i+j} \rangle$ from the costrain $\langle \mathcal{O}^{\dagger} \mathcal{O} \rangle > 0$ writing

$$ \mathcal{O} = \sum_i^{K} c_i x^i $$

We have that $M$ must be positive definite. So we compute this matrix for various parameter ad chek the sign of all eigenvalues.

Double well

Anharmonic oscillator