Bootstrapping Calabi-Yau Quantum Mechanics

Abstract

Recently, a novel bootstrap method for numerical calculations in matrix models and quantum mechanical systems is proposed. We apply the method to certain quantum mechanical systems derived from some well-known local toric Calabi-Yau geometries, where the exact quantization conditions have been conjecturally related to topological string theory. We find that the bootstrap method provides a promising alternative for the precision numerical calculations of the energy eigenvalues. An improvement in our approach is to use a larger set of two-dimensional operators instead of one-dimensional ones. We also apply our improved bootstrap methods to some non-relativistic models in the recent literature and demonstrate better numerical accuracies.

Bootstrapping the deuteron

“…[28], the sizes of the allowable regions decrease quickly as K increases and shrink to a number of points approximately at large Ks. Similar strategies are also used in many other quantum mechanical bootstrap studies [29][30][31][32][33][34][35]. However, as we mention before, for general potentials such as those encountered in lowenergy nuclear physics, it is not easy to work out useful recursive relations.…”

“…Mathematically, this is done by requiring the so-called bootstrap matrix to be always positive semidefinite, with its matrix elements obtained via some recursive relations (see Section II A for details). Their bootstrap method is soon applied to other quantum mechanical problems, such as harmonic oscillator, hydrogen, double-well potential, and Bloch bands [29][30][31][32][33][34][35]. Besides intelligential novelty, it is possible that these studies may eventually lead to an alternative formulation of quantum mechanics that will be important in the future.…”

The null bootstrap

“…When g is small, the Hermitian case (10) can be viewed as a deformation of the harmonic oscillator, so it seems natural to find qualitatively similar results. 20 By contrast, the non-Hermitian case (11) has no mass term and cannot be viewed as a deformed harmonic oscillator. 21 Without knowing much about the lowering operators and spectra, it is a leap of faith to apply the null bootstrap to the non-Hermitian system (11).…”

“…One of the impressive results is the most precise determination of the 3d Ising critical exponents [6][7][8][9]. More recently, the positivity principle 1 has also been applied to the studies of matrix models and quantum mechanical systems [10][11][12][13][14][15][16][17][18][19][20][21][22][23].…”

“…In[20], the corresponding results are more precise because they define K as max{m, n}. The degrees of their polynomials are twice of ours 15.…”