Analysis of Quartic Potential Problems Using Fredholm Integral Operators with Airy Functions

Fredholm integral operators that commute with the Hamiltonians of certain quantum mechanical systems with quartic potentials are introduced in closed analytical form. The operators are expressed in terms of the Airy function, and their eigenvalues fall off exponentially. Using these operators we obtain an exact dual description in terms of infinite one-dimensional chains. The systems discussed include the anharmonic quartic oscillator as well as multivariable potentials and higher dimensional systems. We compute a few moments of the Fredholm operators, using matrix models in the higher dimensional case, and we show how those can be used to approximate the energy spectrum and obtain estimates of the asymptotic behavior of the wavefunctions.