Higher-order airy functions of the first kind and spectral properties of the massless relativistic quartic anharmonic oscillator
2014-12-17T12:07:01Z (GMT) by
This thesis consists of two parts. In the first part, we study a class of special functions Aik (y), k = 2, 4, 6, · · · generalising the classical Airy function Ai(y) to higher orders and in the second part, we apply expressions and properties of Ai4(y) to spectral problem of a specific operator. The first part is however motivated by latter part. We establish regularity properties of Aik (y) and particularly show that Aik (y) is smooth, bounded, and extends to the complex plane as an entire function, and obtain pointwise bounds on Aik (y) for all k. Some analytic properties of Aik (y) are also derived allowing one to express Aik (y) as a finite sum of certain generalised hypergeometric functions. We further obtain full asymptotic expansions of Aik (y) and their first derivative Ai'(y) both for y > 0 and for y < 0. Using these expansions, we derive expressions for the negative real zeroes of Aik (y) and Ai'(y). Using expressions and properties of Ai4(y), we extensively study spectral properties of a non-local operator H whose physical interpretation is the massless relativistic quartic anharmonic oscillator in one dimension. Various spectral results for H are derived including estimates of eigenvalues, spectral gaps and trace formula, and a Weyl-type asymptotic relation. We study asymptotic behaviour, analyticity, and uniform boundedness properties of the eigenfunctions ψn(x) of H. The Fourier transforms of these eigenfunctions are expressed in two terms, one involving Ai4(y) and another term derived from Ai4(y) denoted by AI4(y). By investigating the small effect generated by AI4(y) this work shows that eigenvalues λn of H are exponentially close, with increasing n ∈ N, to the negative real zeroes of Ai4(y) and those of its first derivative Ai'4(y) arranged in alternating and increasing order of magnitude. The eigenfunctions ψn(x) are also shown to be exponentially well-approximated by the inverse Fourier transform of Ai4(|y| − λn) in its normalised form.