On the Borel summability of WKB solutions of certain Schrödinger-type differential equations

Existence and Uniqueness of Exact WKB Solutions for Second-Order Singularly Perturbed Linear ODEs

“…However, although the existence of exact WKB solutions in classes of examples has been established, a general existence theorem for second-order linear ODEs has remained unavailable. Contributions towards such a general theory include Bodine, Dunster, Lutz, and Schäfke [Dun01, DLS93, BS02], Giller and Milczarski [GM01], Koike and Takei [KT13], Ferreira, López, and Sinusía [FLPS14,FLS15], as well as most recently by Nemes [Nem21] whose preprint appeared at roughly the same time as our previous work [Nik20] that underpins our results here. Our paper contributes to this long line of work by establishing a general theory of existence and uniqueness of exact WKB solutions, which generalises the relevant results from the aforementioned works (see Remark 5.33 for a discussion).…”

“…For the equation ( 85), the leading-order characteristic discriminant D 0 = 1, so condition (1) of (5.2) is vacuously true. The boundedness Conditions 1.1 and 1.2 in [Nem21] imply in particular that the coefficients a 1 , a 2 are bounded on Ω ± 0 , which by the discussion in Example 5.5 means condition (2) of Theorem 5.2 is met. So Theorem 1.1 in [Nem21] follows.…”

“…The boundedness Conditions 1.1 and 1.2 in [Nem21] imply in particular that the coefficients a 1 , a 2 are bounded on Ω ± 0 , which by the discussion in Example 5.5 means condition (2) of Theorem 5.2 is met. So Theorem 1.1 in [Nem21] follows. Finally, the first assertion of Theorem 2.1 in [Nem21] is a special case of Lemma 5.14.…”

“…The following proposition asserts that this existence result is a corollary of our main theorem. [Nem21].…”

Overview in Summabilities: Summation Methods for Divergent Series, Ramanujan Summation and Fractional Finite Sums

“…The Borel SM has a wide range of applications, playing an important role in asymptotic analysis and semiclassical methods. For example, it is used in the context of Wentzel-Kramers-Brillouin (WKB) theory to find approximate solutions to certain linear differential equations [49][50][51] and in the study of the 1-D Schrödinger equation [52][53][54]. Moreover, the resurgence theory [55][56][57][58][59] is an important generalization of Borel SM.…”