THE COMPLEX AIRY OPERATOR ON THE LINE WITH A SEMI-PERMEABLE BARRIER

Denis S Grebenkov (1) , Bernard Helffer (2, 3) , Raphael Henry (3)

1 LPMC - Laboratoire de physique de la matière condensée
2 LMJL - Laboratoire de Mathématiques Jean Leray
3 LM-Orsay - Laboratoire de Mathématiques d'Orsay

We consider a suitable extension of the complex Airy operator, −d 2 /dx 2 + ix, on the real line with a transmission boundary condition at the origin. We provide a rigorous definition of this operator and study its spectral properties. In particular, we show that the spectrum is discrete, the space generated by the generalized eigenfunctions is dense in L2 (completeness), and we analyze the decay of the associated semi-group. We also present explicit formulas for the integral kernel of the resolvent in terms of Airy functions, investigate its poles, and derive the resolvent estimates.

Mots clés: Airy operator; transmission boundary condition; spectral theory; Bloch-Torrey; AMS subject classifications 35P10; 47A10; 47A75

Domaines: Physique mathématique [math-ph]

Denis S Grebenkov, Bernard Helffer, Raphael Henry. THE COMPLEX AIRY OPERATOR ON THE LINE WITH A SEMI-PERMEABLE BARRIER . SIAM Journal on Mathematical Analysis, 2017, 49 (3). ⟨hal-01456380⟩