On the bound states of Schrödinger operators with δ-interactions on conical surfaces

Vladimir Lotoreichik (1) , Thomas Ourmières-Bonafos (2)

1 Department of Theoretical Physics, Nuclear Physics Institute
2 BCAM - Basque Center for Applied Mathematics

In dimension greater than or equal to three, we investigate the spectrum of a Schrödinger operator with a δ-interaction supported on a cone whose cross section is the sphere of co-dimension two. After decomposing into fibers, we prove that there is discrete spectrum only in dimension three and that it is generated by the axisymmetric fiber. We get that these eigenvalues are non-decreasing functions of the aperture of the cone and we exhibit the precise logarithmic accumulation of the discrete spectrum below the threshold of the essential spectrum.

Mots clés: Schrödinger operator; δ-interaction; existence of bound states; spectral asymptotics; conical and hyperconical surfaces

Domaines: Théorie spectrale [math.SP]

Vladimir Lotoreichik, Thomas Ourmières-Bonafos. On the bound states of Schrödinger operators with δ-interactions on conical surfaces. Communications in Partial Differential Equations, 2016, ⟨10.1080/03605302.2016.1168843⟩. ⟨hal-01217526⟩