On the critical value of the coupling constant in exterior elliptic problems

ABSTRACT

We consider exterior elliptic problems with coefficients stabilizing at infinity and study the critical value ${\text{β}}_{cr}$ of the coupling constant (the coefficient at the potential) that separates operators with a discrete spectrum and those without it. The dependence of ${\text{β}}_{cr}$ on the boundary condition and on the distance between the boundary and the support of the potential is described. The discrete spectrum of a non-symmetric operator with the FKW boundary condition (that appears in diffusion processes with traps) is also investigated.

Keywords:

• Exterior elliptic problem
• eigenvalue
• coupling constant
• critical value
• spectrum

2010 Mathematics Subject Classifications:

• 35J25
• 35P15
• 47A10
• 47D07

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The second author was supported by the NSF [grant number DMS-1714402] and the Simons Foundation [grant number 527180].