Third-order operators with three-point conditions associated with Boussinesq's equation

Abstract

We consider a non-self-adjoint third-order operator on the interval $[0,2]$ with real 1-periodic coefficients and three-point Dirichlet conditions at the points 0, 1 and 2. The eigenvalues of this operator consist an auxiliary spectrum for the inverse spectral problem associated with the good Boussinesq equation. We determine eigenvalue asymptotics at high energy and the trace formula for the operator.

COMMUNICATED BY:

• Adrian Constantin

Keywords:

• Good Boussinesq equation
• third-order operator
• multi-point problem
• spectral asymptotics
• trace formula

1991 Mathematics Subject Classifications:

• 47E05
• 34L20
• 34L40

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

A. Badanin was supported by the RFBR [grant number 19-01-00094]. E. Korotyaev was supported by the RSF [grant number 18-11-00032].