TY - JOUR

T1 - Trace Formulas for a Complex KdV Equation

AU - Коротяев, Евгений Леонидович

PY - 2024/3/1

Y1 - 2024/3/1

N2 - Faddeev and Zakharov determined the trace formulas for the KdV equation with real initial conditions in 1971. We reprove these results for the KdV equation with complex initial conditions. The Lax operator is a Schrödinger operator with complex-valued potentials on the line. The operator has essential spectrum on the half-line plus eigenvalues (counted with algebraic multiplicity) in the complex plane without the positive half-line. We determine series of trace formulas. Here we have a new term: a singular measure, which is absent for real potentials. Moreover, we estimate of sum of the imaginary part of eigenvalues plus the singular measure in terms of the norm of potentials. The proof is based on classical results about the Hardy spaces.

AB - Faddeev and Zakharov determined the trace formulas for the KdV equation with real initial conditions in 1971. We reprove these results for the KdV equation with complex initial conditions. The Lax operator is a Schrödinger operator with complex-valued potentials on the line. The operator has essential spectrum on the half-line plus eigenvalues (counted with algebraic multiplicity) in the complex plane without the positive half-line. We determine series of trace formulas. Here we have a new term: a singular measure, which is absent for real potentials. Moreover, we estimate of sum of the imaginary part of eigenvalues plus the singular measure in terms of the norm of potentials. The proof is based on classical results about the Hardy spaces.

UR - https://www.mendeley.com/catalogue/6b6961d4-694a-3ea7-8fe9-eccc33416e8a/

U2 - 10.1134/s106192084010096

DO - 10.1134/s106192084010096

M3 - статья

VL - 31

SP - 112

EP - 131

JO - Russian Journal of Mathematical Physics

JF - Russian Journal of Mathematical Physics

SN - 1061-9208

IS - 1

ER -