ASYMPTOTICS OF THE EIGENVALUES AND REGULARIZED TRACE OF THE FIRST-ORDER STURM–LIOUVILLE OPERATOR WITH Δ-Potential

A general trace formula for a differential operator on a segment with zero order coefficient perturbed by a finite signed measure

“…Theorem 2.3. For every sequence R = R ℓ → ∞ separated from |λ 0 N | 1 n the following relations hold: (2) are strongly regular (see, e.g., [10]) then for ℓ large enough there is exactly one value |λ 0 N | 1 2 between R ℓ and R ℓ+1 ;…”

“…Thus, for q ∈ M[a, b] the regularized trace becomes non-linear functional of q. For δ-potential this effect was slightly generalized in [2].…”

“…Thus, h j is the jump of Q at the point x j . Denote by L 0 the operator generated by the differential expression ℓ 0 = (−i) n D n and regular boundary conditions (2). The eigenvalues of L 0 are denoted by {λ 0 N } ∞ N=1 .…”

On the trace formula for higher-order ODO

“…Thus, for q ∈ M[a, b] the regularized trace becomes non-linear functional of q. In [5] this effect was obtained for δ-potential and some other boundary conditions. See also [1,Theorem 1] for a similar effect in a different problem.…”