One-dimensional Schrodinger operator with a negative parameter and its applications to the study of the approximation numbers of a singular hyperbolic operator

Abstract

In this paper we use the one-dimensional Schr?dinger operator with a negative parameter to the study of the approximation numbers of a hyperbolic type singular operator. Estimates for the distribution function of the approximation numbers are obtained.

Estimates of singular numbers (s$$ s $$‐numbers) and eigenvalues of a mixed elliptic‐hyperbolic type operator with parabolic degeneration

“…Lemmas 4.3 and 4.4 are proved in exactly the same way as Lemmas 2.5–2.6 in Muratbekov and Muratbekov 19 …”

“…. Now, repeating the computations and arguments from Lemma 2.7, 19 we obtain the proof of Lemma 4.5. □…”

“…Proof Taking Lemma 3.5 into account and using the computations used in the proof of Lemma 2.4, 19 we prove the inclusion $$$ M\subseteq {\tilde{M}}_{c_0} $$$. The inclusion $$$ {\tilde{\tilde{M}}}_{c_0^{-1}}\subseteq M $$$ is proved by the same method as Lemma 2.4 in Muratbekov and Muratbekov 19 . Lemma 4.2 is proved.…”

“…It is easy to see that $$$ \tilde{M}\subset {L}_2^{\prime}\left(\mathbb{R},{K}_n(y)\right) $$$, $$$ \tilde{\tilde{M}}\subset {L}_2^2\left(\mathbb{R},{Q}_n(y)\right) $$$. Now, repeating the computations and arguments from Lemma 2.7, 19 we obtain the proof of Lemma 4.5. □…”

“…Here, we note that the boundedness of the operators q (𝑦) ( l n,𝛼 + 𝜇 I ) −1 and ( I − A 𝜇,𝛼 ) −1 follows from Lemmas 2.4 and 3.3. From (19), according to the definition of the operator norm, we have…”