Criteria for eigenvalues embedded into the absolutely continuous spectrum of perturbed Stark type operators

Liu, Wencaı

doi:10.1016/j.jfa.2018.11.010

Cited by 14 publications

(27 citation statements)

References 40 publications

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“…It is well known that for any

0 < α < 2

σ_{normaless} (H_{0}) = σ_{normalac} (H_{0}) = R

and H 0 does not have any eigenvalue. The criteria for the perturbation such that the associated perturbed Stark type operator has single eigenvalue, finitely many eigenvalues or countably many eigenvalues have been obtained in [29].…”

Section: Introductionmentioning

confidence: 99%

“…Define

P \subset R

P = {E \in R : - u^{''} - x^{α} u + q u = E u has an L^{2} ({double-struckR}^{+}) solution} .

In [29], the author proved that Theorem [29, Theorem 1.5] Let a be given by

a = \underset{x \to \infty}{lim sup} x^{1 - \frac{α}{2}} false| q (x) false| .

Then we have

a \geq \frac{2 - α}{\sqrt{2}} {(# P)}^{\frac{1}{2}} .

…”

Section: Introductionmentioning

confidence: 99%

“…Theorem [29, Theorem 1.6] For any

{E_{j}}_{j = 1}^{N} \subset R

and any

{θ_{j}}_{j = 1}^{N} \subset false[0, π false]

, there exist potentials

q \in C^{\infty} false[0, + \infty false)

such that

\underset{x \to \infty}{lim sup} x^{1 - \frac{α}{2}} false| q (x) false| \leq false(2 - α false) e^{2 \sqrt{\ln N}} N,

and for any

j = 1, 2, \dots, N

- u^{''} - x^{α} u + q u = E_{j} u

has an

L^{2} ({double-struckR}^{+})

solution u with the boundary condition

\frac{u^{'} false(0 false)}{u (0)} = \tan θ_{j} .

…”

Section: Introductionmentioning

confidence: 99%

“…Remark Theorem 1.6 gives better bounds than those in Theorem 1.1, but less regularity in potentials. Applying additional piecewise constructions in the proof of Theorem 1.6, it is possible to show that the upper bound in (1.5) can be improved to

(2 - α) N

. We refer the readers to the critical case of [29, Theorem 1.2 ] for details. In the main part of this paper, we only consider the relations between L 2 solutions and limit bound of

q (x)

. In the last section, we will mention similar results presented in the bounds of integrals. …”

Section: Introductionmentioning

confidence: 99%

“…Before that, Wigner–von Neumann type functions can only create one L 2 solution [41]. Recently, there have been several important developments on the problem of embedded eigenvalues for Schrödinger operators, Laplacians on manifolds or other models [12, 14–17, 25, 27, 29–32, 34]. For perturbed Stark type operators, under the rational independence assumption of set

{E_{j}}

, Naboko and Pushnitskii [36] constructed operators with given a set

{E_{j}}

as embedded eigenvalues.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Sharp bounds for finitely many embedded eigenvalues of perturbed Stark type operators

Liu

2020

Mathematische Nachrichten

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For perturbed Stark operators Hu=−u′′−xu+qu, the author has proved that lim supx→∞x12false|q(x)false| must be larger than 12N12 in order to create N linearly independent eigensolutions in L2(double-struckR+) [29]. In this paper, we apply generalized Wigner–von Neumann type functions to construct embedded eigenvalues for a class of Schrödinger operators, including a proof that the bound 12N12 is sharp.

show abstract

“…It is well known that for any

0 < α < 2

σ_{normaless} (H_{0}) = σ_{normalac} (H_{0}) = R

Section: Introductionmentioning

confidence: 99%

“…Define

P \subset R

P = {E \in R : - u^{''} - x^{α} u + q u = E u has an L^{2} ({double-struckR}^{+}) solution} .

In [29], the author proved that Theorem [29, Theorem 1.5] Let a be given by

a = \underset{x \to \infty}{lim sup} x^{1 - \frac{α}{2}} false| q (x) false| .

Then we have

a \geq \frac{2 - α}{\sqrt{2}} {(# P)}^{\frac{1}{2}} .

…”

Section: Introductionmentioning

confidence: 99%

“…Theorem [29, Theorem 1.6] For any

{E_{j}}_{j = 1}^{N} \subset R

and any

{θ_{j}}_{j = 1}^{N} \subset false[0, π false]

, there exist potentials

q \in C^{\infty} false[0, + \infty false)

such that

\underset{x \to \infty}{lim sup} x^{1 - \frac{α}{2}} false| q (x) false| \leq false(2 - α false) e^{2 \sqrt{\ln N}} N,

and for any

j = 1, 2, \dots, N

- u^{''} - x^{α} u + q u = E_{j} u

has an

L^{2} ({double-struckR}^{+})

solution u with the boundary condition

\frac{u^{'} false(0 false)}{u (0)} = \tan θ_{j} .

…”

Section: Introductionmentioning

confidence: 99%

(2 - α) N

. We refer the readers to the critical case of [29, Theorem 1.2 ] for details. In the main part of this paper, we only consider the relations between L 2 solutions and limit bound of