schrodinger operator
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2022 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Fan Wu ◽  
Linlin Fu ◽  
Jiahao Xu

<p style='text-indent:20px;'>For <inline-formula><tex-math id="M2">\begin{document}$ C^2 $\end{document}</tex-math></inline-formula> cos-type potentials, large coupling constants, and fixed <inline-formula><tex-math id="M3">\begin{document}$ Diophantine $\end{document}</tex-math></inline-formula> frequency, we show that the density of the spectral points associated with the Schrödinger operator is larger than 0. In other words, for every fixed spectral point <inline-formula><tex-math id="M4">\begin{document}$ E $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M5">\begin{document}$ \liminf\limits_{\epsilon\to 0}\frac{|(E-\epsilon,E+\epsilon)\bigcap\Sigma_{\alpha,\lambda\upsilon}|}{2\epsilon} = \beta $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M6">\begin{document}$ \beta\in [\frac{1}{2},1] $\end{document}</tex-math></inline-formula>. Our approach is a further improvement on the papers [<xref ref-type="bibr" rid="b15">15</xref>] and [<xref ref-type="bibr" rid="b17">17</xref>].</p>


2021 ◽  
Vol 33 (1) ◽  
pp. 155-178
Author(s):  
N. Filonov

The Schrödinger operator − Δ + V ( x , y ) -\Delta + V(x,y) is considered in a cylinder R m × U \mathbb {R}^m \times U , where U U is a bounded domain in R d \mathbb {R}^d . The spectrum of such an operator is studied under the assumption that the potential decreases in longitudinal variables, | V ( x , y ) | ≤ C ⟨ x ⟩ − ρ |V(x,y)| \le C \langle x\rangle ^{-\rho } . If ρ > 1 \rho > 1 , then the wave operators exist and are complete; the Birman invariance principle and the limiting absorption principle hold true; the absolute continuous spectrum fills the semiaxis; the singular continuous spectrum is empty; the eigenvalues can accumulate to the thresholds only.


2021 ◽  
Vol 2070 (1) ◽  
pp. 012023
Author(s):  
J.I. Abdullaev ◽  
Sh.H. Ergashova ◽  
Y.S. Shotemirov

Abstract We consider a Hamiltonian of a system of two bosons on a three-dimensional lattice Z 3 with a spherically simmetric potential. The corresponding Schrödinger operator H(k) this system has four invariant subspaces L(123), L(1), L(2) and L(3). The Hamiltonian of this system has a unique bound state over each invariant subspace L(1), L(2) and L(3). The corresponding energy values of these bound states are calculated exactly.


2021 ◽  
Vol 2070 (1) ◽  
pp. 012017
Author(s):  
J.I. Abdullaev ◽  
A.M. Khalkhuzhaev

Abstract We consider a three-particle discrete Schrödinger operator Hμγ (K), K 2 T3 associated to a system of three particles (two fermions and one different particle) interacting through zero range pairwise potential μ > 0 on the three-dimensional lattice Z 3. It is proved that the operator Hμγ (K), ||K|| < δ, for γ > γ0 has at least two eigenvalues in the gap of the essential spectrum for sufficiently large μ > 0.


2021 ◽  
Vol 15 (8) ◽  
Author(s):  
Burak Hatinoğlu ◽  
Jerik Eakins ◽  
William Frendreiss ◽  
Lucille Lamb ◽  
Sithija Manage ◽  
...  

AbstractWe discuss the problem of unique determination of the finite free discrete Schrödinger operator from its spectrum, also known as the Ambarzumian problem, with various boundary conditions, namely any real constant boundary condition at zero and Floquet boundary conditions of any angle. Then we prove the following Ambarzumian-type mixed inverse spectral problem: diagonal entries except the first and second ones and a set of two consecutive eigenvalues uniquely determine the finite free discrete Schrödinger operator.


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