Dancer–Fuc̆ik spectrum for fractional Schrödinger operators with a steep potential well on RN

2019 ◽  
Vol 189 ◽  
pp. 111565 ◽  
Author(s):  
Zhisu Liu ◽  
Haijun Luo ◽  
Zhitao Zhang
Keyword(s):  
Schrödinger Operators ◽  
Schrodinger Operators ◽  
Potential Well ◽  
Steep Potential Well

Localization of Eigenvalues by a Modification of Müller′s Variational Principle in Some Exactly Solvable Cases

1983 ◽  
Vol 38 (5) ◽  
pp. 493-496 ◽  
Author(s):  
Heinz K. H. Siedentop
Keyword(s):  
Variational Principle ◽  
Lower Bounds ◽  
Schrödinger Operators ◽  
Three Dimensional ◽  
Finite Depth ◽  
Upper And Lower Bounds ◽  
Schrodinger Operators ◽  
Potential Well ◽  
Rough Approximation ◽  
Exactly Solvable

Upper and lower bounds on the eigenvalues of Schrödinger operators with simple one and a simple three dimensional potential (well of finite depth, spherical δ-potential) are given by means of a modification of Müller′s variational principle. The estimates, comparing them with the exact eigenvalues, show a localization of the eigenvalues even in a rough approximation for the trial operator.

scholarly journals Supersymmetric Cluster Expansions and Applications to Random Schrödinger Operators

2021 ◽  
Vol 24 (1) ◽  
Author(s):  
Luca Fresta
Keyword(s):  
Density Of States ◽  
Local Density ◽  
Schrödinger Operators ◽  
Random Schrödinger Operators ◽  
Schrodinger Operators ◽  
Local Density Of States ◽  
Weak Disorder ◽  
Cluster Expansions ◽  
Strong Disorder ◽  
Lifshitz Tail

AbstractWe study discrete random Schrödinger operators via the supersymmetric formalism. We develop a cluster expansion that converges at both strong and weak disorder. We prove the exponential decay of the disorder-averaged Green’s function and the smoothness of the local density of states either at weak disorder and at energies in proximity of the unperturbed spectrum or at strong disorder and at any energy. As an application, we establish Lifshitz-tail-type estimates for the local density of states and thus localization at weak disorder.

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