scholarly journals Semiclassical spectral estimates for Schrödinger operators at a critical energy level. Case of a degenerate minimum of the potential

2008 ◽  
Vol 341 (2) ◽  
pp. 1170-1180
Author(s):  
Brice Camus
Keyword(s):  
Energy Level ◽  
Schrödinger Operators ◽  
Critical Energy ◽  
Schrodinger Operators ◽  
Spectral Estimates ◽  
Degenerate Minimum

scholarly journals Resonance free domain for a system of Schrödinger operators with energy-level crossings

2020 ◽  
pp. 2150007
Author(s):  
Kenta Higuchi
Keyword(s):  
Energy Level ◽  
Differential Operators ◽  
Schrödinger Operators ◽  
Schrodinger Operators ◽  
Level Crossings ◽  
Diagonal Part ◽  
Free Domain ◽  
Small Interactions ◽  
Directed Cycles

We consider a [Formula: see text] system of 1D semiclassical differential operators with two Schrödinger operators in the diagonal part and small interactions of order [Formula: see text] in the off-diagonal part, where [Formula: see text] is a semiclassical parameter and [Formula: see text] is a constant larger than [Formula: see text]. We study the absence of resonance near a non-trapping energy for both Schrödinger operators in the presence of crossings of their potentials. The width of resonances is estimated from below by [Formula: see text] and the coefficient [Formula: see text] is given in terms of the directed cycles of the generalized bicharacteristics induced by two Hamiltonians.

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