ISSN:
1089-7658
Source:
AIP Digital Archive
Topics:
Mathematics
,
Physics
Notes:
It is proved that if H=−∇2+q(x)≥0, Im q=0, ||q(x)||≤c(1+||x||)−a, c=const〉0, a〉2, then zero is not an eigenvalue of H. An example is given of H≥0, with zero a resonance (half-bound state) and q=q(||x||) compactly supported and integrable. An example of a potential q=O(r−2) is known, for which H≥0 and zero is an eigenvalue. This shows that a〉2 is the optimal condition for zero not to be an eigenvalue of H≥0. If the condition H≥0 does not hold and H is an operator in L2(R3), then zero can be an eigenvalue even if q∈C∞0. If H is an operator in L2(R1) or in L2(R1+), R1+ =[0,∞), then zero cannot be an eigenvalue of H provided that a〉2; here conditions H≥0 and Im q=0 can be dropped. Global estimates of the Green's function of H from below and above are given.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1063/1.527817
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