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- W2109118548 abstract "Given a 1-periodic real potential q∈L1(R/Z). We use λ0(q) to denote the smallest 1-periodic eigenvalue of the Hill's equation x″+(λ+q(t))x=0. Let B1[r] be the ball centered at 0 of radius r in the L1 space L1(R/Z). It is trivial that sup{λ0(q):q∈B1[r]}=r for all r⩾0. Based on continuity of λ0(q) in q with the weak topology and continuous differentiability of λ0(q) in q with the L1 norm ‖⋅‖1, we will apply scaling technique, variational approach to extremal values in Lp balls, singular integrals and the limiting approach as p↓1 to obtain (i) λ0(q) is bounded for q in any bounded set of (L1(R/Z),‖⋅‖1), and (ii) the minimal valueL1(r):=inf{λ0(q):q∈L1(R/Z),‖q‖1⩽r}=inf{λ0(q):q∈L1(R/Z),‖q‖1=r} is simply Z0−1(r), where Z0(x):=2−xtanh(−x/2), x⩽0. The extremal values of the smallest Neumann eigenvalues for potentials in L1 balls are also found explicitly." @default.
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- W2109118548 date "2009-06-01" @default.
- W2109118548 modified "2023-10-12" @default.
- W2109118548 title "Extremal values of smallest eigenvalues of Hill's operators with potentials in <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML altimg=si1.gif overflow=scroll><mml:msup><mml:mi>L</mml:mi><mml:mn>1</mml:mn></mml:msup></mml:math> balls" @default.
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- W2109118548 doi "https://doi.org/10.1016/j.jde.2009.03.016" @default.
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