The Essential Self-adjointness of Schrodinger Operators with Oscillating Potentials


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New Zealand Journal of Mathematics

Vol. 46, (2016), Pages 65-72

Adam D. Ward

NZ Institue for Advanced Study,

Massey University,

Private Bag 102 904,

North Shore MSC,

0745 Auckland,

New Zealand.

Abstract Let Ω be a domain in \mathbb{R}^m with non-empty boundary and let H = − Δ + V be a Schrodinger operator defined on C^\infty_0(\Omega) where V = V_1 + V_2 \in L_\infty^{loc}(\Omega) is a real valued potential, V_2 \in L_\infty(\Omega) and

V_1(x) \, \geq \, \frac{1 - \mu_2(\Omega)}{d(x)^2}.

Here d(x) is the Euclidean distance to the boundary of the domain and μ2(Ω) is the non-negative variational constant associated to the L2-Hardy inequality. In [1] it was shown that H is essentially self-adjoint and that the condition described by the equation above is optimal on certain geometrically simple domains. In this paper investigate the essential self-adjointness of Schrodinger operators with potentials that oscillate around the critical limit described by the above equation.

Keywords Essential Self-adjointness of Schrodinger Operators, $L_p$-Hardy Inequalities.

Classification (MSC2000) Primary: 47B25, Secondary: 26D10..

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