# The Essential Self-adjointness of Schrodinger Operators with Oscillating Potentials

### From NZJM

**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 with non-empty boundary and let *H* = − Δ + *V* be a Schrodinger operator defined on where is a real valued potential, and

Here *d*(*x*) is the Euclidean distance to the boundary of the domain and μ_{2}(Ω) is the non-negative variational constant associated to the *L*_{2}-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..