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Suppose I consider the operator $$ -\Delta+V$$ for some potential $V(x)$ for $(x,y)\in\mathbb{R}^2$, as the closure of the corresponding operator on smooth compactly supported functions. If I assume that $$ V(tx,ty)=t^4V(x,y) $$ are there any criteria that make the operator non self-adjoint? For instance, if I take $$ V(x,y)=x^4+y^4-\lambda x^2y^2 $$ then for $\lambda\leq 2$ the operator is self-adjoint but what happens for $\lambda>2$? I'm thinking, is there some condition in terms of positivity at $\infty$ in all directions that's equivalent to self adjointness?

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