How can one demonstrate the consistent self-adjointness of a Hamiltonian operator under an imposed Dirichlet boundary condition? In this context, the operator in question is the Berry-Keating program coupled with a number operator, as the squeezing parameter tends to infinity. Initially, as the squeezing parameter approaches infinity, the operator confines the wave function predominantly to the x-direction. Subsequently, the imposition of the Dirichlet boundary condition enforces the wave function to vanish everywhere except at the point y = 0. However, the problem is that the potential restriction of the operator's domain due to the boundary condition, which might result in the operator losing its property of being self-adjoint.
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