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Unlike the situation for the 1d Dirac delta derivative Schrodinger pseudopotential (SPP) and the 2d Dirac delta SPP, where the indeterminacy originatesfrom a lack of scale in the first and both a lack of scale as well as the wavefunction not being well defined at the support of the generalized function SPP;the obstruction in 3d Euclidean space for the Schrodinger equation with theDirac delta as a SPP only comes from the wave function (the $L^2$ bound satesolution) being singular at the compact point support of the Dirac deltafunction (measure). The problem is solved here in a completely mathematicallyrigorous manner with no recourse to renormalization nor regularization. Themethod involves a distributionally generalized version of the Schrodingertheory as developed by the author, which regards the formal symbol "$H\psi$" asan element of the space of distributions, the topological dual vector space tothe space of smooth functions with compact support. Two main facts come tolight. The first is the bound state energy of such a system can be calculatedin a well-posed context, the value of which agrees with both the mathematicaland theoretical physics literature. The second is that there is then a rigorousdistributional version of the Hellmann-Feynman theorem.

The Energy Eigenvalue for the Singular Wave Function of the Three  Dimensional Dirac Delta Schrodinger Potential via Distributionally  Generalized Quantum Mechanics