Fatih Erman
On the Completeness of Energy Eigenfunctions for Renormalized Quantum Systems
We consider Hamiltonians of the form $H_0 -\alpha \delta_a$ in two or three dimensions, where $\delta_a$ is the delta function supported at a point $a$, and the spectrum of $H_0$ contains only discrete spectrum, which is bounded from below. For instance, $H_0$ could be the harmonic oscillator Hamiltonian or free Hamiltonian for a particle moving intrinsically on a $D=2,3$ dimensional compact manifold with $Ric_g(\cdot, \cdot) \geq (D-1) \kappa g(\cdot, \cdot)$. We first summarize the formula for the resolvent (or Green's function, which is the integral kernel of resolvent) by the renormalization procedure. Then, we give an argument about how the pole structure of the full Green’s function $G(x, y|E)$ is rearranged to form new poles and how the poles of $G_0(x, y|E)$ are removed in general. We then give the proof of the orthonormality and completeness of the eigenfunctions of the Hamiltonian by using a contour deformation in the complex energy plane under the assumption that $H_0$ has the complete set of eigenfunctions. This will allow us to write the Hamiltonian operator explicitly as an integral operator after the renormalization procedure. The complete set of eigenfunctions ensures that the resulting Hamiltonian is essentially self-adjoint. We finally discuss one interesting application of this explicit formula, where the support of the delta function is suddenly changed. This is a joint work with O. Teoman Turgut.
