I am a physicist, and I have found a term of character:
$$ \xi_{n',n} = \int{\delta(x-\frac{a}{2})H_{n'}(x)H_{n}(x-a)e^{-x^2}e^{-(x-a)^2}dx} $$
$$ \xi_{n',n} = H_{n'}\left(\frac{a}{2}\right)H_{n}\left(\frac{-a}{2}\right)e^{-a^2/2} = (-1)^n H_{n'}\left(\frac{a}{2}\right)H_{n}\left(\frac{a}{2}\right)e^{-a^2/2}$$
Of course, the Hermite polynomials themselves form an orthogonal basis, as do the basis states of this harmonic oscillator system, but at a point they can have nonzero overlap. Are there any paths to simplifying $\xi$? I am only aware of properties regarding the inner product, and not elementwise product of the physicist's hermite polynomials.