In evaluating the path integral of the harmonic oscillator, we had to perform a Gaussian integral for which the factor in the exponent was a Hermitian matrix \(\mathbb{A}\) sandwiched between an \(N\)-dimensional real vector \(\vec{x}\). We had shown that

\[\begin{equation*}\int_{-\infty}^{\infty} d \vec{x} e^{-\vec{x}^{\top} \mathbb{A} \vec{x}}=\pi^{N / 2}(\operatorname{det} \mathbb{A})^{-1 / 2} \tag{11.157}\end{equation*}\]

We then took the \(N \rightarrow \infty\) limit to determine the path integral of the harmonic oscillator. However, we didn't have to discretize the time steps; we can directly evaluate the continuous infinity of position integrals directly. In that case we would have

\[\begin{equation*}\int_{-\infty}^{\infty}[d x] \exp \left[-\int_{0}^{T} d t x\left(\frac{d^{2}}{d t^{2}}+\omega^{2}\right) x\right]=\left(\operatorname{det} \frac{\frac{d^{2}}{d t^{2}}+\omega^{2}}{\pi}\right)^{-1 / 2} . \tag{11.158}\end{equation*}\]

But what does the determinant of a differential operator mean?
(a) Determine the eigenvalues of the harmonic oscillator differential operator. That is, determine the values \(\lambda\) for a function of time \(f_{\lambda}(t)\) such that
\[\begin{equation*}\left(\frac{d^{2}}{d t^{2}}+\omega^{2}\right) f_{\lambda}(t)=\lambda f_{\lambda}(t) \tag{11.159}\end{equation*}\]
Make sure to enforce the boundary conditions that \(f_{\lambda}(t=0)=f_{\lambda}(t=\) \(T)=0\).
(b) The determinant is the product of eigenvalues of an operator. Take all values \(\lambda\) established in the previous part and multiply them together. What is the determinant of this harmonic oscillator operator?

Leonard Euler first showed that the sine function can be expressed as an infinite product of its roots, analogous to a finite-order polynomial:
\[\begin{equation*}\sin x=x \prod_{n=1}^{\infty}\left(1-\frac{x^{2}}{n^{2} \pi^{2}}\right) \tag{11.160}\end{equation*}\]
(c) How does this expression compare to the result we found in Sec. 11.5.4 from the limit of the discrete operator?