= Solution
Write the <Hamiltonian operator> as $\widehat H=\widehat p^2/(2m)+V(\widehat q)$, with $[\widehat q,\widehat p]=i$, and define the <quantum-mechanical propagator> by $U(\beta,\alpha;T)=\langle\beta|e^{-iT\widehat H}|\alpha\rangle$. Use $N$ intervals of length $\varepsilon=T/N$, endpoint coordinates $q_0=\alpha,q_N=\beta$, and the <Trotter product formula>. Inserting the completeness relations of the <position eigenstates> between the short-time factors gives
$$
U=\lim_{N\to\infty}\int\prod_{j=1}^{N-1}dq_j\prod_{j=1}^N\langle q_j|e^{-i\varepsilon\widehat p^2/(2m)}e^{-i\varepsilon V(\widehat q)}|q_{j-1}\rangle.
$$
Normalize <momentum eigenstates> by $\langle q|p\rangle=e^{ipq}/\sqrt{2\pi}$. The <Gaussian integral> over the intermediate <momentum> is
$$
\int\frac{dp}{2\pi}\exp\left(ip(q_j-q_{j-1})-\frac{i\varepsilon p^2}{2m}\right)=\left(\frac{m}{2\pi i\varepsilon}\right)^{1/2}\exp\left(\frac{im(q_j-q_{j-1})^2}{2\varepsilon}\right).
$$
The square-root branch is fixed by continuation from $\operatorname{Im}\varepsilon<0$. Consequently the precise <time-sliced configuration-space path integral> is
$$
\boxed{U=\lim_{N\to\infty}\left(\frac{m}{2\pi i\varepsilon}\right)^{N/2}\int_{\mathbb R^{N-1}}\prod_{j=1}^{N-1}dq_j\exp\left[i\sum_{j=1}^N\left\{\frac{m(q_j-q_{j-1})^2}{2\varepsilon}-\varepsilon V(q_{j-1})\right\}\right].}
$$
This formula defines the notation $\int\mathcal Dq\,e^{iS[q]}$; it is not an unnormalized infinite product of ordinary integrals. For real time take the boundary value from $T-i\delta$, $\delta>0$, or the corresponding oscillatory-distribution limit. For a self-adjoint, bounded-below <Hamiltonian operator> under the usual product-formula hypotheses, the operator limit supplies the definition even where the coordinate kernel is a distribution. The discrete exponent approaches the <action> with <kinetic term> $m\dot q^2/2$, and the prefactor is part of the <functional measure>. A different consistent slicing gives the same kernel for this separated kinetic-plus-potential Hamiltonian.
For the <quantum harmonic oscillator>, expand the <action> about a <classical path> $x$, writing $q=x+\xi$ with homogeneous <Dirichlet boundary conditions> on $\xi$. The only mixed term is
$$
m\int_0^T(\dot x\dot\xi-\omega^2x\xi)\,dt=m[\dot x\xi]_0^T-m\int_0^T(\ddot x+\omega^2x)\xi\,dt=0.
$$
The boundary term vanishes and the <Euler-Lagrange equation> sets the remaining integrand to zero. Thus $S[x+\xi]=S[x]+S[\xi]$ exactly, rather than only to quadratic approximation. Translation of each integrated coordinate has unit <Jacobian determinant>; in the continuum measure convention of the question this is $\mathcal Dq=\mathcal D\xi$. The <classical-path factorization of a quadratic path integral> follows:
$$
\boxed{U(\beta,\alpha;T)=e^{iS[x]}F(T),\qquad F(T)=\int_{\xi(0)=\xi(T)=0}\mathcal D\xi\,e^{iS[\xi]}.}
$$
The fluctuation <action>, measure and boundary values depend on $T,m,\omega$ but not on $\alpha,\beta$. For real $T$ with $\sin\omega T\ne0$ the boundary-value <classical path> exists uniquely. At the conjugate times $\omega T=k\pi$, generic endpoints admit no such path and the kernel is a delta distribution; the formula is interpreted by the same regulated boundary limit, not as an ordinary finite prefactor times a nonexistent classical solution.
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