On various versions of the harmonic oscillator

Question

The standard $n$-dimensional harmonic oscillator is the operator $ \mathcal H=\frac{1}{2}\sum_{1\le j\le n}(D_j^2+x_j^2), \text{ $D_j=-i\partial_{x_j}$}, $ and its spectral decomposition is $$ \mathcal H=\sum_{k\ge 0}(\frac{1}{2}+k)\mathbb P_{k,n}, \quad \mathbb P_{k,n}=\sum_{ \alpha\in \mathbb N^n,\alpha_1+\dots+\alpha_n=k} \mathbb P_{\alpha_1}\otimes\dots\otimes\mathbb P_{\alpha_n}, $$ where $\mathbb P_{\alpha_j}$ stands for the orthogonal projection onto the one-dimensional Hermite function with level $\alpha_j$. Now let us consider for $µ=(µ_1,\dots, µ_n)$ with $\mu_j>0$, the operator $$ \mathcal H_µ=\frac{1}{2}\sum_{1\le j\le n}µ_j(D_j^2+x_j^2). $$ With the notation $\vert µ\vert=\sum_{1\le j\le n} µ_j$ and $µ\cdot \alpha=\sum_{1\le j\le n}µ_j\alpha_j$, we have $$ \mathcal H_µ=\sum_{\alpha\in \mathbb N^n}(\frac{\vert µ\vert}2+µ\cdot\alpha) (\mathbb P_{\alpha_1}\otimes\dots\otimes\mathbb P_{\alpha_n}), $$ so that the eigenspaces are the same as for $\mathcal H$ but the arithmetic properties of $µ$ make possible that all eigenvalues $(\frac{\vert µ\vert}2+µ\cdot\alpha)$ are simple. What are the typical examples and standard references on this topic?