= Solution
The spatial shift in the PDF also requires a boundary value beyond the supplied right endpoint: if $m$ is the last interior index, the newest $m+2$ value is outside the interval. Thus \b[the printed Dirichlet method has no specified boundary closure, and no unique stability range can be assigned to it as stated].
For completeness, the literal interior formula has an exact <von Neumann stability analysis> on an infinite or periodic grid. Substitution of a <Fourier mode> $U_m^n=\xi^ne^{im\theta}$ gives
$$
A(\theta)\xi^2-4\xi+1=0,\qquad
A(\theta)=3e^{2i\theta}+8\mu\sin^2(\theta/2).
$$
For the <root of a polynomial> tending to one as $\theta\to0$, expansion gives
$$
\xi=1-3i\theta-(\mu+3/2)\theta^2+O(\theta^3),
\qquad
|\xi|^2=1+(6-2\mu)\theta^2+O(\theta^4).
$$
Consequently $\mu<3$ is unstable: sufficiently small nonzero frequencies have a <root of a polynomial> of <modulus> greater than one, and arbitrarily fine periodic grids contain such frequencies.
Conversely, if $\mu\geq3$ and $s=\cos\theta$, then
$$
\operatorname{Re}A-3
=6(s-1)^2+4(\mu-3)(1-s)\geq0.
$$
For $A=u+iv$ with $u\geq3$,
$$
(|A|^2-1)^2-16|A-1|^2
=(u-1)^2(u-3)(u+5)+2(u^2-9)v^2+v^4\geq0.
$$
Since $|A|\geq3$, the <complex quadratic Schur criterion> yields both <roots of a polynomial> in the <unit disk>, strictly inside except at $\theta=0$, where they are $1$ and $1/3$. Uniform <stability> follows directly from the <BDF2 discrete energy identity>: the modal recurrence is $3v^{n+2}-4v^{n+1}+v^n=-(A-3)v^{n+2}$, whose <energy> decreases because $\operatorname{Re}(A-3)\geq0$. Summing the modal energies proves a bound independent of the grid and even of $\mu\geq3$; no eigenvector-separation assumption is needed. This is the <stability of a spatially shifted BDF2 stencil>. Thus the literal periodic-grid result is
$$
\boxed{\mu\geq3\quad\text{for the printed shifted stencil on a periodic grid}}.
$$
This cannot establish stability of an unspecified <Dirichlet boundary condition> closure. It also does not restore <consistency of a numerical method>.
For the intended unshifted <backward differentiation formula>, a spatial <eigenvalue> $-\beta\leq0$ gives
$$
(3+2k\beta)\xi^2-4\xi+1=0.
$$
The <Schur stability criterion> gives the <root condition for a multistep method> for every $k\beta\geq0$, with strict inequalities for $\beta>0$. A useful uniform <energy method> proof, including the repeated interior roots that a bare separation argument would miss, is the <BDF2 discrete energy identity>. With $\delta^2U=U^{n+2}-2U^{n+1}+U^n$ and
$$
\mathcal E_n=\|U^{n+1}\|_h^2+\|2U^{n+1}-U^n\|_h^2,
$$
the corrected <Dirichlet discrete Laplacian> scheme satisfies
$$
\mathcal E_{n+1}-\mathcal E_n+\|\delta^2U\|_h^2
+4k\|D_+U^{n+2}\|_h^2=0.
$$
This follows by taking the <inner product> of $3U^{n+2}-4U^{n+1}+U^n=2kL_hU^{n+2}$ with $U^{n+2}$ and using <summation by parts>. It controls both time levels uniformly, regardless of the mesh ratio. Hence the corrected method is stable for
$$
\boxed{\text{every }\mu>0}.
$$
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