Corollary. The oscillator Hamiltonian [fdip-007W]
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  • November 1, 2026

For the normalized transfer operator \(\widehat T=T/\lambda _0\) of Theorem [fdip-007U], the transfer Hamiltonian is the spectral logarithm \(H=-\log \widehat T\). Explicitly, if \(f=\sum _{n\ge 0}c_nh_n^s\), its domain and action are \[\begin {aligned} D(H)&=\left \{f:\sum _{n=0}^\infty (n\omega )^2|c_n|^2<\infty \right \},\\ Hf&=\sum _{n=0}^\infty n\omega c_nh_n^s. \end {aligned}\] It is nonnegative and self-adjoint, with eigenvalues \(n\omega \). Its zero eigenspace is the line spanned by \(h_0^s\), and its mass gap is \(\omega \). It is unbounded even though \(\widehat T\) is bounded and injective.

On the finite Hermite span, it has the differential expression \[Hf=\frac \omega {2s} \left (-f''+s^2x^2f-sf\right ).\] The closure of this expression on that span is the above spectral operator. This is the harmonic oscillator with its ground energy subtracted. The time step has length one; a specified step \(a_t>0\) instead gives \(H/a_t\) and gap \(\omega /a_t\).

Proof. 

  • November 1, 2026

The unitary coefficient map \(f\mapsto (c_n)\) identifies \(\widehat T\) with multiplication by \(q^n>0\) on \(\ell ^2\). Its negative logarithm is multiplication by \(n\omega \) on the stated maximal domain. A real diagonal operator on that domain is self-adjoint; its quadratic form is nonnegative. The listed eigenspaces, spectrum and gap follow directly, and \(\|Hh_n^s\|=n\omega \) proves unboundedness. Finite coefficient truncations converge both in norm and after applying \(H\), so the finite Hermite span is a core.

Differentiating the Hermite generating function and comparing coefficients gives \(\operatorname {He}_n''(u)-u\operatorname {He}_n'(u) +n\operatorname {He}_n(u)=0\). Substitution into the definition of \(h_n^s\) yields \[-(h_n^s)''+s^2x^2h_n^s=(2n+1)s\,h_n^s.\] The displayed differential expression therefore acts as \(n\omega \) on every basis vector. The core statement proves that its closure is exactly \(H\).

TauCeti's TauCeti.hermiteFunction_oscillator proves the pointwise differential identity at \(s=1\) for the same normalized Hermite functions. Dilation gives the identity above for \(s>0\); the spectral logarithm, its domain and the closure statement are supplied by the preceding argument.

The logarithmic prescription follows [wingate2025lattice, Eq. (5.5), p. 4], with the leading transfer eigenvalue removed explicitly. It constructs the Hamiltonian for a Gaussian model of finite spatial volume. No limit in time, spatial volume or lattice spacing is needed.