Corollary. The physical spectral Hamiltonian [fdip-0085]
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  • November 1, 2026

For the physical transfer operator, let \(\lambda _0=\|T_{\mathrm {phys}}\|>0\) and fix a time step \(a_t>0\). There is an orthonormal eigenbasis \((\psi _j)\) with \(0<\lambda _j\le \lambda _0\), including an eigenvector at \(\lambda _0\). Define \[\widehat T=\frac {T_{\mathrm {phys}}}{\lambda _0},\qquad E_j=\frac 1{a_t}\log \frac {\lambda _0}{\lambda _j}.\] The transfer Hamiltonian is \(H=-a_t^{-1}\log \widehat T\), with the explicit domain and action \[\begin {aligned} D(H)&=\left \{f=\sum _jc_j\psi _j: \sum _jE_j^2|c_j|^2<\infty \right \},\\ Hf&=\sum _jE_jc_j\psi _j. \end {aligned}\] It is nonnegative and self-adjoint, and \(e^{-a_tH}=\widehat T\). If the physical Hilbert space is infinite dimensional, \(H\) is unbounded. This assertion does not specify the multiplicity of its zero eigenspace.

Proof. 

  • November 1, 2026

The spectral theorem for compact self-adjoint operators applies to the positive trace-class operator of Theorem [fdip-0083]. Injectivity removes its zero eigenspace, so the positive eigenvectors form a complete orthonormal basis. Its nonzero norm is an eigenvalue. In the finite-dimensional case the list is finite; otherwise its eigenvalues tend to zero.

The eigenspace-density assertion is formalized by ContinuousLinearMap.orthogonalComplement_iSup_eigenspaces_eq_bot: for a compact symmetric operator on a complete real or complex inner-product space, the span of all eigenspaces has zero orthogonal complement. This reference supplies that spectral premise; the logarithm, its domain and its core are addressed below.

Under the unitary coefficient map \(f\mapsto (c_j)\), the normalized transfer is multiplication by \(\lambda _j/\lambda _0\). Its spectral logarithm is therefore multiplication by the real nonnegative sequence \(E_j\) on the stated maximal domain. This diagonal operator is self-adjoint; finite coefficient truncations form a core. Exponentiating on the basis proves the transfer identity. If the list is infinite, \(\lambda _j\to 0\) implies \(E_j\to \infty \), proving unboundedness.

This is the same normalization and domain construction as the Gaussian oscillator, without its explicit eigenvalues. It supplies a Hamiltonian at finite spatial volume. A unique vacuum, limits in time or volume, and a continuum field theory are separate questions, not conclusions of this spectral construction.