In classical mechanics, Newton-equivalent Hamiltonians yield the same equations of motion and hence identical dynamics. In the quantum context, they instead generally yield different discrete energy spectra in the confined case and different scattering amplitudes in the unconfined case.

1.
A.
Degasperis
and
S. N. M.
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Newton-equivalent Hamiltonians for the harmonic oscillator
,”
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,
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(
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).
2.
F. Calogero and A. Degasperis, Spectral Transform and Solitons: Tools to Solve and Investigate Nonlinear Evolution Equations (North–Holland, Amsterdam, 1982).
3.
See for example, Eq. (1.3.1.-3) of Ref. 2.
4.
See, for example, Ref. 2, p. 434.
5.
See, for example, Ref. 2, p. 433.
6.
A. Messiah, Quantum Mechanics (North–Holland, Amsterdam, 1961), Vol. I, pp. 69–70.
7.
G. Galilei, Discorsi e dimostrazioni matematiche intorno a due nuove scienze (Elzevir, Leiden, 1638);
English translation: H. Crew and A. de Salvio, Dialogues Concerning Two New Sciences (Macmillan, New York, 1914) and (Dover, New York, 1954).
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