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Open Dartmouth: Faculty Open Access Scholarship


Some recent efforts to reformulate analytic number theory in terms of Hamiltonian eigenspectra has led to some developments in non-Hermitian operator theory. Herein we examine analytic number theory using Hamilton’s equations from quantum mechanics. Borrowing from the second axiom of Kol- mogorov, the eigenfunctions of these equations can be treated as a chaotic quantum system in a rigged Hilbert space, much like the harmonic oscillator is for integrable quantum systems. As such, herein we perform a symmetrization procedure from the recent developments of non-Hermitian operator theory to obtain a Hermitian analogue using a similarity transformation, and provide an analytical expression for the eigenvalues of the results using Green’s functions. A nontrivial expression for the eigensolution of the Hamilton eigenequation is also obtained. A Gelfand triplet is then used to ensure that the eigensolution is well defined. The holomorphicity of the resulting eigenspectrum is demon- strated, and a second quantization of the resulting Schrodinger equation is performed. From the holomorphicity of the eigensolution, a general solution is also obtained by performing an invariant similarity transformation.