Acceleration-dependent hamiltonian for the zeros of the riemann zeta function: Spectral properties and self-adjoint realizations of the new operator

An acceleration-dependent Hamiltonian operator [Formula: see text] and its corresponding Hamiltonian operator [Formula: see text] are introduced based on the new notion of quantum acceleratum operator. The new Hamiltonian operator consists of two independent parts: the Berry–Keating (BK) operator [Formula: see text] and the new operator part [Formula: see text] which depends on the acceleratum operator [Formula: see text]. The Hamiltonian [Formula: see text] offers new features for the zeros of the Riemann zeta function. The operator couples [Formula: see text] with a higher-order interaction involving the position and acceleratum operators, introducing nonlinear effects beyond conventional canonical mechanics. The structure of this coupling determines whether the eigenvalue spectrum remains discrete or transitions to a continuous regime, reflecting the balance between confinement and scaling symmetry. Establishing conditions for spectral discreteness provides key insight into the existence of bound states, the boundedness of energy levels, and the physical realizability of acceleration-dependent quantum dynamics. Such analysis extends the mathematical framework of self-adjoint operators and spectral theory to higher-order Hamiltonians, offering new perspectives on quantization in extended phase-space formulations and on the emergence of discrete energy spectra in nonstandard dynamical systems.