  We introduce a Hamiltonian to address the Hilbert-P\'olya conjecture. The
eigenfunctions of the introduced Hamiltonian, subject to the Dirichlet boundary
conditions on the positive half-line, vanish at the origin by the nontrivial
zeros of the Riemann zeta function. Consequently, the eigenvalues are
determined by these nontrivial Riemann zeros. If the Riemann hypothesis (RH) is
true, the eigenvalues become real and represent the imaginary parts of the
nontrivial zeros. Conversely, if the Hamiltonian is self-adjoint, or more
generally, admits only real eigenvalues, then the RH follows. In our attempt to
demonstrate the latter, we establish the existence of a similarity
transformation of the introduced Hamiltonian that is self-adjoint on the domain
specified by an appropriate boundary condition, which is satisfied by the
eigenfunctions through the vanishing of the Riemann zeta function. Our result
can be extended to a broader class of functions whose zeros lie on the critical
line.
