  A time operator is a Hermitian operator that is canonically conjugate to a
given Hamiltonian. For a particle in 1-dimension, a Hamiltonian conjugate
operator in position representation can be obtained by solving a hyperbolic
second-order partial differential equation, known as the time kernel equation,
with some boundary conditions. One possible solution is the time of arrival
operator. Here, we are interested in finding other Hamiltonian conjugates by
further studying the boundary conditions. A modified form of the time kernel
equation is also considered which gives an even bigger solution space.
