  In this paper, we consider the one-dimensional semirelativistic
Schr\"{o}dinger equation for a particle interacting with $N$ Dirac delta
potentials. Using the heat kernel techniques, we establish a resolvent formula
in terms of an $N \times N$ matrix, called the principal matrix. This matrix
essentially includes all the information about the spectrum of the problem. We
study the bound state spectrum by working out the eigenvalues of the principal
matrix. With the help of the Feynman-Hellmann theorem, we analyze how the bound
state energies change with respect to the parameters in the model. We also
prove that there are at most $N$ bound states and explicitly derive the bound
state wave function. The bound state problem for the two-center case is
particularly investigated. We show that the ground state energy is bounded
below, and there exists a self-adjoint Hamiltonian associated with the
resolvent formula. Moreover, we prove that the ground state is nondegenerate.
The scattering problem for $N$ centers is analyzed by exactly solving the
semirelativistic Lippmann-Schwinger equation. The reflection and the
transmission coefficients are numerically and asymptotically computed for the
two-center case. We observe the so-called threshold anomaly for two
symmetrically located centers. The semirelativistic version of the
Kronig-Penney model is shortly discussed, and the band gap structure of the
spectrum is illustrated. The bound state and scattering problems in the
massless case are also discussed. Furthermore, the reflection and the
transmission coefficients for the two delta potentials in this particular case
are analytically found. Finally, we solve the renormalization group equations
and compute the beta function nonperturbatively.
