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The quantum propagator and characteristic equation in the presence of a chainof $\delta$-potentials are obtained in the rectangular, cylindrical andspherical coordinate systems. The simplicity and efficiency of the method isillustrated via examples. As an application, the characteristic equation of aquantum harmonic oscillator confined to an infinite box is obtained. The rootsof the characteristic equation, determining the energy eigenvalues of therestricted oscillator, are calculated approximately and compared with theexisting numerical data.

Quantum propagator and characteristic equation in the presence of a chain of $\delta$-potentials