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The quantum behavior of charge carriers in semiconductor structures is oftendescribed in terms of the effective mass Schr\"{o}dinger equation, neglectingthe rapid fluctuations of the wave function on the scale of the atomic lattice.For systems with piecewise-constant mass and potential energy, this amounts tosolving a set of Helmholtz equations with wavenumbers dictated by the physicalparameters of each homogeneous subregion. Making use of the Green functionmethod, the system of differential equations can be expressed in boundaryintegral form to enable efficient numerical solution. In the present study,this strategy is applied in combination with a Galerkin technique to computethe energy spectrum and the wave functions of the electron in a mesoscopicstructure composed of two regions. The proposed formulation differs from thosepresented before for the same scenario in that it implements a symmetricdiscretization of the four Helmholtz boundary integral operators, which leadsto compact expressions and very accurate results.

Symmetric Galerkin boundary element method for computing the quantum states of the electron in a piecewise-uniform mesoscopic system