Point Interactions With Bias Potentials

Abstract

We develop an approach on how to define single-point interactions under the application of external fields. The essential feature relies on an asymptotic method based on the one-point approximation of multi-layered heterostructures that are subject to bias potentials. In this approach , the zero-thickness limit of the transmission matrices of specific structures is analyzed and shown to result in matrices connecting the two-sided boundary conditions of the wave function at the origin. The reflection and transmission amplitudes are computed in terms of these matrix elements as well as biased data. Several one-point interaction models of two-and three-terminal devices are elaborated. The typical transistor in the semiconductor physics is modeled in the "squeezed limit" as a δ-and a δ -potential and referred to as a "point" transistor. The basic property of these one-point interaction models is the existence of several extremely sharp peaks as an applied voltage tunes, at which the transmission amplitude is non-zero, while beyond these resonance values, the heterostructure behaves as a fully reflecting wall. The location of these peaks referred to as a "resonance set" is shown to depend on both system parameters and applied voltages. An interesting effect of resonant transmission through a δ-like barrier under the presence of an adjacent well is observed. This transmission occurs at a countable set of the well depth values.

Supersymmetric Partners of the One-Dimensional Infinite Square Well Hamiltonian

“…We are interested now in getting more information on the parity preserving extensions of H 0 . Then, if we use ( 6) in (17) we obtain that Re(A B * ) sin s = 0. Hence, either…”

“…Although the idea of self-adjoint extensions of symmetric (or Hermitian) operators on (infinite dimensional) Hilbert spaces is not yet very popular among physicists, it is, however, possible to find recent papers on the topic [16][17][18][19][20][21][22][23][24].…”

Regularization of a strong-weak duality between pointlike interactions in one dimension

“…But in contrast with bosons, the interpretation of this pointlike potential between fermions in terms of distributions or as the zero-range limit of smooth potentials is problematic and has attracted attention in the past [9][10][11][12][13][14][15][16][17][18][19][20][21][22][23]. Regularizations of related pointlike interactions is still a topic of recent research [24][25][26][27][28][29][30][31][32][33][34]. At this day, the Cheon-Shigehara potential [18] provides the most physical and operational interpretation of this interaction, since it is built as the zero-range limit of a Hermitian potential.…”

The Schrödinger particle on the half-line with an attractive $δ$-interaction: bound states and resonances

“…In relation with one-dimensional point interactions, we must say that there is an extensive literature concerning both mathematical and physical properties of such potentials [1,2,3,4,5,6,7,8,9,10,11,12,13]. From the mathematical point of view, such one-dimensional singular interactions/potentials may be associated with self-adjoint extensions of the differential operator −d 2 /dx 2 on a given domain where this operator is symmetric with equal non-zero deficiency indices.…”