A resonant tunneling effect of an extremely thin potential well on the transmission of charged particles through a planar heterostructure with an arbitrary potential profile is investigated in a squeezing limit as the well width tends to zero. In this limit, the transmission probability through the structure is shown to be blocked for all the parameter values of the well, except a resonance set of Lebesgue measure zero. The peak-to-valley ratio is shown to increase crucially with the squeezing of the well: the thinner is its thickness, the resonant peaks become sharper and localized at isolated points. In contrast, a discrete spectrum of the heterostructure (if any) exists both on the resonance set and beyond it; however, the squeezing scenario here turns out to be quite interesting and sophisticated.

The spectrum of a one-dimensional pseudospin-one Hamiltonian with a three-component potential is studied for two configurations: (i) all the potential components are constants over the whole coordinate space and (ii) the profile of some components is of a rectangular form. In case (i), it is illustrated how the structure of three (lower, middle and upper) bands depends on the configuration of potential strengths including the appearance of flat bands at some special values of these strengths. In case (ii), the set of two equations for finding bound states is derived. The spectrum of bound-state energies is shown to depend crucially on the configuration of potential strengths. Each of these configurations is specified by a single strength parameter V . The bound-state energies are calculated as functions of the strength V and a one-point approach is developed realizing correspondent point interactions. For different potential configurations, the energy dependence on the strength V is described in detail, including its one-point approximation. From a whole variety of bound-state spectra, four characteristic types are singled out.

A heterostructure composed of N parallel homogeneous layers is studied in the limit as their widths l 1 , …, l N shrink to zero. The problem is investigated in one dimension and the piecewise constant potential in the Schrödinger equation is given by the strengths V 1 , …, V N as functions of l 1 , …, l N , respectively. The key point is the derivation of the conditions on the functions V 1 ( l 1 ), …, V N ( l N ) for realizing a family of one-point interactions as l 1 , …, l N tend to zero along available paths in the N -dimensional space. The existence of equations for a squeezed structure, the solution of which determines the system parameter values, under which the non-zero tunneling of quantum particles through a multi-layer structure occurs, is shown to exist and depend on the paths. This tunneling appears as a result of an appropriate cancellation of divergences.

Two approaches are developed for the study of the bound states of a one-dimensional Dirac equation with the potential consisting of N delta-function centers. One of these uses Green's function method. This method is applicable to a finite number N of delta-point centers, reducing the bound state problem to finding the energy eigenvalues from the determinant of a 2 N x 2 N matrix. The second approach starts with the matrix for a single delta-center that connects the two-sided boundary conditions for this center. This connection matrix is obtained from the squeezing limit of a piecewise constant approximation of the delta-function. Having then the connection matrices for each center, the transmission matrix for the whole system is obtained by multiplying the one-center connection matrices and the free transfer matrices between neighbor centers. An equation for bound state energies is derived in terms of the elements of the total transfer matrix. Within both approaches, the transcendental equations for bound state energies are derived, the solutions to which depend on the strength of delta-centers and the distance between them, and this dependence is illustrated by numerical calculations. The bound state energies for the potentials composed of one, two, and three delta-centers (N = 1, 2, 3) are computed explicitly. The principle of strength additivity is analyzed in the limits as the delta-centers merge at a single point or diverge to infinity.

A heterostructure composed of two parallel homogeneous layers is studied in the limit as their widths l 1 and l 2 , and the distance between them r shrinks to zero simultaneously. The problem is investigated in one dimension and the squeezing potential in the Schrödinger equation is given by the strengths V 1 and V 2 depending on the layer thickness. A whole class of functions V 1 ( l 1 ) and V 2 ( l 2 ) is specified by certain limit characteristics as l 1 and l 2 tend to zero. The squeezing limit of the scattering data a ( k ) and b ( k ) derived for the finite system is shown to exist only if some conditions on the system parameters V j , l j , j = 1, 2, and r take place. These conditions appear as a result of an appropriate cancellation of divergences. Two ways of this cancellation are carried out and the corresponding two resonance sets in the system parameter space are derived. On one of these sets, the existence of non-trivial bound states is proven in the squeezing limit, including the particular example of the squeezed potential in the form of the derivative of Dirac’s delta function, contrary to the widespread opinion on the non-existence of bound states in δ ′-like systems. The scenario how a single bound state survives in the squeezed system from a finite number of bound states in the finite system is described in detail.

A heterostructure composed of two parallel homogeneous layers is studied in the limit as their width and the distance between them shrinks to zero simultaneously. The problem is considered in one dimension and the squeezing potential in the Schrödinger equation is chosen in the form of a piecewise constant function. As a result, two families of point interactions with bound state energy are realized from this structure. The specific feature of these interactions is the resonant-tunneling transmission of electrons through one-point singular potentials under certain conditions described by transcendental equations. The solutions to these equations define so-called resonance sets of Lebesgue’s measure zero. A particular example is the potential in the form of the derivative of Dirac’s delta function. For a whole family of point interactions including this example, the existence of a bound state is proven, contrary to the widespread opinion on the non-existence of bound states in δ'-like systems.

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 delta- and a delta'-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 delta-like barrier under the presence of an adjacent well is observed. This transmission occurs at a countable set of the well depth values.

A “one-point” approximation is proposed to investigate the transmission of electrons through the extra thin heterostructures composed of two parallel plane layers. The typical example is the bilayer for which the squeezed potential profile is the derivative of Dirac’s delta function. The Schr¨odinger equation with this singular one-dimensional profile produces a family of contact (point) interactions each of which (called a “distributional” б′-potential) depends on the way of regularization. The discrepancies widely discussed so far in the literature regarding the family of delta derivative potentials are eliminated using a two-scale power-connecting parametrization of the bilayer potential that enables one to extend the family of distributional б′-potentials to a whole class of “generalized” б′-potentials. In a squeezed limit of the bilayer structure to zero thickness, the resonant tunneling through this structure is shown to occur in the form of sharp peaks located on the sets of Lebesgue’s measure zero (called resonance sets). A four-dimensional parameter space is introduced for the representation of these sets. The transmission on the complement sets in the parameter space is shown to be completely opaque.

The physical interpretation of the appearance of resonant transmission through single-point barriers is discussed on the basis of a double-layer heterostructure in the squeezing limit as both the thickness of the layers and the distance between them tend to zero simultaneously. In this limit, the electron transmission through a barrier-well structure is derived to be non-zero at certain discrete values of the system parameters forming the so-called resonance set, while beyond this set, the structure behaves as a perfectly reflecting wall. The origin of this phenomenon is shown to result from the reflection coefficients at the interfaces in the inter-layer space. The transmission amplitude is computed as a set function defined on the trihedral angle surface in a three-dimensional parameter space.

The so-called delta'-interaction as a particular example in Kurasov's distribution theory developed on the space of discontinuous (at the point of singularity) test functions, is identified with the diagonal transmission matrix, continuously depending on the strength of this interaction. On the other hand, in several recent publications, the delta'-potential has been shown to be transparent at some discrete values of the strength constant and opaque beyond these values. This discrepancy is resolved here on the simple physical example, namely the heterostructure consisting of two extremely thin layers separated by infinitesimal distance. In the three-scale squeezing limit as the thickness of the layers and the distance between them simultaneously tend to zero, a whole variety of single-point interactions is realized. The key point is the generalization of the delta'-interaction to the family for which the resonance sets appear in the form of a countable number of continuous two-dimensional curves. In this way, the connection between Kurasov's delta'-interaction and the resonant-tunneling point interactions is derived and the splitting of the resonance sets for tunneling plays a crucial role. (C) 2018 Elsevier Inc. All rights reserved.

Several families of one-point interactions are derived from the system consisting of two and three delta-potentials which are regularized by piecewise constant functions. In physical terms such an approximating system represents two or three extremely thin layers separated by some distance. The two-scale squeezing of this heterostructure to one point as both the width of delta-approximating functions and the distance between these functions simultaneously tend to zero is studied using the power parameterization through a squeezing parameter epsilon -> 0, so that the intensity of each delta-potential is c(j) = a(j) epsilon(1-mu), aj is an element of R, j = 1, 2, 3, the width of each layer l = epsilon and the distance between the layers r = c epsilon(tau), c > 0. It is shown that at some values of the intensities a(1), a(2) and a(3), the transmission across the limit point potentials is non-zero, whereas outside these (resonance) values the one-point interactions are opaque splitting the system at the point of singularity into two independent subsystems. Within the interval 1 < mu < 2, the resonance sets consist of two curves on the (a(1), a(2))-plane and three surfaces in the (a(1), a(2), a(3))-space. As the parameter mu approaches the value mu = 2, three types of splitting the one-point interactions into countable families are observed.

Families of one-point interactions are derived from the system consisting of regularized two- and three-delta potentials using different paths of the convergence of corresponding transmission matrices in the squeezing limit. This limit is controlled by the relative rate of shrinking the width of delta-like functions and the distance between these functions using the power parameterization: width l =ε^μ -1, μ∈ [2, ∞] (for width) and r = ε^τ, τ∈ [1, ∞] (for distance). It is shown that at some values of real coefficients (intensities a_1, a_2 and a_3) at the delta potentials, the transmission across the limit point interactions is non-zero, whereas outside these (resonance) values the one-point interactions are opaque splitting the system at the point of singularity into two independent subsystems. The resonance sets of intensities at which a non-zero transmission occurs are proved to be of four types depending on the way of squeezing the regularized system to one point. In its turn, on these sets the limit one-point interactions are observed to be either single- or multiple-resonant-tunnelling potentials also depending on the squeezing way. In the two-delta case the resonance sets are curves on the (a_1,a_2)-plane and surfaces in the (a_1,a_2,a_3)-space for the three-delta system. A new phenomenon of furcation of single-valued resonance sets to multi-valued ones is observed under approaching the parameter μ >2 to the value μ =2.

A family of non-trivial one-point interactions is constructed by means of a two-scale approximation with spatially symmetric piecewise constant functions. The approximation is realized through two powers of a squeezing parameter. The convergence is considered in terms of finite-range transmission matrices which are given in an explicit form. The limiting matrix that connects the two-sided boundary conditions at the point of singularity appears to be the unity matrix defined at a countable set in the space of intensities of the potential. Everywhere beyond this 'resonance' set, the limiting point interaction separates the system into two subsystems with boundary conditions of the Dirichlet type. In other words, any interaction of this family behaves as a fully transparent barrier on a resonance set and beyond this set as a fully reflecting wall. Among these interactions a special regularizing sequence that converges to the second derivative of Dirac's delta function in the sense of distributions is found. Because of this example, it is reasonable to term all the point interactions with the properties described above as 'resonant-tunnelling delta ''-potentials'.

A zero-thickness limit for two-terminal and three-terminal devices from the quantum electronics domain is analysed. The study is focused on heterostructures composed of a single barrier with, adjacent, one or two prewells. The point interactions obtained in this limit are shown to be described by a family of 'resonant' diagonal matrices that connect the two-sided boundary conditions at the device origin, which are a subclass of the whole four-parameter family of point interactions. Transmission through such a device is absent almost everywhere, except at a few points, whose number and position can be controlled by a gate voltage applied externally to the barrier subsystem. It is remarkable that the existence of resonances in the zero-thickness limit occurs only if a squeezing sequence is constructed in a 'delta'-like' way. In this case, the delta'-limit describes adequately the resonant behaviour of a barrier-well heterostructure with realistic parameters. Simple analytical expressions obtained for resonance sets are supported by direct numerical calculations of the transmission being in agreement with the results of experiments on semiconductor devices. The zero-thickness delta'-like limiting procedure can be used in the design of nanodevices or contacting quantum wires.

A zero-thickness limit of three-layer heterostructures under two bias voltages applied externally, where one of which is supposed to be a gate parameter, is studied. As a result, an effect of controllable resonant tunnelling of electrons through single-point potentials is shown to exist. Therefore the limiting structure may be termed a “point triode” and considered in the theory of point interactions as a new object. The simple limiting analytical expressions adequately describe the resonant behaviour in the transistor with realistic parameter values and thus one can conclude that the zero-range limit of multi-layer structures may be used in fabricating nanodevices. The difference between the resonant tunnelling across single-point potentials and the Fabry–Pérot interference effect is also emphasized.

Two rectangular models described by the one-dimensional Schroedinger equation with sharply localized potentials are suggested. The potentials have a multi-layer thin structure being composed from adjacent barriers and wells. Their peculiar tunneling properties are studied in considerable detail. Particularly, in the zero-range limit when the potentials are squeezed to a single point, sharp peaks with total transmission are observed at certain (positive and negative) quantized values of the potential strength constant forming infinite discrete sets. Beyond these sets, the barrier-well structures behave as a perfectly reflecting wall. The transcendental equations with respect to potential strengths, the solutions of which determine transmission (resonance) sets, are derived. In this regard, both the models are exactly solvable. The energy dependence of an incident particle is shown to reveal a resonance behavior, being completely different from that observed in a typical double barrier structure.

Restricting ourselves to a simple rectangular approximation but using properly a two-scale regularization procedure, additional resonant tunneling properties of the one-dimensional Schrödinger operator with a delta derivative potential are established, which appear to be lost in the zero-range limit. These "intrinsic" properties are complementary to the main already proved result that different regularizations of Dirac's delta function produce different limiting self-adjoint operators. In particular, for a given regularizing sequence, a one-parameter family of connection condition matrices describing bound states is constructed. It is proposed to consider the convergence of transfer matrices when the potential strength constant is involved into the regularization process, resulting in an extension of resonance sets for the transmission across a δ′-barrier.

A simple one-dimensional lattice model is suggested to describe the experimentally observed plateau in force-stretching diagrams for some macromolecules. This chain model involves the nearest-neighbor interaction of a Morse-like potential (required to have a saturation branch) and a harmonic second-neighbor coupling. Under an external stretching applied to the chain ends, the intersite Morse-like potential results in the appearance of a double-well potential within each chain monomer, whereas the interaction between the second neighbors provides a homogeneous bistable (degenerate) ground state, at least within a certain part of the chain. As a result, different conformational changes occur in the chain under the external forcing. The transition regions between these conformations are described as topological solitons. With a strong second-neighbor interaction, the solitons describe the transition between the bistable ground states. However, the key point of the model is the appearance of a heterogenous structure, when the second-neighbor coupling is sufficiently weak. In this case, a part of the chain has short bonds with a single-well potential, whereas the complementary part admits strongly stretched bonds with a double-well potential. This case allows us to explain the existence of a plateau in the force-extension diagram for DNA and α-helix protein. Finally, the soliton dynamics are studied in detail.

Recently, the non-zero transmission of a quantum particle through the one-dimensional singular potential given in the form of the derivative of Dirac's delta function, lambda delta' (x), with lambda is an element of R, being a potential strength constant, has been discussed by several authors. The transmission occurs at certain discrete values of. forming a resonance set {lambda(n)}(n=1)(infinity) For lambda is not an element of {lambda(n)}(n=1)(infinity) this potential has been shown to be a perfectly reflecting wall. However, this resonant transmission takes place only in the case when the regularization of the distribution delta'(x) is constructed in a specific way. Otherwise, the delta'-potential is fully non-transparent. Moreover, when the transmission is non-zero, the structure of a resonant set depends on a regularizing sequence Delta(epsilon)'(x) that tends to delta'(x) in the sense of distributions as epsilon -> 0. Therefore, from a practical point of view, it would be interesting to have an inverse solution, i.e. for a given (lambda) over bar is an element of R, to construct such a regularizing sequence Delta(epsilon)'(x) that the delta'-potential at this value is transparent. If such a procedure is possible, then this value (lambda) over bar has to belong to a corresponding resonance set. This paper is devoted to solving this problem and, as a result, the family of regularizing sequences is constructed by tuning adjustable parameters in the equations that provide a resonance transmission across the delta'-potential. This construction can be realized if each regularizing sequence Delta(epsilon)'(x) depends on lambda is an element of R and this is a key point of our approach. Next, we can solve the inverse problem if the regularization is constructed from rectangles. Since in some cases the renormalization procedure Delta(epsilon)'(x) -> delta'(x) leads to the existence of an effective delta-interaction, it is reasonable from the beginning to consider the linear combination V(x) = eta delta(x) + lambda delta'(x) with (eta, lambda) is an element of R-2.

A family of point interactions of the dipole type is studied in one dimension using a regularization by rectangles in the form of a barrier and a well separated by a finite distance. The rectangles and the distance are parametrized by a squeezing parameter epsilon -> 0 with three powers mu, nu and tau describing the squeezing rates for the barrier, the well and the distance, respectively. This parametrization allows us to construct a whole family of point potentials of the dipole type including some other point interactions, such as e. g. delta-potentials. Varying the power tau, it is possible to obtain in the zero-range limit the following two cases: (i) the limiting delta'-potential is opaque (the conventional result obtained earlier by some authors) or (ii) this potential admits a resonant tunneling (the opposite result obtained recently by other authors). The structure of resonances (if any) also depends on a regularizing sequence. The sets of the{mu, nu, tau}-space where a non-zero ( resonant or non-resonant) transmission occurs are found. For all these cases in the zero-range limit the transfer matrix is shown to be of the form Lambda = ((chi)(g) (0)(chi-1)) with real parameters chi and g depending on a regularizing sequence. Those cases when chi not equal 1 and g not equal 0 mean that the corresponding delta'-potential is accompanied by an effective delta-potential.