Abstract We introduce a new exactly solvable model in quantum mechanics that describes the propagation of particles through a potential field created by regularly spaced δ ′ -type point interactions, which model the localized dipoles often observed in crystal structures. We refer to the corresponding potentials as δ θ ′ -combs, where the parameter θ represents the contrast of the resonant wave at zero energy and determines the interface conditions in the Hamiltonians. We explicitly calculate the scattering matrix for these systems and prove that the transmission probability exhibits sharp resonance peaks while rapidly decaying at other frequencies. Consequently, Hamiltonians with δ θ ′ -comb potentials act as quantum filters, permitting tunneling only for specific wave frequencies. Furthermore, for each θ > 0, we construct a family of regularized Hamiltonians approximating the ideal model and prove that their transmission probabilities have a similar structure, thereby confirming the physical realizability of the band-pass filtering effect.
Abstract We introduce a new exactly solvable model in quantum mechanics that describes the propagation of particles through a potential field created by regularly spaced δ ′ -type point interactions, which model the localized dipoles often observed in crystal structures. We refer to the corresponding potentials as δ θ ′ -combs, where the parameter θ represents the contrast of the resonant wave at zero energy and determines the interface conditions in the Hamiltonians. We explicitly calculate the scattering matrix for these systems and prove that the transmission probability exhibits sharp resonance peaks while rapidly decaying at other frequencies. Consequently, Hamiltonians with δ θ ′ -comb potentials act as quantum filters, permitting tunneling only for specific wave frequencies. Furthermore, for each θ > 0, we construct a family of regularized Hamiltonians approximating the ideal model and prove that their transmission probabilities have a similar structure, thereby confirming the physical realizability of the band-pass filtering effect.
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.
Two rectangular models described by the one-dimensional Schr\"odinger equation with sharply localized potentials are suggested. The potentials have a multilayer 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.
This issue comprises two dozen research papers which are all in one sense or another devoted to models in which the interaction is singular and sharply localized; a typical example is a quantum particle interacting with a family of δ-type potentials. Such an idealization usually makes analysis of their properties considerably easier, sometimes allowing us to reduce it to a simple algebraic problem—this is why one speaks about solvable models. The subject can be traced back to the early days of quantum mechanics; however, the progress in this field was slow and uneven until the 1960s, mostly because singular interactions are often difficult to deal with mathematically and intuitive arguments do not work. After overcoming the initial difficulties the `classical' theory of point interactions was developed, and finally summarized in 1988 in a monograph by Albeverio, Gesztesy, Høegh-Krohn, and Holden, which you will find quoted in numerous places within this issue. A reliable way to judge theories is to observe the progress they make within one or two decades. In this case there is no doubt that the field has witnessed a continuous development and covered areas which nobody had thought of when the subject first emerged. The reader may see it in the second edition of the aforementioned book which was published by AMS Chelsea only recently and contained a brief survey of these new achievements. It is no coincidence that this topical issue appears at the same time; it has been conceived as its counterpart and a forum at which fresh results in the field can demonstrated. Let us briefly survey the contents of the issue. While the papers included have in common the basic subject, they represent a broad spectrum philosophically as well as technically, and any attempt to classify them is somewhat futile. Nevertheless, we will divide them into a few groups. The first comprises contributions directly related to the usual point-interaction ideology. M Correggi and one of the editors study a toy model of a decay under the influence of a time-periodic δ potential. E Demiralp describes the spectrum of a spherical harmonic oscillator amended with a concentric family of δ-shell interactions. Another of the editors presents an isoperimetric problem for point interactions arranged at vertices of a polygon. W Huddell and R Hughes show how singular perturbations of a one-dimensional Dirac operator can be approximated by regular potentials, and J Brasche constructs a family of Hamiltonians in which the singular interaction has a more complicated support, namely a Brownian path. Finally, B Pavlov and I Antoniou apply the singular perturbation technique to another classical Hamiltonian, that of a generalized Friedrichs model; no matter that the unperturbed observable is called momentum in their paper. The three papers in the following group are distinguished by the fact that they consider systems which are fully or partially periodic. F Bentosela and M Tater analyse scattering on a crystalline `slab' modelled by point interactions distributed periodically on a finite number of parallel plates. E de Prunelé studies evolution of wavepackets in crystal models of different geometries, and M Avdonin et al discuss a simple model of a spin-dependent scattering on a one-dimensional array of quantum dots. The next group of papers is devoted to a topic which was untouched at the time of the aforementioned first edition, namely quantum graphs, which became a subject of interest after numerous applications of such systems to semiconductor, carbon and other nanostructures. Most contributions here deal with the `usual' model in which the Hamiltonian is a Schrödinger operator supported by the graph. P Kuchment describes spectral properties of such graphs, in particular periodic ones and those with decorations. S Albeverio and K Pankrashkin present a modification of Krein's formula which is suitable for constructing Hamiltonians of quantum graphs using boundary conditions at vertices directly. Two papers are devoted to inverse problems in this context: M Harmer studies inverse scattering for the matrix Schrödinger operator on the halfline with applications to star graphs, while P Kurasov and M Nowaczyk give a mathematically rigorous version of the known Gutkin–Smilansky result on the inverse spectral problem. The paper by O Post contributes to the question of how graphs can be approximated by more realistic `fat' graphs, and describes a class leading to disconnected quantum graphs. Finally, S Kondej and one of the editors study scattering in the context of `leaky' graphs which takes quantum tunnelling into account. While most results in this field describe one-particle Hamiltonians, more complicated systems have also been studied. In this issue we have three examples. C Cacciapuito, R Carlone, and R Figari discuss decoherence in a simple model of two particles, one heavy and one light, interacting through a δ potential; they give a rigorous meaning to a formula derived by Joos and Zeh. A related model by R Figari and A Teta is used to describe ionization. M Hallnäs, E Langmann, and C Paufler treat a true N -body situation, namely a model of one-dimensional gas of distinguishable particles interacting through generalized point interactions; they write the Bethe ansatz and present the solution of a particular case. The last group is a collection of contributions which in one sense or another are outside quantum mechanics, either modifying its postulates or applying it to a different physical situation. The latter applies to the paper of D Noja and A Posilicano in which they study nonlinear wave equations with point perturbations and show the existence of a solution to the Cauchy problem. F Coutinho et al discuss one-dimensional point interactions with energy-dependent coupling constant, S Albeverio and S Kuzhel examine a class of point interactions which are not symmetric but P-symmetric, where P is the parity operator, and M Znojil and V Jakubský consider a `double-well' PT-symmetric model with two δ interactions with an imaginary coupling. The last two papers present mathematical constructions. Yu Shondin demonstrates a way to define an interaction more singular than the usual δ potentials obtained by means of self-adjoint extensions, and V Koshmanenko presents a construction of strongly singular perturbations leading to rather unusual `Hamiltonians'.
We investigate numerically the time-dependent Schrödinger equation on a two-dimensional comb in the presence of a localized Dirac delta potential on the backbone. The geometric constraint and the point interaction are both regularized via narrow Gaussian approximations, and the resulting equation is discretized with second-order finite differences and integrated with a fourth-order Runge–Kutta scheme on a finite domain. Two initial states are considered—a Gaussian wave packet and a comb-like wave function—and two barrier configurations are studied: a potential centered at the origin and one displaced along the backbone. In both cases, the Dirac delta interaction suppresses the probability density at the barrier position and reorganizes the wave packet into a persistent split-peak profile, in contrast to the single spreading maximum observed in the free case. When the barrier is displaced, due to reflection from the barrier, the left–right symmetry is broken and the density remains asymmetrically concentrated near the interaction site. To elucidate the results further, we also analyze the second moment, which undoubtedly confirms that the anomalous dynamics features, such as the subdiffusive behavior on comb-like structures in classical systems are retained in the quantum case as well.
Abstract Quantum tunneling through potential barriers plays a central role in diverse physical phenomena ranging from nuclear fusion and radioactive decay to semiconductor devices and superconducting technologies. In this work, we calculate tunneling probabilities and wave functions for an arbitrary number of potential barriers with diverse functional forms, including square, Gaussian, parabolic, and Lorentzian profiles, using a general numerical framework. The numerical results are first validated by comparison with exact analytical solutions for double, triple and quadruple-square barrier systems, demonstrating excellent agreement. The results reveal that for a system with N barriers, a single resonance peak splits into ( N − 1 ) distinct peaks, which eventually merge into an energy band as N becomes large, in a manner fully consistent with the existing literature. A key additional result is the incorporation of absorption effects through an imaginary component of the potential, providing a realistic description of particle trapping and leading to a progressive suppression of resonant transmission. For sufficiently strong absorption, the resonant structures are completely washed out. Further, if the barriers become non-uniform or disordered, the regular resonance pattern is also observed to be completely washed out. These findings demonstrate the broad applicability of the approach to complex potential landscapes and offer insights relevant to both fundamental quantum mechanics and the design of next-generation quantum devices.
Scattering of a quantum particle with internal structure is fundamentally different from that of a point particle and shows quantum effects such as the modification of transmission due to tunnelling and trapping of the particle. As in a preceding paper (Shore et al 2014 New J. Phys. 17 013046 ) we consider a model of a symmetric, rigid rotor travelling through an aperture in a thin but impenetrable screen which is perpendicular to both the direction of motion and the rotation axis. We determine the quantum mechanical properties of this two-dimensional geometrical model using a quasi one-dimensional scattering problem with unconventional boundaries. Our calculations rely on finding the Green's function, which has a direct connection to the scattering matrix. Evaluated on a discrete lattice the Hamiltonian is 'dressed' by a self-energy correction that takes into account the open boundary conditions in an exact way. We find that the passage through the aperture can be suppressed or enhanced as a result of the rotational motion. These effects manifest themselves through resonances in the transmission probability as a function of incident energy and symmetry of the incident wavefunction. We determine the density-of-states to reveal the mode structure of resonant states and to exhibit the lifetimes of temporary trapping within the aperture.