One-dimensional Point Interactions Research Articles

Approximation of one-dimensional relativistic point interactions by regular potentials revised

We show that the one-dimensional Dirac operator with quite general point interaction may be approximated in the norm resolvent sense by the Dirac operator with a scaled regular potential of the form $1/\varepsilon~h(x/\varepsilon)\otimes B$, where $B$ is a suitable $2\times 2$ matrix. Moreover, we prove that the limit does not depend on the particular choice of $h$ as long as it integrates to a constant value.

An optimization problem for finite point interaction families

We consider the spectral problem for a family of N point interactions of the same strength confined to a manifold with a rotational symmetry, a circle or a sphere, and ask for configurations that optimize the ground state energy of the corresponding singular Schrödinger operator. In case of the circle the principal eigenvalue is sharply maximized if the point interactions are distributed at equal distances. The analogous question for the sphere is much harder and reduces to a modification of Thomson problem; we have been able to indicate the unique maximizer configurations for . We also discuss the optimization for one-dimensional point interactions on an interval with periodic boundary conditions. We show that the equidistant distributions give rise to maximum ground state eigenvalue if the interactions are attractive, in the repulsive case we get the same result for weak and strong coupling and we conjecture that it is valid generally.

The spectrum of the Schrödinger–Hamiltonian for trapped particles in a cylinder with a topological defect perturbed by two attractive delta interactions

In this paper, we exploit the technique used in [Albeverio and Nizhnik, On the number of negative eigenvalues of one-dimensional Schrödinger operator with point interactions, Lett. Math. Phys. 65 (2003) 27; Albeverio, Gesztesy, Hoegh-Krohn and Holden, Solvable Models in Quantum Mechanics (second edition with an appendix by P. Exner, AMS Chelsea Series 2004); Albeverio and Kurasov, Singular Perturbations of Differential Operators: Solvable Type Operators (Cambridge University Press, 2000); Fassari and Rinaldi, On the spectrum of the Schrödinger–Hamiltonian with a particular configuration of three one-dimensional point interactions, Rep. Math. Phys. 3 (2009) 367; Fassari and Rinaldi, On the spectrum of the Schrödinger–Hamiltonian of the one-dimensional harmonic oscillator perturbed by two identical attractive point interactions, Rep. Math. Phys. 3 (2012) 353; Albeverio, Fassari and Rinaldi, The Hamiltonian of the harmonic oscillator with an attractive-interaction centered at the origin as approximated by the one with a triple of attractive-interactions, J. Phys. A: Math. Theor. 49 (2016) 025302; Albeverio, Fassari and Rinaldi, Spectral properties of a symmetric three-dimensional quantum dot with a pair of identical attractive [Formula: see text]-impurities symmetrically situated around the origin II, Nanosyst. Phys. Chem. Math. 7(5) (2016) 803; Albeverio, Fassari and Rinaldi, Spectral properties of a symmetric three-dimensional quantum dot with a pair of identical attractive [Formula: see text]-impurities symmetrically situated around the origin, Nanosyst. Phys. Chem. Math. 7(2) (2016) 268] to deal with delta interactions in a rigorous way in a curved spacetime represented by a cosmic string along the [Formula: see text] axis. This mathematical machinery is applied in order to study the discrete spectrum of a point-mass particle confined in an infinitely long cylinder with a conical defect on the [Formula: see text] axis and perturbed by two identical attractive delta interactions symmetrically situated around the origin. We derive a suitable approximate formula for the total energy. As a consequence, we found the existence of a mixing of states with positive or zero energy with the ones with negative energy (bound states). This mixture depends on the radius [Formula: see text] of the trapping cylinder. The number of quantum bound states is an increasing function of the radius [Formula: see text]. It is also interesting to note the presence of states with zero total energy (quasi free states). Apart from the gravitational background, the model presented in this paper is of interest in the context of nanophysics and graphene modeling. In particular, the graphene with double layer in this framework, with the double layer given by the aforementioned delta interactions and the string on the [Formula: see text]-axis modeling topological defects connecting the two layers. As a consequence of these setups, we obtain the usual mixture of positive and negative bound states present in the graphene literature.

Some properties of one-dimensional point interactions

In this review paper we shall give a brief account of some properties of so called point interactions that describe a potentials spatially localized. A different families of one-dimensional point interactions are investigated and then relevant tunneling times are calculated. Than we demonstrate how a bound state in the continuum may be generated applying supersymmetric quantum mechanics. Finally, it is shown how the latter method may be used to tailor phase rigidity of one-dimensional point interactions.

PHASE RIGIDITY OF POINT INTERACTIONS

In this paper, we investigate phase rigidity of one-dimensional point interactions. With the aid of supersymmetric quantum mechanics (SUSYQM) we generate family of isospectral potentials describing point interactions. We than demonstrate that for SUSYQM generated bound states in the continuum (BIC) phase rigidity is always zero, while for bound states from discrete part of spectrum phase rigidity may vary from zero to unity, depending on the strength of point interaction.

GENERALIZED POINT INTERACTION AND TUNNELING TIMES

The scattering properties of four-parameter one-dimensional point interactions also define time aspect of these interactions. In this paper we investigate how different families of point interactions affect the relevant tunneling times. Particular emphasis is given to various interpretations of so called δ' potential.

One-dimensional point interaction with Griffiths’ boundary conditions

Griffiths proposed a pair of boundary conditions that define a point interaction in one dimensional quantum mechanics. The conditions involve the nth derivative of the wave function where n is a non-negative integer. We re-examine the interaction so defined and explicitly confirm that it is self-adjoint for any even value of n and for n = 1. The interaction is not self-adjoint for odd n > 1. We then propose a similar but different pair of boundary conditions with the nth derivative of the wave function such that the ensuing point interaction is self-adjoint for any value of n.

On the spectrum of the Schrödinger Hamiltonian with a particular configuration of three one-dimensional point interactions

In this note we investigate in detail the spectrum of the Schrodinger Hamiltonian with a configuration of three equally spaced one-dimensional point interactions (Dirac distributions), with the external ones having the same negative coupling constant. It will be seen that despite its simplicity, such a toy model exhibits a fairly rich variety of spectral combinations when the two coupling constants and the separation distance are manipulated. By analysing the equation determining the square root of the absolute value of the ground state energy and those determining the same quantity for the two possible excited states, we explicitly calculate the eigenvalues for all possible values of the separation distance and the two coupling constants. As a result of our analysis, we provide the conditions in terms of the three parameters in order to have the emergence of such excited states. Furthermore, we use our findings in order to get the confirmation of the fact that the Hamiltonian with such a configuration of three simple point interactions whose coupling constants undergo a special scaling in terms of the vanishing separation distance, converges in the norm resolvent sense to the Hamiltonian with an attractive δ′-interaction centred at the origin, as was shown by Exner and collaborators making the result previously obtained by Cheon et al. mathematically rigorous.

One-dimensional point interaction with three complex parameters

For a pair of non-Hermitian Hamiltonian H and its Hermitian adjoint H†, there are situations in which their eigenfunctions form a biorthogonal system. We illustrate such a situation by means of a one-particle system with a one-dimensional point interaction in the form of the Fermi pseudo-potential. The interaction consists of three terms with three strength parameters gi (i = 1, 2 and 3), which are all complex. This complex point interaction is neither Hermitian nor -invariant in general. The S-matrix for the transmission–reflection problem constructed with H (or with H†) in the usual manner is not unitary, but it conforms to the pseudo-unitarity that we define. The pseudo-unitarity is closely related to the biorthogonality of the eigenfunctions. The eigenvalue spectrum of H with the complex interaction is generally complex but there are cases where the spectrum is real. In such a case H and H† form a pseudo-Hermitian pair.

Wigner functions, contact interactions, and matching

Quantum mechanics in phase space (or deformation quantization) appears to fail as an autonomous quantum method when infinite potential walls are present. The stationary physical Wigner functions do not satisfy the normal eigen equations, the ★-eigen equations, unless an ad hoc boundary potential is added [N.C. Dias, J.N. Prata, J. Math. Phys. 43 (2002) 4602 (quant-ph/0012140)]. Alternatively, they satisfy a different, higher-order, “★-eigen-★ equation”, locally, i.e. away from the walls [S. Kryukov, M.A. Walton, Ann. Phys. 317 (2005) 474 (quant-ph/0412007)]. Here we show that this substitute equation can be written in a very simple form, even in the presence of an additional, arbitrary, but regular potential. The more general applicability of the ★-eigen-★ equation is then demonstrated. First, using an idea from [D.B. Fairlie, C.A. Manogue, J. Phys. A 24 (1991) 3807], we extend it to a dynamical equation describing time evolution. We then show that also for general contact interactions, the ★-eigen-★ equation is satisfied locally. Specifically, we treat the most general possible (Robin) boundary conditions at an infinite wall, general one-dimensional point interactions, and a finite potential jump. Finally, we examine a smooth potential, that has simple but different expressions for x positive and negative. We find that the ★-eigen-★ equation is again satisfied locally. It seems, therefore, that the ★-eigen-★ equation is generally relevant to the matching of Wigner functions; it can be solved piece-wise and its solutions then matched.

Characterization of one-dimensional point interactions for the Schrödinger operator by means of boundary conditions

We find a new representation of all boundary conditions corresponding to the so-called point interactions. We show that there is one-to-one correspondence between one-dimensional point interactions and boundary conditions of the form with α,β,η satisfying |η| = |α|2 + |β|2 = 1.

Relativistic point interactions: Approximation by smooth potentials

We show that the four-parameter family of one-dimensional relativistic point interactions studied by Benvegnù and Da̧browski may be approximated in the strong resolvent sense by smooth, local, short-range perturbations of the Dirac Hamiltonian. In addition, we prove that the nonrelativistic limits correspond to the Schrödinger point interactions studied extensively by the author and Paul Chernoff.

The regulated four-parameter one-dimensional point interaction

The general four-parameter point interaction in one-dimensional quantum mechanics is regulated. It allows the exact solution, but not the perturbative one. We conjecture that this is due to the interaction not being asymptotically free. We then propose a different breakup of unperturbed theory and interaction, which now is asymptotically free but leads to the same physics. The corresponding regulated potential can be solved both exactly and perturbatively, in agreement with the conjecture.

One-dimensional point interactions

One-dimensional point interactions

Some remarks on the δ′-interaction in one dimension

We discuss the existence and the physical properties of the one-dimensional δ′- interaction Hamiltonian. We show that the so called δ′-interaction Hamiltonian which appears in the literature represents a self-adjoint realization of the heuristic operator - d 2 dx 2 +β|δ′(x)〉 〈δ′(x)| with a renormalized coupling constant β. We investigate also a possible self-adjoint realization of the formal Hamiltonian - d 2 dx 2 +βδ′ (x). We show that this realization coincides with the ordinary one-dimensional point interaction Hamiltonian H A = - d 2 dx 2 +Aδ(x).