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In this article, we provide an expansion (up to the fourth order of the coupling constant) of the energy of the ground state of the Hamiltonian of a quantum mechanical particle moving inside a parabolic well in the x-direction and constrained by the presence of a two-dimensional impurity, modelled by an attractive two-dimensional isotropic Gaussian potential. By investigating the associated Birman–Schwinger operator and exploiting the fact that such an integral operator is Hilbert–Schmidt, we use the modified Fredholm determinant in order to compute the energy of the ground state created by the impurity.

In this note we consider a one-dimensional quantum mechanical particle constrained by a parabolic well perturbed by a Gaussian potential. As the related Birman-Schwinger operator is trace class, the Fredholm determinant can be exploited in order to compute the modified eigenenergies which differ from those of the harmonic oscillator due to the presence of the Gaussian perturbation. By taking advantage of Wang's results on scalar products of four eigenfunctions of the harmonic oscillator, it is possible to evaluate quite accurately the two lowest-lying eigenvalues as functions of the coupling constant $\lambda$., Comment: 15 pages, 4 figures

In this paper we adapt the mathematical machinery presented in \cite{P1} to get, by means of the Laplace-Beltrami operator, the discrete spectrum of the Hamiltonian of the Schr\"{o}dinger operator perturbed by an actractive 3D delta interaction in a Friedmann flat universe. In particular, as a consequence of the treatment in \cite{P1} suitable for a Minkowski spacetime, the discrete spectrum of the ground state and the first exited state in the above mentioned cosmic framework can be regained. Thus, the coupling constant $\lambda$ must be choosen as a function of the cosmic comooving time $t$ as ${\lambda}/a^{2}(t)$, with ${\lambda}$ be the one of the static Hamiltonian studied in \cite{P1}. In this way we can introduce a time dependent delta interaction which is relevant in a primordial universe, where $a(t)\rightarrow 0$ and becomes negligible at late times, with $a(t)>>1$. We investigate, with the so obtained model, quantum effects provided by point interactions in a strong gravitational regime near the big bang. In particular, as a physically interesting application, we present a method to depict, in a semi-classical approximation, a test particle in a (non commutative) quantum spacetime where, thanks to Planckian effects, the initial classical singularity disappears and as a consequnce a ground state with negative energy emerges. Conversely, in a scenario where the scale factor $a(t)$ follows the classical trajectory, this ground state is instable and thus physically cannot be carried out., Comment: Final version published on Mathematical Physics, Analysis and Geometry

We consider the one-dimensional Hamiltonian with a V-shaped potential H 0 = 1 2 − d 2 d x 2 + | x | , decorated with a point impurity of either δ -type, or local δ ′ -type or even nonlocal δ ′ -type, thus yielding three exactly solvable models. We analyse the behaviour of the change in the energy levels when an interaction of the type − λ δ ( x ) or − λ δ ( x − x 0 ) is switched on. In the first case, even energy levels, pertaining to antisymmetric bound states, remain invariant with respect to λ even though odd energy levels, pertaining to symmetric bound states, decrease as λ increases. In the second, all energy levels decrease when the factor λ increases. A similar study has been performed for the so-called nonlocal δ ′ interaction, requiring a coupling constant renormalisation, which implies the replacement of the form factor λ by a renormalised form factor β . In terms of β , odd energy levels are unchanged. However, we show the existence of level crossings: after a fixed value of β the energy of each even level, with the natural exception of the first one, becomes lower than the constant energy of the previous odd level. Finally, we consider an interaction of the type − λ δ ( x ) + μ δ ′ ( x ) , and analyse in detail the discrete spectrum of the resulting self-adjoint Hamiltonian.

We propose a new approach to the problem of finding the eigenvalues (energy levels) in the discrete spectrum of the one-dimensional Hamiltonian with an attractive Gaussian potential by using the well-known Birman-Schwinger technique. However, in place of the Birman-Schwinger integral operator we consider an isospectral operator in momentum space, taking advantage of the unique feature of this potential, that is to say its invariance under Fourier transform. Given that such integral operators are trace class, it is possible to determine the energy levels in the discrete spectrum of the Hamiltonian as functions of the coupling constant with great accuracy by solving a finite number of transcendental equations. We also address the important issue of the coupling constant thresholds of the Hamiltonian, that is to say the critical values of λ for which we have the emergence of an additional bound state out of the absolutely continuous spectrum.

In this note we consider a quantum mechanical particle moving inside an infinitesimally thin layer constrained by a parabolic well in the $x$-direction and, moreover, in the presence of an impurity modelled by an attractive Gaussian potential. We investigate the Birman-Schwinger operator associated to a model assuming the presence of a Gaussian impurity inside the layer and prove that such an integral operator is Hilbert-Schmidt, which allows the use of the modified Fredholm determinant in order to compute the bound states created by the impurity. Furthermore, we consider the case where the Gaussian potential degenerates to a $\delta$-potential in the $x$-direction and a Gaussian potential in the $y$-direction. We construct the corresponding self-adjoint Hamiltonian and prove that it is the limit in the norm resolvent sense of a sequence of corresponding Hamiltonians with suitably scaled Gaussian potentials. Satisfactory bounds on the ground state energies of all Hamiltonians involved are exhibited., Comment: 14 pages, 2 figures

In this paper, a model of 3D Helmholtz resonator with two close point-like windows is considered. The Dirichlet condition is assumed at the boundary. The model is based on the theory of self-adjoint extensions of symmetric operators in Pontryagin space. The model is explicitly solvable and allows one to obtain the equation for resonances (quasi-eigenvalues) in an explicit form. A proper choice of the model parameter leads to the coincidence of the model solution with the main term of the asymptotics (in the window width) of the realistic solution, corresponding to small windows. A regularization is suggested to obtain a realistic limiting result for two merging windows.

In this note we continue our analysis of the behaviour of self-adjoint Hamiltonians with a pair of identical point interactions symmetrically situated around the origin perturbing various types of "free Hamiltonians" as the distance between the two centres shrinks to zero. In particular, by making the coupling constant to be renormalised dependent also on the separation distance between the centres of the two point interactions, we prove that also in two dimensions it is possible to define the unique self-adjoint Hamiltonian that, differently from the one studied in detail in Albeverio's monograph on point interactions, behaves smoothly as the separation distance vanishes. In fact, we rigorously prove that such a twodimensional Hamiltonian converges in the norm resolvent sense to the one of the negative two-dimensional Laplacian perturbed by a single attractive point interaction situated at the origin having double strength, thus making this two-dimensional model similar to its one-dimensional analogue (not requiring the renormalisation procedure).

Producción Científica, In this brief presentation, some striking differences between level crossings of eigenvalues in one dimension (harmonic or conic oscillator with a central nonlocal δ’-interaction) or three dimensions (isotropic harmonic oscillator with a three-dimensional delta located at the origin) and those occurring in the two-dimensional analogue of these models will be highlighted., Ministerio de Economía, Industria y Competitividad (Project MTM2014-57129-C2-1-P), Junta de Castilla y León - FEDER (programa de apoyo a proyectos de investigación - Ref. VA057U16)

We study three solvable two-dimensional systems perturbed by a point interaction centered at the origin. The unperturbed systems are the isotropic harmonic oscillator, a square pyramidal potential and a combination thereof. We study the spectrum of the perturbed systems. We show that, while most eigenvalues are not affected by the point perturbation, a few of them are strongly perturbed. We show that for some values of one parameter, these perturbed eigenvalues may take lower values than the immediately lower eigenvalue, so that level crossings occur. These level crossings are studied in some detail.

In this paper the self-adjoint Hamiltonian of the one-dimensional harmonic oscillator perturbed by two identical attractive point interactions (delta distributions) situated symmetrically with respect to the equilibrium position of the oscillator is rigorously defined by means of its resolvent (Green's function). The equations determining the even and odd eigenvalues of the Hamiltonian are explicitly provided in order to shed light on the behaviour of such energy levels both with respect to the separation distance between the point interaction centres and to the coupling constant.

In this paper we exploit the technique used in \cite{A}-\cite{5b} to deal with delta interactions in a rigorous way in a curved spacetime represented by a cosmic string along the $z$ 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 $z$ 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 $R$ of the trapping cylinder. The number of quantum bound states is an increasing function of the radius $R$. 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 $z-$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., Comment: Version published on International Journal of Geometric Methods in Modern Physics

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.

In this presentation, we wish to provide an overview of the spectral features for the self-adjoint Hamiltonian of the three-dimensional isotropic harmonic oscillator perturbed by either a single attractive delta-interaction centered at the origin or by a pair of identical attractive delta-interactions symmetrically situated with respect to the origin. Given that such Hamiltonians represent the mathematical model for quantum dots with sharply localized impurities, we cannot help having the renowned article by Bruning, Geyler and Lobanov [1] as our key reference. We shall also compare the spectral features of the aforementioned three-dimensional models with those of the self-adjoint Hamiltonian of the harmonic oscillator perturbed by an attractive delta'-interaction in one dimension, fully investigated in [2, 3], given the existence in both models of the remarkable spectral phenomenon called "level crossing". The rigorous definition of the self-adjoint Hamiltonian for the singular double well model will be provided through the explicit formula for its resolvent (Green's function). Furthermore, by studying in detail the equation determining the ground state energy for the double well model, it will be shown that the concept of "positional disorder", introduced in [1] in the case of a quantum dot with a single delta-impurity, can also be extended to the model with the twin impurities in the sense that the greater the distance between the two impurities is, the less localized the ground state will be. Another noteworthy spectral phenomenon will also be determined; for each value of the distance between the two centers below a certain threshold value, there exists a range of values of the strength of the twin point interactions for which the first excited symmetric bound state is more tightly bound than the lowest antisymmetric bound state. Furthermore, it will be shown that, as the distance between the two impurities shrinks to zero, the 3D-Hamiltonian with the singular double well (requiring renormalization to be defined) does not converge to the one with a single delta-interaction centered at the origin having twice the strength, in contrast to its one-dimensional analog for which no renormalization is required. It is worth stressing that this phenomenon has also been recently observed in the case of another model requiring the renormalization of the coupling constant, namely the one-dimensional Salpeter Hamiltonian perturbed by two twin attractive delta-interactions symmetrically situated at the same distance from the origin.

The ground state energyE0(λ)ofHλ=-d2/dx2-λe-x2is computed for small values ofλby means of an approximation of an integral operator in momentum space. Such an approximation leads to a transcendental equation for whichϵ0(λ)=|E0(λ)|1/2is the root.

We consider Schrodinger operators of the form Hλ=−Δ+V+λW on L2(Rν) (ν=1, 2, or 3) with V periodic, W short range, and λ a real non-negative parameter. Then the continuous spectrum of Hλ has the typical band structure consisting of intervals, separated by gaps. In the gaps there may be discrete eigenvalues of Hλ that are functions of the parameter λ. Let (a,b) be a gap and E(λ)∈(a,b) an eigenvalue of Hλ. We study the asymptotic behavior of E(λ) as λ approaches a critical value λ0, called a coupling constant threshold, at which the eigenvalue either emerges from or is absorbed into the continuous spectrum. A typical question is the following: Assuming E(λ)↓a as λ↓λ0, is E(λ)−a∼c(λ−λ0)α for some α>0 and c≠0, or is there an expansion in some other quantity? As one expects from previous work in the case V=0, the answer strongly depends on ν.

We study the behaviour of eigenenergies of the operator H(λ(g).g) = H0 + λ(g)x2(1 + gx2) with H0 = −d2dx2 + x2 and λ(g). g > 0, as functions of the parameter η = g−12 near g = ∞ when λ(g) = g12, 1 = 1, 2, 3. It will be shown that, while in the first two cases the eigenvalues can be expressed as power series of η > 0, in the third case we have a divergent behaviour due to the presence of a term equal to 1η. Furthermore, apart from such a divergent term, in this case the eigenvalues approximate those of the harmonic oscillator with an attractive δ-type interaction generated by the potential 1(1 + x2) by means of a suitable scaling in η.

The energy of the ground state and that of the first excited level of the harmonic oscillator perturbed by the potential λ x 2 1 + gx 2 (with both λ and g positive) are determined in the range of small λ and large g by means of a perturbation argument for the Fredholm determinant of the Birman-Schwinger kernel of the harmonic oscillator perturbed by the potential − λ g (1 + gx 2 . It is seen that the eigenvalues are given by smooth functions of the two parameters λ and η = g − 1 2 in the neighbourhood of (0,0).

In this note we provide an alternative way of defining the self-adjoint Hamiltonian of the harmonic oscillator perturbed by an attractive delta'-interaction, of strength beta, centred at 0 (the bottom of the confining parabolic potential), that was rigorously defined in a previous paper by means of a 'coupling constant renormalisation'. Here we get the Hamiltonian as a norm resolvent limit of the harmonic oscillator Hamiltonian perturbed by a triple of attractive delta-interactions, thus extending the Cheon-Shigehara approximation to the case in which a confining harmonic potential is present.

We rigorously define the self-adjoint one-dimensional Salpeter Hamiltonian perturbed by an attractive of strength centred at the origin, by explicitly providing its resolvent. Our approach is based on a ‘coupling constant renormalization’, a technique used first heuristically in quantum field theory and implemented in the rigorous mathematical construction of the self-adjoint operator representing the negative Laplacian perturbed by the in two and three dimensions. We show that the spectrum of the self-adjoint operator consists of the absolutely continuous spectrum of the free Salpeter Hamiltonian and an eigenvalue given by a smooth function of the parameter The method is extended to the model with two twin attractive deltas symmetrically situated with respect to the origin in order to show that the discrete spectrum of the related self-adjoint Hamiltonian consists of two eigenvalues, namely the ground state energy and that of the excited antisymmetric state. We investigate in detail the dependence of these two eigenvalues on the two parameters of the model, that is to say both the aforementioned strength and the separation distance. With regard to the latter, a remarkable phenomenon is observed: differently from the well-behaved Schrodinger case, the 1D-Salpeter Hamiltonian with two identical Dirac distributions symmetrically situated with respect to the origin does not converge, as the separation distance shrinks to zero, to the one with a single centred at the origin having twice the strength. However, the expected behaviour in the limit (in the norm resolvent sense) can be achieved by making the coupling of the twin deltas suitably dependent on the separation distance itself.

We rigorously define the self-adjoint Hamiltonian of the harmonic oscillator perturbed by an attractive ??-interaction, of strength ?, centred at 0 (the bottom of the confining parabolic potential), by explicitly providing its resolvent. Our approach is based on a ?coupling constant renormalization?, related to a technique originated in quantum field theory and implemented in the rigorous mathematical construction of the self-adjoint operator representing the negative Laplacian perturbed by the ?-interaction in two and three dimensions. The way the ??-interaction enters in our Hamiltonian corresponds to the one originally discussed for the free Hamiltonian (instead of the harmonic oscillator one) by P S?ba. It should not be confused with the ??-potential perturbation of the harmonic oscillator discussed, e.g., in a recent paper by Gadella, Glasser and Nieto (also introduced by P S?ba as a perturbation of the one-dimensional free Laplacian and recently investigated in that context by Golovaty, Hryniv and Zolotaryuk). We investigate in detail the spectrum of our perturbed harmonic oscillator. The spectral structure differs from that of the one-dimensional harmonic oscillator perturbed by an attractive ?-interaction centred at the origin: the even eigenvalues are not modified at all by the ??-interaction. Moreover, all the odd eigenvalues, regarded as functions of ?, exhibit the rather remarkable phenomenon called ?level crossing? after first producing the double degeneracy of all the even eigenvalues for the value (B( ?, ?) being the beta function).

We prove that it is possible to define the self‐adjoint operator which gives sense to the merely formal expression −Δ−∑y∈Lλδ(⋅−y) (where L is a certain lattice of R3) as the limit when e→0+ in the resolvent sense of the net He =−Δ+∑y∈L λ(e)e−2V(⋅−y/e) λ(e) being a real‐valued, C∞ [0,1] function with λ(0)=1 and V∈L ∞ is such that supp V is contained in the Wigner–Seitz cell. By using the direct integral decomposition, we reduce the problem to the convergence of the reduced Hamiltonian He (θ)=−Δθ +λ(e)e−2V(⋅/e). In order to find the limit when e→0+ of [He(θ)−E]−1, we also study the properties of its integral kernel.

It is shown that there exist bound states of the operator H±λ=−(d2/dx2) +∑m∈Zδ(⋅−(2m+1)π)±λW, W being an L1(−∞,+∞) non‐negative function, in every sufficiently far gap of the spectrum of H0=−d2/dx2 +∑m∈Zδ(⋅−(2m+1)π). Such an operator represents the Schrodinger Hamiltonian of a Kronig–Penney‐type crystal with a localized impurity. The analyticity of the greatest (resp. lowest) eigenvalue of Hλ (resp. H−λ) occurring in a spectral gap as a function of the coupling constant λ when W is assumed to have an exponential decay is also proven.