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arXiv:2312.15126

The Dirac Delta as a Singular Potential for the 2D Schrodinger Equation

Michael Maroun

Published Dec 23, 2023• 0 citations
arXivHTMLPDF

Abstract

In the framework of distributionally generalized quantum theory, the object $Hψ$ is defined as a distribution. The mathematical significance is a mild generalization for the theory of para- and pseudo-differential operators (as well as a generalization of the weak eigenvalue problem), where the $ψ$-do symbol (which is not a proper linear operator in this generalized case) can have its coefficient functions take on singular distributional values. Here, a distribution is said to be singular if it is not L$^p(\mathbb{R}^d)$ for any $p\geq 1$. Physically, the significance is a mathematically rigorous method, which does not rely upon renormalization or regularization of any kind, while producing bound state energy results in agreement with the literature. In addition, another benefit is that the method does not rely upon self-adjoint extensions of the Laplace operator. This is important when the theory is applied to non-Schrodinger systems, as is the case for the Dirac equation and a necessary property of any finite rigorous version of quantum field theory. The distributional interpretation resolves the need to evaluate a wave function at a point where it fails to be defined. For $d=2$, this occurs as $K_o(a|x|)δ(x)$, where $K_o$ is the zeroth order MacDonald function. Finally, there is also the identification of a missing anomalous length scale, owing to the scale invariance of the formal symbol(ic) Hamiltonian, as well as the common identity for the logarithmic function, with $a,\,b\in\mathbb{R}^+$, $\log(ab)=\log(a)+\log(b)$, which loses unitlessness in its arguments. Consequently, the energy or point spectrum is generalized as a family (set indexed by the continuum) of would-be spectral values, called the C-spectrum.

DOI
10.48550/arxiv.2312.15126

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Unlike the situation for the 1d Dirac delta derivative Schrodinger pseudo potential (SPP) and the 2d Dirac delta SPP, where the indeterminacy originates from a lack of scale in the first and both a lack of scale as well as the wave function not being well defined at the support of the generalized function SPP; the obstruction in 3d Euclidean space for the Schrodinger equation with the Dirac delta as a SPP only comes from the wave function (the $L^2$ bound sate solution) being singular at the compact point support of the Dirac delta function (measure). The problem is solved here in a completely mathematically rigorous manner with no recourse to renormalization nor regularization. The method involves a distributionally generalized version of the Schrodinger theory as developed by the author, which regards the formal symbol "$Hψ$" as an element of the space of distributions, the topological dual vector space to the space of smooth functions with compact support. Two main facts come to light. The first is the bound state energy of such a system can be calculated in a well-posed context, the value of which agrees with both the mathematical and theoretical physics literature. The second is that there is then a rigorous distributional version of the Hellmann-Feynman theorem.

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Singular interactions in quantum mechanics: solvable models

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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. 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Schrodinger operators with very singular potentials fail to produce reasonable spectra, because the usual boundary conditions are insufficient to make them self-adjoint. If one regularizes a singular potential at a length , the bound state energies diverge as → 0. A meaningful spectrum can be restored by picking a self-adjoint extension of the operator, by renormalizing the theory, or by allowing nonunitary time evolution. We show that renormalizations of the 1/r potential fall into two classes: those that are asymptotically equivalent to self-adjoint extensions, and those that reduce to treatments of delta functions. We also apply the apparatus of self-adjoint extensions and renormalization to clarify aspects of anomalous symmetry breaking, supersymmetric quantum mechanics, and one-dimensional quantum mechanics. Lines composed upon reading a thesis abstract by S. Gopalakrishnan Think a moment, dear reader, of that long-legged fly, The sprung meniscus underfoot, And what asymptotes invisible to well-trained eyes Still break the surface at its root. How a singular potential picks tight-bounded energy Apart till it no longer recognizes Its own mirrored face, its extensions, or symmetry; Unless one cautiously renormalizes. You have tried in these pensive pages to show the stuff Such theory’s made of, its quiddity When logic is applied to the diamond rough, And quantum notes ring purely. May Apollo’s clarity pierce through your reasoning And self-adjointed truth and fact Elevate your thought to that sense so pleasing When truth recovers nature’s pact.

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It is explicitly shown the weak points (being not rigorous) in the reasoning of Dirac, of Landau, of Von Neumann, of Weyl, of Fadeev and of others, deriving many fundamental relations, which later became classical and are being commonly used in Quantum Theory up to now. It is because of their derivations were made when functional analysis was not yet sufficiently developed. Those weak points are of two following reasons:1. Unbounded property of the physical operators is not yet paid by attention.2. The “Dirac delta function” defined by Dirac is not always distinguished clearly with the Dirac distribution defined by Schwartz and with the generalized functions defined by Gel'fand.Particularly, the formulation of the closure relation by using equality of the functionals is not rigorous. Consequently, it is necessary to improve Quantum Theory.

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In the framework of distributionally generalized quantum theory, the object HψHψHψ is defined as a distribution. The mathematical significance is a mild generalization for the theory of para- and pseudo-differential operators (as well as a generalization of the weak eigenvalue problem), where the ψψψ-do symbol (which is not a proper linear operator in this generalized case) can have its coefficient functions take on singular distributional values. Here, a distribution is said to be singular if it is not Lp(Rd)^p(\mathbb{R}^d)p(Rd) for any p≥1p\geq 1p≥1. Physically, the significance is a mathematically rigorous method, which does not rely upon renormalization or regularization of any kind, while producing bound state energy results in agreement with the literature. In addition, another benefit is that the method does not rely upon self-adjoint extensions of the Laplace operator. This is important when the theory is applied to non-Schrodinger systems, as is the case for the Dirac equation and a necessary property of any finite rigorous version of quantum field theory. The distributional interpretation resolves the need to evaluate a wave function at a point where it fails to be defined. For d=2d=2d=2, this occurs as Ko(a∣x∣)δ(x)K_o(a|x|)δ(x)Ko​(a∣x∣)δ(x), where KoK_oKo​ is the zeroth order MacDonald function. Finally, there is also the identification of a missing anomalous length scale, owing to the scale invariance of the formal symbol(ic) Hamiltonian, as well as the common identity for the logarithmic function, with a, b∈R+a,\,b\in\mathbb{R}^+a,b∈R+, log⁡(ab)=log⁡(a)+log⁡(b)\log(ab)=\log(a)+\log(b)log(ab)=log(a)+log(b), which loses unitlessness in its arguments. Consequently, the energy or point spectrum is generalized as a family (set indexed by the continuum) of would-be spectral values, called the C-spectrum.
Unlike the situation for the 1d Dirac delta derivative Schrodinger pseudo potential (SPP) and the 2d Dirac delta SPP, where the indeterminacy originates from a lack of scale in the first and both a lack of scale as well as the wave function not being well defined at the support of the generalized function SPP; the obstruction in 3d Euclidean space for the Schrodinger equation with the Dirac delta as a SPP only comes from the wave function (the L2L^2L2 bound sate solution) being singular at the compact point support of the Dirac delta function (measure). The problem is solved here in a completely mathematically rigorous manner with no recourse to renormalization nor regularization. The method involves a distributionally generalized version of the Schrodinger theory as developed by the author, which regards the formal symbol "HψHψHψ" as an element of the space of distributions, the topological dual vector space to the space of smooth functions with compact support. Two main facts come to light. The first is the bound state energy of such a system can be calculated in a well-posed context, the value of which agrees with both the mathematical and theoretical physics literature. The second is that there is then a rigorous distributional version of the Hellmann-Feynman theorem.
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'.
Much of the literature on point interactions in quantum mechanics has focused on the differential form of Schrödinger’s equation. This paper, in contrast, investigates the integral form of Schrödinger’s equation. While both forms are known to be equivalent for smooth potentials, this is not true for distributional potentials. Here, we assume that the potential is given by a distribution defined on the space of discontinuous test functions. First, by using Schrödinger’s integral equation, we confirm a seminal result by Kurasov, which was originally obtained in the context of Schrödinger’s differential equation. This hints at a possible deeper connection between both forms of the equation. We also sketch a generalisation of Kurasov’s [J. Math. Anal. Appl. 201(1), 297–323 (1996)] result to hypersurfaces. Second, we derive a new closed-form solution to Schrödinger’s integral equation with a delta prime potential. This potential has attracted considerable attention, including some controversy. Interestingly, the derived propagator satisfies boundary conditions that were previously derived using Schrödinger’s differential equation. Third, we derive boundary conditions for “super-singular” potentials given by higher-order derivatives of the delta potential. These boundary conditions cannot be incorporated into the normal framework of self-adjoint extensions. We show that the boundary conditions depend on the energy of the solution and that probability is conserved. This paper thereby confirms several seminal results and derives some new ones. In sum, it shows that Schrödinger’s integral equation is a viable tool for studying singular interactions in quantum mechanics.
We revisit the discussion about the boundary condition at the origin in the Schrödinger radial equation for central potentials. We give a transparent and convincing reason for demanding the radial part R(r) of the wave function to be finite at r = 0, showing that if R(0) diverges the complete wave function ψ does not satisfy the full Schrödinger equation. If R(r) is singular, we show that the corresponding ψ follows an equation similar to Schrödinger's, but with an additional term involving the Dirac delta function or its derivatives at the origin. Although, in general, understanding some of our arguments requires certain knowledge of the theory of distributions, the important case of a behavior R ∝ 1/r near r = 0, which gives rise to a normalizable ψ, is especially simple: The origin of the Dirac delta term is clearly demonstrated by using a slight modification of the usual spherical coordinates. The argument can be easily followed by undergraduate physics students.
Schrodinger operators with very singular potentials fail to produce reasonable spectra, because the usual boundary conditions are insufficient to make them self-adjoint. If one regularizes a singular potential at a length , the bound state energies diverge as → 0. A meaningful spectrum can be restored by picking a self-adjoint extension of the operator, by renormalizing the theory, or by allowing nonunitary time evolution. We show that renormalizations of the 1/r potential fall into two classes: those that are asymptotically equivalent to self-adjoint extensions, and those that reduce to treatments of delta functions. We also apply the apparatus of self-adjoint extensions and renormalization to clarify aspects of anomalous symmetry breaking, supersymmetric quantum mechanics, and one-dimensional quantum mechanics. Lines composed upon reading a thesis abstract by S. Gopalakrishnan Think a moment, dear reader, of that long-legged fly, The sprung meniscus underfoot, And what asymptotes invisible to well-trained eyes Still break the surface at its root. How a singular potential picks tight-bounded energy Apart till it no longer recognizes Its own mirrored face, its extensions, or symmetry; Unless one cautiously renormalizes. You have tried in these pensive pages to show the stuff Such theory’s made of, its quiddity When logic is applied to the diamond rough, And quantum notes ring purely. May Apollo’s clarity pierce through your reasoning And self-adjointed truth and fact Elevate your thought to that sense so pleasing When truth recovers nature’s pact.
It is explicitly shown the weak points (being not rigorous) in the reasoning of Dirac, of Landau, of Von Neumann, of Weyl, of Fadeev and of others, deriving many fundamental relations, which later became classical and are being commonly used in Quantum Theory up to now. It is because of their derivations were made when functional analysis was not yet sufficiently developed. Those weak points are of two following reasons:1. Unbounded property of the physical operators is not yet paid by attention.2. The “Dirac delta function” defined by Dirac is not always distinguished clearly with the Dirac distribution defined by Schwartz and with the generalized functions defined by Gel'fand.Particularly, the formulation of the closure relation by using equality of the functionals is not rigorous. Consequently, it is necessary to improve Quantum Theory.