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Singular factorizations, self-adjoint extensions, and applications to quantum many-body physics

Edwin Langmann A, Ari Arievich Laptev, Cornelius Paufler

Published Jan 1, 2005• 4 citations
OpenAlex

Abstract

We study self-adjoint operators defined by factorizing second order differential operators in first order ones. We discuss examples where such factorizations introduce singular interactions into simple quantum mechanical models like the harmonic oscillator or the free particle on the circle. The generalization of these examples to the many-body case yields quantum models of distinguishable and interacting particles in one dimensions which can be solved explicitly and by simple means. Our considerations lead us to a simple method to construct exactly solvable quantum many-body systems of Calogero-Sutherland type. 1. Introduction. In this paper we define and study peculiar self-adjoint extensions of simple differential operators like the one defining the quantum mechanical harmonic oscillator. We also find the explicit solution of some of the resulting quantum mechanical models and discuss their physical interpretations. These examples provide amusing illustrations that the

Related papers

Singular factorizations, self-adjoint extensions and applications to quantum many-body physics

Edwin Langmann A, Ari Arievich Laptev, Cornelius Paufler

We study self-adjoint operators defined by factorizing second-order differential operators in first-order ones. We discuss examples where such factorizations introduce singular interactions into simple quantum-mechanical models such as the harmonic oscillator or the free particle on the circle. The generalization of these examples to the many-body case yields quantum models of distinguishable and interacting particles in one dimensions which can be solved explicitly and by simple means. Our considerations lead us to a simple method to construct exactly solvable quantum many-body systems of Calogero–Sutherland type.

arXiv:math-ph/0510046•Journal of Physics A Mathematical and General•Jan 18, 2006•3 citations

Exactly solved quantum many-body systems in one dimension

Martin A. Hallnäs

This thesis is devoted to the study of various examples of exactly solved quantum many-body systems in one-dimension. It is divided into two parts: the first provides background and complementary results to the second, which consists of three scientific papers. The first paper concerns a particu- lar extension, corresponding to the root system CN, of the delta-interaction model. We prove by construction that its exact solution, even in the gen- eral case of distinguishable particles, can be obtained by the coordinate Bethe ansatz. We also elaborate on the physical interpretation of this model. It is well known that the delta-interaction is included in a four parameter family of local interactions. In the second paper we interpret these parameters as cou- pling constants of certain momentum dependent interactions and determine all cases leading to a many-body system of distinguishable particles which can be solved by the coordinate Bethe ansatz. In the third paper we consider the so-called rational Calogero-Sutherland model, describing an arbitrary number of particles on the real line, confined by a harmonic oscillator potential and interacting via a two-body interaction proportional to the inverse square of the inter-particle distance. We construct a novel solution algorithm for this model which enables us to obtain explicit formulas for its eigenfunctions. We also show that our algorithm applies, with minor changes, to all extensions of the rational Calogero-Sutherland model which correspond to a classical root system.

•Jan 1, 2005•0 citations

Singular interactions in quantum mechanics: solvable models

Gianfausto Dell’Antonio, Pavel Exner, Vladimir A. Geyler

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'.

Journal of Physics A Mathematical and General•May 18, 2005•6 citations

Quantum many-body systems exactly solved by special functions

Martin A. Hallnäs

This thesis concerns two types of quantum many-body systems in one dimension exactly solved by special functions: firstly, systems with interactions localised at points and solved by the (coordinate) Bethe ansatz; secondly, systems of Calogero-Sutherland type, as well as certain recently introduced deformations thereof, with eigenfunctions given by natural many-variable generalisations of classical (orthogonal) polynomials. The thesis is divided into two parts. The first provides background and a few complementary results, while the second presents the main results of this thesis in five appended scientific papers. In the first paper we consider two complementary quantum many-body systems with local interactions related to the root systems CN, one with delta-interactions, and the other with certain momentum dependent interactions commonly known as delta-prime interactions. We prove, by construction, that the former is exactly solvable by the Bethe ansatz in the general case of distinguishable particles, and that the latter is similarly solvable only in the case of bosons or fermions. We also establish a simple strong/weak coupling duality between the two models and elaborate on their physical interpretations. In the second paper we consider a well-known four-parameter family of local interactions in one dimension. In particular, we determine all such interactions leading to a quantum many-body system of distinguishable particles exactly solvable by the Bethe ansatz. We find that there are two families of such systems: the first is described by a one-parameter deformation of the delta-interaction model, while the second features a particular one-parameter combination of the delta and the delta-prime interactions. In papers 3-5 we construct and study particular series representations for the eigenfunctions of a family of Calogero-Sutherland models naturally associated with the classical (orthogonal) polynomials. In our construction, the eigenfunctions are given by linear combinations of certain symmetric polynomials generalising the so-called Schur polynomials, with explicit and rather simple coefficients. In paper 5 we also generalise certain of these results to the so-called deformed Calogero-Sutherland operators.

•Jan 1, 2007•1 citation

Self-adjoint differential operators associated with self-adjoint differential expressions

Boris Leonidovich Voronov, Igor Viktorovich Tyutin, Dmitri M Gitman

Considerable attention has been recently focused on quantum-mechanical systems with boundaries and/or singular potentials for which the construction of physical observables as self-adjoint (s.a.) operators is a nontrivial problem. We present a comparative review of various methods of specifying ordinary s.a. differential operators generated by formally s.a. differential expressions based on the general theory of s.a. extensions of symmetric operators. The exposition is untraditional and is based on the concept of asymmetry forms generated by adjoint operators. The main attention is given to a specification of s.a. extensions by s.a. boundary conditions. All the methods are illustrated by examples of quantum-mechanical observables like momentum and Hamiltonian. In addition to the conventional methods, we propose a possible alternative way of specifying s.a. differential operators by explicit s.a. boundary conditions that generally have an asymptotic form for singular boundaries. A comparative advantage of the method is that it allows avoiding an evaluation of deficient subspaces and deficiency indices. The effectiveness of the method is illustrated by a number of examples of quantum-mechanical observables.

•Mar 21, 2006•20 citations

Oscillator representations for self-adjoint Calogero Hamiltonians

Dmitri M Gitman, Igor Viktorovich Tyutin, Boris Leonidovich Voronov

In the article arXiv:0903.5277 [quant-ph], we have presented a mathematically rigorous quantum-mechanical treatment of a one-dimensional motion of a particle in the Calogero potential V (x) = αx −2. In such a way, we have described all possible s.a. operators (s.a. Hamiltonians) associated with the formal differential expression ˇH = −d 2 x + αx−2 for the Calogero Hamiltonian. Here, we discuss a new aspect of the problem, the so-called oscillator representation for the Calogero Hamiltonians. As it is know, operators of the form ˆ N = â + â and  = ââ+ are called operators of oscillator type. Oscillator type operators obey several useful properties in case if the elementary operator â and â + are densely defined. It turns out that some s.a. Calogero Hamiltonians are oscillator type operators. We describe such Hamiltonians and find the corresponding mutually adjoint elementary operators. 1

Journal of Physics A Mathematical and Theoretical•Sep 29, 2011•6 citations
We study self-adjoint operators defined by factorizing second order differential operators in first order ones. We discuss examples where such factorizations introduce singular interactions into simple quantum mechanical models like the harmonic oscillator or the free particle on the circle. The generalization of these examples to the many-body case yields quantum models of distinguishable and interacting particles in one dimensions which can be solved explicitly and by simple means. Our considerations lead us to a simple method to construct exactly solvable quantum many-body systems of Calogero-Sutherland type. 1. Introduction. In this paper we define and study peculiar self-adjoint extensions of simple differential operators like the one defining the quantum mechanical harmonic oscillator. We also find the explicit solution of some of the resulting quantum mechanical models and discuss their physical interpretations. These examples provide amusing illustrations that the
We study self-adjoint operators defined by factorizing second-order differential operators in first-order ones. We discuss examples where such factorizations introduce singular interactions into simple quantum-mechanical models such as the harmonic oscillator or the free particle on the circle. The generalization of these examples to the many-body case yields quantum models of distinguishable and interacting particles in one dimensions which can be solved explicitly and by simple means. Our considerations lead us to a simple method to construct exactly solvable quantum many-body systems of Calogero–Sutherland type.
This thesis is devoted to the study of various examples of exactly solved quantum many-body systems in one-dimension. It is divided into two parts: the first provides background and complementary results to the second, which consists of three scientific papers. The first paper concerns a particu- lar extension, corresponding to the root system CN, of the delta-interaction model. We prove by construction that its exact solution, even in the gen- eral case of distinguishable particles, can be obtained by the coordinate Bethe ansatz. We also elaborate on the physical interpretation of this model. It is well known that the delta-interaction is included in a four parameter family of local interactions. In the second paper we interpret these parameters as cou- pling constants of certain momentum dependent interactions and determine all cases leading to a many-body system of distinguishable particles which can be solved by the coordinate Bethe ansatz. In the third paper we consider the so-called rational Calogero-Sutherland model, describing an arbitrary number of particles on the real line, confined by a harmonic oscillator potential and interacting via a two-body interaction proportional to the inverse square of the inter-particle distance. We construct a novel solution algorithm for this model which enables us to obtain explicit formulas for its eigenfunctions. We also show that our algorithm applies, with minor changes, to all extensions of the rational Calogero-Sutherland model which correspond to a classical root system.
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'.
This thesis concerns two types of quantum many-body systems in one dimension exactly solved by special functions: firstly, systems with interactions localised at points and solved by the (coordinate) Bethe ansatz; secondly, systems of Calogero-Sutherland type, as well as certain recently introduced deformations thereof, with eigenfunctions given by natural many-variable generalisations of classical (orthogonal) polynomials. The thesis is divided into two parts. The first provides background and a few complementary results, while the second presents the main results of this thesis in five appended scientific papers. In the first paper we consider two complementary quantum many-body systems with local interactions related to the root systems CN, one with delta-interactions, and the other with certain momentum dependent interactions commonly known as delta-prime interactions. We prove, by construction, that the former is exactly solvable by the Bethe ansatz in the general case of distinguishable particles, and that the latter is similarly solvable only in the case of bosons or fermions. We also establish a simple strong/weak coupling duality between the two models and elaborate on their physical interpretations. In the second paper we consider a well-known four-parameter family of local interactions in one dimension. In particular, we determine all such interactions leading to a quantum many-body system of distinguishable particles exactly solvable by the Bethe ansatz. We find that there are two families of such systems: the first is described by a one-parameter deformation of the delta-interaction model, while the second features a particular one-parameter combination of the delta and the delta-prime interactions. In papers 3-5 we construct and study particular series representations for the eigenfunctions of a family of Calogero-Sutherland models naturally associated with the classical (orthogonal) polynomials. In our construction, the eigenfunctions are given by linear combinations of certain symmetric polynomials generalising the so-called Schur polynomials, with explicit and rather simple coefficients. In paper 5 we also generalise certain of these results to the so-called deformed Calogero-Sutherland operators.
Considerable attention has been recently focused on quantum-mechanical systems with boundaries and/or singular potentials for which the construction of physical observables as self-adjoint (s.a.) operators is a nontrivial problem. We present a comparative review of various methods of specifying ordinary s.a. differential operators generated by formally s.a. differential expressions based on the general theory of s.a. extensions of symmetric operators. The exposition is untraditional and is based on the concept of asymmetry forms generated by adjoint operators. The main attention is given to a specification of s.a. extensions by s.a. boundary conditions. All the methods are illustrated by examples of quantum-mechanical observables like momentum and Hamiltonian. In addition to the conventional methods, we propose a possible alternative way of specifying s.a. differential operators by explicit s.a. boundary conditions that generally have an asymptotic form for singular boundaries. A comparative advantage of the method is that it allows avoiding an evaluation of deficient subspaces and deficiency indices. The effectiveness of the method is illustrated by a number of examples of quantum-mechanical observables.
In the article arXiv:0903.5277 [quant-ph], we have presented a mathematically rigorous quantum-mechanical treatment of a one-dimensional motion of a particle in the Calogero potential V (x) = αx −2. In such a way, we have described all possible s.a. operators (s.a. Hamiltonians) associated with the formal differential expression ˇH = −d 2 x + αx−2 for the Calogero Hamiltonian. Here, we discuss a new aspect of the problem, the so-called oscillator representation for the Calogero Hamiltonians. As it is know, operators of the form ˆ N = â + â and  = ââ+ are called operators of oscillator type. Oscillator type operators obey several useful properties in case if the elementary operator â and â + are densely defined. It turns out that some s.a. Calogero Hamiltonians are oscillator type operators. We describe such Hamiltonians and find the corresponding mutually adjoint elementary operators. 1