One-parameter Family Of Self-adjoint Extensions Research Articles

Spectral theory of the thermal Hamiltonian: 1D case

In 1964 J. M. Luttinger introduced a model for the quantum thermal transport. In this paper we study the spectral theory of the Hamiltonian operator associated with Luttinger’s model, with a special focus at the one-dimensional case. It is shown that the (so called) thermal Hamiltonian has a one-parameter family of self-adjoint extensions and the spectrum, the time-propagator group and the Green function are explicitly computed. Moreover, the scattering by convolution-type potentials is analyzed. Finally, also the associated classical problem is completely solved, thus providing a comparison between classical and quantum behavior. This article aims to be a first contribution in the construction of a complete theory for the thermal Hamiltonian.

Quantum gravitational collapse as a Dirac particle on the half line

We show that the quantum dynamics of a thin spherical shell in general relativity is equivalent to the Coulomb-Dirac equation on the half line. The Hamiltonian has a one-parameter family of self-adjoint extensions with a discrete energy spectrum $|E| < m$, and a continuum of scattering states for $|E|>m$, where $m$ is the rest mass of the shell and $E$ is the Arnowitt-Deser-Misner mass. For sufficiently large $m$, the ground state energy level is negative. This suggests that classical positivity of energy does not survive quantization. The scattering states provide a realization of singularity avoidance. We speculate on the consequences of these results for black hole radiation.

Normal-mode analysis for scalar fields in BTZ black-hole background

We analyze the possibility of inequivalent boundary conditions for a scalar field propagating in the BTZ black-hole space-time. We find that for certain ranges of the black-hole parameters, the Klein–Gordon operator admits a one-parameter family of self-adjoint extensions. For this range, the BTZ space-time is not quantum mechanically complete. We suggest a physically motivated method for determining the spectra of the Klein–Gordon operator.

Inequivalent quantizations of the rational Calogero model with a Coulomb type interaction

We consider the inequivalent quantizations of a N-body rational Calogero model with a Coulomb type interaction. It is shown that for a certain range of the coupling constants, this system admits a one-parameter family of self-adjoint extensions. We analyze both the bound and scattering state sectors and find novel solutions of this model. We also find the ladder operators for this system, with which the previously found solutions can be constructed.

On Painlevé VI transcendents related to the Dirac operator on the hyperbolic disk

Dirac Hamiltonian on the Poincaré disk in the presence of an Aharonov–Bohm flux and a uniform magnetic field admits a one-parameter family of self-adjoint extensions. We determine the spectrum and calculate the resolvent for each element of this family. Explicit expressions for Green’s functions are then used to find Fredholm determinant representations for the tau function of the Dirac operator with two branch points on the Poincaré disk. Isomonodromic deformation theory for the Dirac equation relates this tau function to a one-parameter class of solutions of the Painlevé VI equation with γ=0. We analyze long-distance behavior of the tau function, as well as the asymptotics of the corresponding Painlevé VI transcendents as s→1. Considering the limit of flat space, we also obtain a class of solutions of the Painlevé V equation with β=0.

DIRAC SPINORS IN SOLENOIDAL FIELD AND SELF-ADJOINT EXTENSIONS OF ITS HAMILTONIAN

We discuss Dirac equation and its solution in the presence of solenoid (infinitely long) field in (3+1) dimensions. Starting with a very restricted domain for the Hamiltonian, we show that a one-parameter family of self-adjoint extensions are necessary to make sure the correct evolution of the Dirac spinors. Within the extended domain bound state (BS) and scattering state (SS) solutions are obtained. We argue that the existence of bound state in such system is basically due to the breaking of classical scaling symmetry by the quantization procedure. A remarkable effect of the scaling anomaly is that it puts an open bound on both sides of the spectrum, i.e. E ∈ (-M, M) for ν2[0, 1)! We also study the issue of relationship between scattering state and bound state in the region ν2 ∈ [0, 1) and recovered the BS solution and eigenvalue from the SS solution.

SELF-ADJOINTNESS OF GENERALIZED MIC–KEPLER SYSTEM

We have studied the self-adjointness of generalized MIC–Kepler Hamiltonian, obtained from the formally self-adjoint generalized MIC–Kepler Hamiltonian. We have shown that for [Formula: see text], the system admits a one-parameter family of self-adjoint extensions and for [Formula: see text] but [Formula: see text], it also has a one-parameter family of self-adjoint extensions.

The quantum oscillator on complex projective space (Lobachewski space) in a constant magnetic field and the issue of generic boundary conditions

We perform a one-parameter family of self-adjoint extensions characterized by the parameter ω0. This allows us to get generic boundary conditions for the quantum oscillator on N-dimensional complex projective space and on its non-compact version, i.e., Lobachewski space in the presence of a constant magnetic field. As a result, we get a family of energy spectra for the oscillator. In our formulation the already known result of this oscillator also belongs to the family. We have also obtained an energy spectrum which preserves all the symmetries (full-hidden symmetry and rotational symmetry) of the oscillator. The method of self-adjoint extensions has also been discussed for a conic oscillator in the presence of the constant magnetic field.

Quantization and conformal properties of a generalized Calogero model

We analyze a generalization of the quantum Calogero model with the underlying conformal symmetry, paying special attention to the two-body model deformation. Owing to the underlying SU(1,1) symmetry, we find that the analytic solutions of this model can be described within the scope of the Bargmann representation analysis, and we investigate its dynamical structure by constructing the corresponding Fock space realization. The analysis from the standpoint of supersymmetric quantum mechanics (SUSYQM), when applied to this problem, reveals that the model is also shape invariant. For a certain range of the system parameters, the two-body generalization of the Calogero model is shown to admit a one-parameter family of self-adjoint extensions, leading to inequivalent quantizations of the system.

Dependence of s‐waves on continuous dimension: The quantum oscillator and free systems

AbstractWavefunctions with rotational symmetry (i.e., zero angular momentum) in D dimensions, are called s‐waves. In quantum quadratic systems (free particle, harmonic and repulsive oscillators), their radial parts obey Schrödinger equations with a fictitious centrifugal (for integer D ≥ 4) or centripetal (for D = 2) potential. These Hamiltonians close into the three‐dimensional Lorentz algebra so(2,1), whose exceptional interval corresponds to the critical range of continuous dimensions 0 &lt; D &lt; 4, where they exhibit a one‐parameter family of self‐adjoint extensions in ℒ︁2(ℜ︁+). We study the characterization of these extensions in the harmonic oscillator through their spectra which – except for the Friedrichs extension – are not equally spaced, and we build their time evolution Green function. The oscillator is then contracted to the free particle in continuous‐D dimensions, where the extension structure is mantained in the limit of continuous spectra. Finally, we compute the free time evolution of the expectation values of the Hamiltonian, dilatation generator, and square radius between three distinct sets of ‘heat’‐diffused localized eigenstates. This provides a simple group‐theoretic description of the purported contraction/expansion of Gaussian‐ring s‐waves in D &gt; 0 dimensions.

On position and momentum operators in theq-oscillator

The position and momentum operators of the q-oscillator (with the main relation aa+ − qa+a = 1) are symmetric but not self-adjoint if q > 1. They have one-parameter family of self-adjoint extensions. These extensions are given explicitly. Their spectra and eigenfunctions are derived. Spectra of different extensions do not intersect. The results show that the creation and annihilation operators a+ and a of the q-oscillator at q > 1 cannot determine a physical system without further more precise definition. In order to determine a physical system we have to choose appropriate self-adjoint extensions of the position and momentum operators.

Further evidence for the conformal structure of a Schwarzschild black hole in an algebraic approach

We study the excitations of a massive Schwarzschild black hole of mass M resulting from the capture of infalling matter described by a massless scalar field. The near-horizon dynamics of this system is governed by a Hamiltonian which is related to the Virasoro algebra and admits a one-parameter family of self-adjoint extensions described by a parameter z∈R. The density of states of the black hole can be expressed equivalently in terms of z or M, leading to a consistent relation between these two parameters. The corresponding black hole entropy is obtained as S=S(0)−32 logS(0)+C, where S(0) is the Bekenstein–Hawking entropy, C is a constant with other subleading corrections exponentially suppressed. The appearance of this precise form for the black hole entropy within our formalism, which is expected on general grounds in any conformal field theoretic description, provides strong evidence for the near-horizon conformal structure in this system.

Bound states in one-dimensional quantum N-body systems with inverse square interaction

We investigate the existence of bound states in N-body Calogero model without the confining term. The effective Hamiltonian of this system admits a one-parameter family of self-adjoint extensions and the bound states occur when most general boundary conditions are considered. Such states are found to exist only when N=3,4 and for certain values of the system parameters.

Hamiltonian self-adjoint extensions for (2+1)-dimensional Dirac particles

We study the stationary problem of a charged Dirac particle in (2+1) dimensions in the presence of a uniform magnetic field B and a singular magnetic tube of flux Φ = 2πκ/e. The rotational invariance of this configuration implies that the subspaces of definite angular momentum l + 1/2 are invariant under the action of the Hamiltonian H. We show that for κ-l⩾1 or κ-l⩽0 the restriction of H to these subspaces, Hl, is essentially self-adjoint, while for 0<κ-l<1 Hl admits a one-parameter family of self-adjoint extensions (SAEs). In the latter case, the functions in the domain of Hl are singular (but square integrable) at the origin, their behaviour being dictated by the value of the parameter γ that identifies the SAE. We also determine the spectrum of the Hamiltonian as a function of κ and γ, as well as its closure.

Boundary conditions for quantum mechanics on cones and fields around cosmic strings

We study the options for boundary conditions at the conical singularity for quantum mechanics on a two-dimensional cone with deficit angle ≦ 2π and for classical and quantum scalar fields propagating with a translationally invariant dynamics in the 1+3 dimensional spacetime around an idealized straight infinitely long, infinitesimally thin cosmic string. The key to our analysis is the observation that minus-the-Laplacian on a cone possesses a one-parameter family of selfadjoint extensions. These may be labeled by a parameterR with the dimensions of length—taking values in [0, ∞). ForR=0, the extension is positive. WhenR≠0 there is a bound state. Each of our problems has a range of possible dynamical evolutions corresponding to a range of allowedR-values. They correspond to either finite, forR=0, or logarithmically divergent, forR≠0, boundary conditions at zero radius. Non-zeroR-values are a satisfactory replacement for the (mathematically ill-defined) notion of δ-function potentials at the cone's apex. We discuss the relevance of the various idealized dynamics to quantum mechanics on a cone with a rounded-off centre and field theory around a “true” string of finite thickness. Provided one is interested in effects at sufficiently large length scales, the “true” dynamics will depend on the details of the interaction of the wave function with the cone's centre (/field with the string etc.) only through a single parameterR (its “scattering length”) and will be well-approximated by the dynamics for the corresponding idealized problem with the sameR-value. This turns out to be zero if the interaction with the centre is purely gravitational and minimally coupled, but non-zero values can be important to model nongravitational (or non-minimally coupled) interactions. Especially, we point out the relevance of non-zeroR-values to electromagnetic waves around superconducting strings. We also briefly speculate on the relevance of theR-parameter in the application of quantum mechanics on cones to 1+2 dimensional quantum gravity with massive scalars.