Generalized uncertainty principle and self-adjoint operators

In this work we explore the self-adjointness of the GUP-modified momentum and Hamiltonian operators over different domains. In particular, we utilize the theorem by von-Neumann for symmetric operators in order to determine whether the momentum and Hamiltonian operators are self-adjoint or not, or they have self-adjoint extensions over the given domain. In addition, a simple example of the Hamiltonian operator describing a particle in a box is given. The solutions of the boundary conditions that describe the self-adjoint extensions of the specific Hamiltonian operator are obtained.     

New Criteria for Self-Adjointness and its Application to Dirac–Maxwell Hamiltonian

We present a new theorem concerning a sufficient condition for a symmetric operator acting in a complex Hilbert space to be essentially self-adjoint. By applying the theorem, we prove that the Dirac–Maxwell Hamiltonian, which describes a quantum system of a Dirac particle and a radiation field minimally interacting with each other, is essentially self-adjoint. Our theorem covers the case where the Dirac particle is in the Coulomb-type potential.     

Constructing quantum observables and self-adjoint extensions of symmetric operators. II. Differential operators

We discuss a problem of constructing self-adjoint ordinary differential operators starting from self-adjoint differential expressions based on the general theory of self-adjoint extensions of symmetric operators outlined in [1]. We describe one of the possible ways of constructing in terms of the closure of an initial symmetric operator associated with a given differential expression and deficient spaces. Particular attention is focused on the features peculiar to differential operators, among them on the notion of natural domain and the representation of asymmetry forms generated by adjoint operators in terms of boundary forms. Main assertions are illustrated in detail by simple examples of quantum-mechanical operators like the momentum or Hamiltonian. __________ Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 8, pp. 3–36, August, 2007.     

Quantum States of a Neutral Massive Fermion with an Anomalous Magnetic Moment in an External Electric Field

The quantum dynamics of a nonrelativistic neutral massive fermion with an anomalous magnetic moment (AMM) is examined in the external electric field of an infinitely long thin homogeneously charged thread in the plane with a normal directed along the thread. The Hamiltonian of the Dirac–Pauli equation for a neutral fermion with AMM is essentially singular in the considered external field and requires a supplementary extension of the definition in order for it to be treated as a self-adjoint quantum-mechanical operator. All one-parameter self-adjoint extensions of the Hamiltonian of the Dirac–Pauli equation in the considered external field are found in the nonrelativistic approximation. The corresponding Hilbert space of squareintegrable functions, including a singularity point of the Hamiltonian, is specified for each self-adjoint extension of the Hamiltonian. The wave functions of free and bound states, as well as discrete energy levels, are determined by the self-adjoint extension method and their correspondence with similar quantities obtained by the physical regularization procedure is discussed. It is shown that energy levels of bound states are simple poles of the scattering amplitude, which should be extended in definition by introducing the self-adjoint extension parameter into it. Expressions for the scattering amplitude and cross-section, depending on the orientation of the initial-state spin of fermion, are obtained.     

On certain non-relativistic quantized fields

We consider a Euclidean invariant interaction Hamiltonian which is a polynomial in smeared Fermion field operators (the smearing function providing an ultraviolet cut-off). By considering Guenin's perturbation series for the time-development of the theory, we show that time-displacements define a one-parameter group of automorphisms of the field algebra att=0, which acts continuously in the time-parameter. Results are obtained for any dimension of space and for both relativistic and nonrelativistic forms for the free Hamiltonian. In special cases the total Hamiltonian is a positive self-adjoint operator in Fock space, thus defining a concrete non-relativistic quantized field with non-trivial particle production.Supported in part by the United States Air Force under contract AFOSR 500-66.     

电磁理论中算符的一些性质

在本文中,把在不均匀各向异性介质中的麦克斯韦方程看作算符,它定义在一个有界区域,可以被理解为微波技术中的谐振腔。但在这腔中充填着铁氧体,等离子体或其他各向异性介质,这些介质在应用中日益重要。文中证明了在某些μ、ε和边界条件下,算符成为对称。而对称性和自伴性在本征函数展开中带来很多方便;此外我们推导了本征振动的正变性和互易定理。如果不满足对称性,引入伴谐振腔的概念,所谓伴谐振腔在几何形状上和原来的腔相同,但ε、μ和边界条件不一样。它和自伴谐振腔在正交性和互易定理上有某些相似之处。     

Gauge invariance and the dirac equation

The gauge invariance of the Dirac equation is reviewed and gauge-invariant operators are defined. The Hamiltonian is shown to be gauge dependent, and an energy operator is defined which is gauge invariant. Gauge-invariant operators corresponding to observables are shown to satisfy generalized Ehrenfest theorems. The time rate of change of the expectation value of the energy operator is equal to the expectation value of the power operator. The virial theorem is proved for a relativistic electron in a time-varying electromagnetic field. The conventional approach to probability amplitudes, using the eigenstates of the unperturbed Hamiltonian, is shown in general to be gauge dependent. A gaugeinvariant procedure for probability amplitudes is given, in which eigenstates of the energy operator are used. The two methods are compared by applying them to an electron in a zero electromagnetic field in an arbitrary gauge. Presented at the Dirac Symposium, Loyola University, New Orleans, May 1981.     

高强度磁场中氢原子性态的研究

采用变分法和微扰法相结合的方法 ,把高强度磁场中氢原子的哈密顿H分为三部分 :球对称哈密顿 ;z分量角动量算符相应部分和非球对称势微扰 ,并用一种特别规定的分解法将哈密顿H中含磁场平方项的势能分解为球对称与非球对称两部分 ,且使非球对称部分引起的一级修正能量值为零 ,并采用一种简便的变分法直接求出B2 对能级的二级修正值 .这一方法不仅计算简单 ,而且提高了计算结果的精度 .计算了在均匀高强度静磁场下氢原子的 11个低能态能级和平均半径 ,讨论了高强度磁场对能级和半径的影响. In this paper we separate the Hamiltonian into three parts: a spherical symmetry Hamiltonian; a z-component of the angular momentum operator, and a non-spherical symmetric potential as the perturbation operator, and provide a propose method by separating the potential containing squared magnetic field B 2 into two parts: spherical symmetric and non-spherical symmetric ones so that the first-order energy correction due to the non-spherical symmetric potential is zero...     

Continuous transition from the extensive to the non-extensive statistics in an agent-based herding model

An infinite bent chain of nanospheres connected by wires is considered. We assume that there are δ-like potentials at the contact points. A solvable mathematical model based on the theory of self-adjoint extensions of symmetric operators is constructed. The spectral equation for the model operator is derived in an explicit form. It is shown that the Hamiltonian has non-empty point spectrum. The positions of the eigenvalues for different values of the system parameters (the length of the connecting wires, the intensities of δ-interactions and the bent angle) are found.     

Intertwining operators for solving differential equations,with applications to symmetric spaces

The use of intertwining operators to solve both ordinary and partial differential equations is developed. Classes of intertwining operators are constructed which transform between Laplacians which are self-adjoint with respect to different non-trivial measures. In the two-dimensional case, the intertwining operator transforms a non-separable partial differential operator to a separable one. As an application, the heat kernels on the rank 1 and rank 2 symmetric spaces are constructed.     

An alternative factorization of the quantum harmonic oscillator and two-parameter family of self-adjoint operators

We introduce an alternative factorization of the Hamiltonian of the quantum harmonic oscillator which leads to a two-parameter self-adjoint operator from which the standard harmonic oscillator, the one-parameter oscillators introduced by Mielnik, and the Hermite operator are obtained in certain limits of the parameters. In addition, a single Bernoulli-type parameter factorization, which is different from the one introduced by M.A. Reyes, H.C. Rosu, and M.R. Gutiérrez [Phys. Lett. A 375 (2011) 2145], is briefly discussed in the final part of this work.     

The Hamiltonian Operator Associated with Some Quantum Stochastic Evolutions

We consider the Hamiltonian operator associated to the quantum stochastic differential equation introduced by Hudson and Parthasarathy to describe a quantum mechanical evolution in the presence of a “quantum noise”. We characterize such a Hamiltonian in the case of arbitrary multiplicity and bounded coefficients: we find an essentially self-adjoint restriction of the operator and, in particular, we provide an explicit construction of a dense set of vectors belonging to its domain. An erratum to this article is available at .     

Precanonical quantization of Yang-Mills fields and the functional Schrödinger representation

Precanonical quantization of pure Yang-Mills fields, which is based on the covariant De Donder-Weyl (DW) Hamiltonian formulation, and its connection with the functional Schrödinger representation in the temporal gauge are discussed. The mass gap problem is related to the finite-dimensional spectral problem for a generalized Clifford-valued magnetic Schrödinger operator which represents the DW Hamiltonian operator.     

The general relativistic hydrogen atom

The general relativistic Dirac equation is formulated in an arbitrary curved space-time using differential forms. These equations are applied to spherically symmetric systems with arbitrary charge and mass. For the case of a black hole (with event horizon) it is shown that the Dirac Hamiltonian is self-adjoint, has essential spectrum the whole real line and no bound states. Although rigorous results are obtained only for a spherically symmetric system, it is argued that, in the presence of any event horizon there will be no bound states. The case of a naked singularity is investigated with the results that the Dirac Hamiltonian is not self-adjoint. The self-adjoint extensions preserving angular momentum are studied and their spectrum is found to consist of an essential spectrum corresponding to that of a free electron plus eigenvalues in the gap (–mc ², +mc ²). It is shown that, for certain boundary conditions, neutrino bound states exist.Supported in part by the National Science Foundation     

General Nth-order differential spectral problem: General structure of the integrable equations,nonuniqueness of recursion operator and gauge invariance

The general scalar Gelfand-Dikij-Zakharov-Shabat spectral problem of arbitrary order is considered within the framework of the AKNS method. The general form of the integrable equations is found. Uncertainties which appear in the construction of recursion operator and transformation properties of the integrable equations under the gauge transformations are considered. The manifestly gauge-invariant formulation of the integrable equations is given. It is shown that the intergrable equations under consideration are Hamiltonian ones with respect to the infinite family of Hamiltonian structures.     

Constructing quantum observables and self-adjoint extensions of symmetric operators. III. Self-adjoint boundary conditions

This paper completes the review of the theory of self-adjoint extensions of symmetric operators for physicists as a basis for constructing quantum-mechanical observables. It contains a comparative presentation of the well-known methods and a newly proposed method for constructing ordinary self-adjoint differential operators associated with self-adjoint differential expressions in terms of self-adjoint boundary conditions. The new method has the advantage that it does not require explicitly evaluating deficient subspaces and deficiency indices (these latter are determined in passing) and that boundary conditions are of explicit character irrespective of the singularity of a differential expression. General assertions and constructions are illustrated by examples of well-known quantum-mechanical operators like momentum and Hamiltonian.     

New Coherent States for the BDS-Hamiltonian

In the paper we construct a new set of coherent states for a deformed Hamiltonian of the harmonic oscillator, previously introduced by Beckers, Debergh, and Szafraniec, which we have called the BDS-Hamiltonian. This Hamiltonian depends on the new creation operator a ⁺, i.e. the usual creation operator displaced with the real quantity . In order to construct the coherent states, we use a new measure in the Hilbert space of the Hamiltonian eigenstates, in fact we change the inner product. This ansatz assures that the set of eigenstates be orthonormalized and complete. In the new inner product space the BDS-Hamiltonian is self-adjoint. Using these coherent states, we construct the corresponding density operator and we find the P-distribution function of the unnormalized density operator of the BDS-Hamiltonian. Also, we calculate some thermal averages related to the BDS-oscillators system which obey the quantum canonical distribution conditions.     

On the Relationship of the Spectra of the Self-adjoint Operator and its Liouville Operator

In this paper, we study the self-adjoint extensions of the Liouville operator and correct the relationship of absolutely and singular continuous spectra between a Hamiltonian and its corresponding Liouvillian. We also give the relationship of the essential and discrete spectra between the two operators.