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Journal of Physics A Mathematical and Theoretical

Oscillator representations for self-adjoint Calogero Hamiltonians

Dmitri M Gitman, Igor Viktorovich Tyutin, Boris Leonidovich Voronov

Published Sep 29, 2011• 6 citations
Publisher

Abstract

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

DOI
10.1088/1751-8113/44/42/425204

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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
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.
We present a mathematically rigorous quantum-mechanical treatment of a one-dimensional non-relativistic motion of a particle in the potential field , For g 2 >0 and g 1 <0, the potential is known as the Kratzer potential V K ( x ) and is usually used to describe molecular energy and structure, interactions between different molecules and interactions between non-bonded atoms. We construct all self-adjoint Schrödinger operators with the potential V ( x ) and represent rigorous solutions of the corresponding spectral problems. Solving the first part of the problem, we use a method of specifying self-adjoint extensions by (asymptotic) self-adjoint boundary conditions. Solving spectral problems, we follow Krein's method of guiding functionals. This work is a continuation of our previous works devoted to the Coulomb, Calogero and Aharonov–Bohm potentials.
The matrix U' contains the full dynamics of a bilinear oscillator, because any observable of the system is an analytical function of the operators that determine the matrix representation. One cannot solve the time evolution equation in the general case, but there are many exactly solvable problems that simulate and enable one to understand the dynamics of more elaborate and realistic quantum-mechanical models. In the case of the solvable model just discussed it is easy to derive analytical expressions for operators in the Heisenberg picture as well as for invariants or constants of the motion. The mathematical treatment of the driven oscillator just discussed is trivial because the Hamiltonian operator belongs to a solvable Lie algebra. This model is a convenient starting point to illustrate the use of the Lie algebraic method, but it is also an oversimplification of most oscillatory phenomena. Nonlinear Hamiltonians are useful in many physical applications such as the simulation of frictional forces and dissipation effects.
Abstract The object of this book is to present the new quantum mechanics in a unified representation which, so far as it is possible and useful, is mathematically rigorous [A]presentation of the mathematical tools necessary for the purposes of this theory will be given, i.e., a theory of Hilbert spaces and the so-called Hermitian [or self-adjoint] operators. To this end, an accurate introduction to unbounded operators is also necessary; that is, an extension of the theory beyond its classical limits (developed by Hilbert and E. Hellinger, F. Riesz, E. Schmidt, 0. Toeplitz). The following may be said regarding the method employed in this type of treatment: as a rule, calculations should be performed with the operators themselves (which represent physical quantities) and not with the matrices, which, after the introduction of a (special and arbitrary) coordinate system in Hilbert space, result from them. This “coordinate free” (i.e., invariant) method, with its strongly geometric language, possesses noticeable formal advantages.