Renormalization of the Inverse Square Potential

Abstract

The quantum-mechanical D-dimensional inverse square potential is analyzed using field-theoretic renormalization techniques. A solution is presented for both the bound-state and scattering sectors of the theory using cutoff and dimensional regularization. In the renormalized version of the theory, there is a strong-coupling regime where quantum-mechanical breaking of scale symmetry takes place through dimensional transmutation, with the creation of a single bound state and of an energydependent s-wave scattering matrix element.PACS numbers: 03.65. Ge, 03.65.Nk, 11.10.Gh, The quantum-mechanical inverse square potential is a singular problem that has generated controversy for decades. For instance, the solution proposed in Ref.[1] failed to give a Hamiltonian bounded from below, and this led to a number of alternative regularization techniques [2-4] based on appropriate parametrizations of the potentialincluding the replacement [5] of self-adjointness by an interpretation of the "fall of the particle to the center" [6]. However, it is generally recognized that the singular nature of this problem lies in that its Hamiltonian, being symmetric but not self-adjoint, admits self-adjoint extensions [7]. Recently, a renormalized solution was presented using field-theoretic techniques [8], but it was just limited to the one-dimensional case and cutoff renormalization.In this Letter (i) we generalize the results of Ref.[8] to D dimensions (including the all-important D = 3 case) using cutoff regularization in configuration space; (ii) present a complete picture of the renormalized theory; and (iii) confirm the same conclusions using dimensional regularization [9]. This problem is crucial for the analysis and interpretation of the point dipole interaction of molecular physics [10,11], and may be relevant in polymer physics [12]. In addition (i) it displays remarkable similarities with the two-dimensional δ-function potential [13][14][15]; (ii) it provides another example of dimensional transmutation [16] in a system with a finite number of degrees of freedom; and (iii) it illustrates the relevance of field-theoretic concepts in quantum mechanics [13][14][15]17].This problem is ideally suited for implementation in configuration space [18], where the radial Schrödinger equation for a particle subject to the r −2 potential in D dimensions [19] reads (withh = 1 and 2m = 1)which is explicitly scale-invariant because λ is dimensionless [20]. In Eq. (1), l is the angular momentum quantum number and λ > 0 corresponds to an attractive potential; with the transformation R l (r) = r −(D−1)/2 u l (r), Eq. (1) If λ were allowed to vary, one would see that the nature of the solutions changes around the critical value λ ( * ) l , for each angular momentum state. For λ < λ ( * ) l (including repulsive potentials), the order s l of the Bessel functions is real, so that the solution regular at the origin is proportional to the Bessel function of the first kind J s l √ E r . However, the same solution fails to satisfy the required behavior at ...

Analytic solution of the wave equation for an electron in the field of a molecule with an electric dipole moment

“…, where x is real and μ, a, b are positive except for a possible pair of complex conjugates with positive real parts [13]. The general recursion relation for these polynomials reads as follows 11) where the weight function is ( 1) …”

“…See, for example, [9] and references therein. Regularization procedures [9,11] and self-adjoint extensions of the Hamiltonian [12] were introduced to handle these irregularities (anomalies) in the spectrum. , where x is real and μ, a, b are positive except for a possible pair of complex conjugates with positive real parts [13].…”

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“…This implies that the system will collapse if η 2 < 0. But re-normalization technique [20,21,22,23] has a remedy for this problem to give a finite ground state, thus making the problem physically realizable. In re-normalization technique, the divergent Hamiltonian is regularized with an ultraviolet cut off ρ = Θ, for example consider an infinite barrier regularized potential…”

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“…Furthermore, Eq. (21) implies the formal infrared collapse of the bound-state spectrum: if the ground-state energy is fixed, the conformal tower of excited states is pushed towards its accumulation point E = 0; however, an effective reinterpretation should still lead to the familiar sequence (20).…”

“…Even though f 0 and a still determine the precise value of E 0 , in this framework, the scale E 0 is either an observable to be adjusted experimentally or a quantity to be determined from additional ultraviolet-specific information. In short, the conformal tower of bound states (20) has the following attributes:…”