Root operators and “evolution” equations 1 Introduction

Root operators and “evolution” equations G. Dattoli and A. Torre ENEA UTAPRAD–MAT Laboratorio di Modellistica Matematica, via E. Fermi 45, 00044 Frascati (Rome), Italy

Root-operator factorization ` a la Dirac provides an effective tool to deal with equations, which are not of evolution type, or are ruled by fractional differential operators, thus eventually yielding evolution-like equations although for a multicomponent vector. We will review the method along with its extension to root operators of degree higher than two. Also, we will show the results obtained by the Dirac-method as well as results from other methods, specifically in connection with evolution-like equations ruled by square-root operators, that we will address to as relativistic evolution equations . Keywords: Dirac equation; evolution equation; root operator

1

Introduction

Evolution equation is a rather generic term, which in Physics signifies any mathematical device useful to describe the evolution of a dynamical system. Here, we will adopt this term to refer to a partial differential equation of the type [1] ∂ V(x, t) = K · V(x, t), ∂t

(1)

for the dependent variable V, designated to characterize the physical state of the dynamical system of concern. Equation (1) specifies in fact the rate of change of V, regarded as a function of the independent “space” and “time” variables (x, t), with respect to the relevant evolution variable “t”. The left hand side is just the first order derivative of V with respect to t, whilst the right hand side, which may be both linear and non linear, only involves V, its derivatives with respect to x, and possibly the independent varables x, t. The meaning of the “state function” V(x, t) depends on the specific problem at hand as also the definite form of the right hand side. Anyway, since the equation only involves the first order time derivative, one expects its solution to be uniquely specified by a single initial condition V(x, t0 ) = V0 (x), so that, known V at a given time t0 , it can be known at any subsequent time t > t0 after integrating the equation.

1

Many physical processes are well modelled by equations of the type (1). The heat equation (HE) [2], the time-dependent Schr¨odinger equation (SE) [3] and the paraxial wave equation (PWE) [4] are substantive examples of evolution equation. The theory concerned with such equations is well established; we will review some basic aspects of it below, thus exemplifying the essential features of the theory concerned in general with evolution-type equations. Many physical processes are equally well modelled by equations, which cannot be traced back to the scheme of Eq. (1). We may face indeed equations involving higher-order derivatives of V with respect to the “time” variable t, or equations containing fractional order derivatives with respect to the “space” variables x. We may think, for instance, of the relativistic heat equation (RHE), as proposed in the telegraph form in [5],   α ∂2 ∂ + u(r, t) = α∇2 u(r, t), (2) C 2 ∂t2 ∂t α and C being the thermal diffusivity and the spead of heat, respectively. We may also think of the Klein-Gordon equation (KGE) [6]   mc 2  1 ∂2 ∇2 − 2 2 − ψ(r, t) = 0, (3) c ∂t ~ and the homogeneous scalar wave equation (WE) [7]   1 ∂2 2 ∇ − 2 2 ϕ(r, t) = 0. c ∂t All of them contain the second-order derivative with respect to time1 . Likewise, we may think of the relativistic Schr¨odinger equation (RSE) [6] ∂ψ(r, t) p 2 4 i~ = m c − ~2 c2 ∇2 ψ(r, t), ∂t

(4)

(5)

which involves indeed the square-root of an operator containing the laplacian ∇2 . The analysis here presented is only an aspect of an investigation, which resorting to different methods, is aimed at establishing if, at which extent and in which form some properties of the evolution equations could be recovered to equations, which are not of evolution type or demand to deal with fractional differential operators, like the aforementioned RHE, KGE,WE and RSE. As shown in [8, 9], the Dirac-like factorization approach conveys a valuable method to tackle with both kinds of difficulties. In fact, the Dirac equation [6, 10]   ∂ mc γj j − I 4×4 ψ = 0, j = 0, 1, 2, 3 (6) ∂x ~ 1

Interestingly, the SE and the PWE can be seen as the non-relativistic and paraxial limit, respectively, of the KGE and the WE.

2

with γ j , j = 0, 1, 2, 3, being the (4 × 4) Dirac matrices and I 4×4 the 4 × 4 unit matrix, offers in a sense the “evolution-like” alternative to the KGE. As is well known, it was originally formulated by Dirac when seeking a relativistically covariant evolution equation for the state function of a quantum particle in the Schr¨odinger-like form, i.e. of the type i~

∂ψ b = Hψ, ∂t

b being a linear Hermitian operator [10]. However, it can also be with the Hamiltonian H understood as following from a “factorization” of the Klein-Gordon equation   mc 2  1 ∂2 2 ∇ − 2 2− I 4×4 ψ = c ∂t ~    mc mc j ∂ j ∂ γ − I 4×4 γ + I 4×4 ψ = 0, ∂xj ~ ∂xj ~ so that one can eventually deal with a first-order derivative with respect to the evolution variable even though in a system of four coupled linear differential equations for the 4component state vector ψ. The Dirac-like factorization approach can as well be effectively applied to deal with evolution equations ruled by fractional differential operators, like that entering the RSE. In fact, allowing to express the power of operators as the sum of operators, it allows for the “disentanglement” of root operators into the sum of operators, and hence, under appropriate conditions, one can overcome the problem of working with fractional differential operators [8]. In addition, the factorization approach to root operators may open new perspectives within the theory of fractional calculus, suggesting, for instance, alternative formulations to already well-established definitions and/or treatments [9]. The plan of the paper is as follows. In Sect. 2 we will briefly recall the main features of the theory concerned with evolution equations, referring to the HE, the SE and the PWE as basic examples. In Sect. 3, we will review the Dirac-like factorization method in connection with root operators. Square, cube and quartic root operators will be treated in some detail, specific properties of the relevant Lie algebra of inherent matrices will be deduced. Then, we will illustrate some applications of the method, specifically in connection with the square and cube root operators. Finally, in Sect. 4 we will focus the discussion on evolution-like equations ruled by square-root operators, that we will address to as relativistic evolution equations. We will show that the solution to such a kind of equations can be expressed in the form of an integral transform in full analogy with the HE, SE and PWE; some properties of it will be deduced and the inherent results in some specific cases will be compared with those obtained from the Dirac-factorization method. The concluding notes of Sect. 5 will close the paper.

3

2

Examples of evolution equation: heat, Schr¨ odinger and paraxial wave equations

The (1+1)D heat equation (HE) [2] ∂ ∂2 u(x, t) = α 2 u(x, t), −∞ < x < ∞, t > 0, ∂t ∂x

(7)

rules the time evolution of the temperature function u(x, t) through an in principle infinitely long homogeneous bar, characterized by the thermal diffusivity α. The relevance of such an equation is not limited to the context of heat propagation, and not only to the context of second order linear evolution equations. For instance, the non linear Burgers’ equation [11] can be converted into the HE by the Hopf-Cole transformation [12,13]. Also, the Black-Scholes equation [14, 15] can be trasformed into the HE; originally proposed in the early 1970’s as a model for investment portfolios, the Black-Scholes equation is now at the heart of the modern financial industry. At a basic level, we write down the (1+1)D SE [3] i ∂ 1 ∂2 ψ(x, t) = − ψ(x, t), ~ ∂t 2m ∂x2

(8)

which, as evolution equation for the wavefunction ϕ(x, t), describes the 1D free motion of a (spinless) particle of mass m. Correspondingly, the 2D PWE [4, 16] ik0

∂ 1 ∂2 ψ(x, z) = − ψ(x, z), ∂z 2n ∂x2

(9)

is the equation of motion for the complex slowly-varying amplitude ψ(x, z) of a monochromatic scalar light field E(x, z, t) = ei(k0 nz−ωt) ψ(x, z), propagating in a homogeneous medium of refractive index n. Here, z denotes the coordinate along the main direction of the field propagation, k0 = 2π/λ0 is the field wave number in the vacuum, and ω the relevant angular frequency, ω = nk0 c. The analogies between the two equations are evident, and may be synthesized by the correspondences 1 t ↔ z, ~ ↔ , m ↔ n. k0 Likewise, the analogy with the HE becomes recognizable once allowing the evolution variable in (8) and (9), or equivalently in (7), to take on a purely imaginary value. The theory concerned with the heat, Schr¨odinger and paraxial wave equations is well established. However, in spite of the afore-evidenced analogies, the three equations have been analysed by different approaches, yielding parallel formulations, that only recently have been merged into each other within the framework of a unifying formalism. 4

2.1

General treatment: Hamiltonian and evolution operators

Evidently, the HE, the SE and the PWE can be recast in the form ∂ϕ(x, ζ) b = Hϕ(x, ζ), ∂ζ

(10)

b (not necessarily Hermitian), which can be introducing a sort of “Hamiltonian” operator H seen as the “generator” of the system evolution. Specifically, the “Hamiltonian” pertaining to the aforementioned equations is simply the free-evolution operator, 2 b= ∂ . H ∂x2

(11)

The solution to (10) can be written in the form b ϕ(x, ζ) = U(ζ)ϕ 0 (x),

(12)

b (not necessarily unitary), introducing the “evolution operator” U b b U(ζ) = eζ H ,

(13)

which acts on the initial data ϕ(x, 0) = ϕ0 (x) to yield the wavefunction at subsequent times. Specifically, for the equations we are considering the evolution operator is ∂2

b U(ζ) = eζ ∂x2 ,

(14)

and the “time-like” variable ζ is intended to be ζ = αt, ζ = the equation under consideration.

2.2

i~t 2m

and ζ =

iz 2k0 n

according to

Evolution operator as Poisson and Fresnel transforms

Under the minimal assumption that initial condition ϕ0 (x) tends to zero sufficiently rapidly as x → ±∞, the evolution operator specializes into the Poisson transform for the HE and into the Fresnel transform for the SE and PWE [19]. Thus, the solutions to the (1+1)D HE follow from the Poisson transform of the initial condition u0 (x) = u(x, 0) according to [2]: Z ∞ (x−y)2 1 u(x, t) = √ e− 4αt u0 (y)dy. (15) 4παt −∞ . The kernel of the transform is the fundamental solution of the HE, S(x, t) = √

x2 1 e− 4αt , 4παt

5

t > 0,

(16)

i.e. the solution corresponding to a “point-like source” at t = 0: u0 (x) = δ(x), signifying a highly concentrated unit heat source; in fact, limS(x, t) = δ(x). t→0

Analogously, the solutions to the (1+1)D SE and the 2D PWE result from the Fresnel transform of the initial condition ψ0 (x) = ψ(x, 0) as Z ∞ (x−y)2 1 ψ(x, ς) = √ ei 4ς ψ0 (y)dy, (17) 4πiς −∞ ~t or ς = 2kz0 n according to whether we are interested in (8) or (9). with ς being ς = 2m The above can be seen as a sort of “imaginary” version of (15). Again, the kernel is the fundamental solution of the equations of concern, x2 1 W (x, ς) = √ ei 4ς = S(x, iς), 4πiς

(18)

being, as before, lim W (x, ς) = δ(x). ς→0

It is worth noting that Eqs. (15) and (17) implicitly assume the equivalence of the repb as an exponential operator (Eq. (14)), involving resentations of the evolution operator U ∂2 the free-Hamiltonian operator ∂x2 , and as a Poisson or Fresnel transform. Evidently, the 2n

2n

ψ0 u0 former requires the implied series of derivatives ot the initial data, ∂∂x2n or ∂∂x2n , to exist and converge to a finite value, whereas the latter is meaningful only if u0 (x) or ψ0 (x) are integrable and the pertinent integrals converge. The discussion of the legitimacy of such an equivalence is out of the topic of the paper [19].

2.3

Polynomial solutions and symmetry transformations

In view of the considerations we will develop in Sec. 4 in connection with the treatment of relativistic evolution equations, we will focus here on two aspects of the theory of the HE, the SE and the PWE: the polynomial solutions and the symmetry transformations. The polynomial solutions vn (x, t) of the HE, referred to as heat polynomials [2], correspond to initial data given by monomials: vn (x, 0) = xn , and hence Z ∞ (x−y)2 1 vn (x, t) = √ e− 4t y n dy. (19) 4πt −∞ For simplicity’s sake, we have set α[m2 /s] = 1. Also, an appropriate multiplying constant of unit value is implied in the initial monomials in order to provide the vn s with the correct dimensions in conformity to those of u(x, t). The heat polynomials realize the power series expansion of the simple exponential solution of the (1+1)D HE [2] E(x, t, χ) = e

χx+χ2 t

=

∞ X χn n=0

6

n!

vn (x, t),

(20)

with χ arbitrary parameter. Accordingly, they explicitly write as [n/2]

vn (x, t) = n!

X j=0

tj x ), xn−2j = (−t)n/2 Hn ( √ j!(n − 2j)! −4t

(21)

with Hn denoting Hermite polynomials [17]. For future use, let us write down the explicit expressions of the first vn s: v0 (x, t) = 1, v1 (x, t) = x, v2 (x, t) = x2 + 2t, v3 (x, t) = x3 + 6xt, v4 (x, t) = x4 + 12x2 t + 12t2 , v5 (x, t) = x5 + 20x3 t + 60xt2 . A further property of the vn s of interest here concerns the raising and lowering operators (or, multiplication and derivative operators), conveyed by the relations [x + 2t∂x ] vn (x, t) = vn+1 (x, t),

∂x vn (x, t) = nvn−1 (x, t),

yielding the differential equation obeyed by the vn s,  2  2t∂x + x∂x − n vn (x, t) = 0,

vn (x, 0) = xn .

(22)

(23)

Recently, polynomial solutions of the 2D PWE have been introduced in full analogy with the HE [19, 20]. In fact, in the light of the afore-mentioned analogy, the optical analogues vn (x, ς) of the heat polynomials are found to be explicitly given by n

vn (x, ς) = (−iς) 2 Hn ( √

x ). −4iς

(24)

Also, they result from the Fresnel transform of monomials, i.e. Z ∞ (x−y)2 1 vn (x, ς) = √ ei 4ς y n dy, 4πiς −∞ and realize the power series expansion of the function 2

E(x, ς, λ) = eiλx−iλ ς ,

(25)

which simply conveys the plane-wave solution of both (9) and (8), corresponding to the inputE(x, 0, λ) = eiλx . The parameter λ is then related to the transverse wavenumber (i.e. spatial frequency) of the field or to the momentum of the particle; the frequency chirping in optics corresponds indeed to the “energy chirping” in quantum mechanics. The investigation of the symmetry transformations, pertaining to the equations we are dealing with, is a very fascinating and fruitful job. Symmetry transformations establish specific rules to pass from one solution to another solution of the same equation. 7

In particular, in relation with the HE, SE and PWE, we may quote the special conformal transformation that allows for an appropriate Gaussian modulation of a solution to get another solution. In fact, it can easily be verified that if u(x, t) is a temperature function, also 2 1 εx εt − x v(x, t) = √ e 4(t+ε) u( , ) (26) t+ε t+ε t+ε is a temperature function for any arbitrary parameter ε. It can be traced back to the x2

Gaussian modulation of the initial data, which then turn from u0 (x) to w0 (x) = e− 4ε u0 (x). Similarly, any solution of the SE or PWE can be transformed into a solution as well according to 2 1 ες εx − x ϕ(x, ς) = √ e 4(ε+iς) ψ( , ). (27) ε + iς ε + iς ε + iς Interestingly, in connection with optical propagation, the modulation of the input function x2

by e− 4ε may signify lensing or Gaussian aperturing according to whether ε is real or purely imaginary [16]. Note that the modulating function in both (26) and (27) is the fundamental solution (28) or (18). Relation (27) confirms the Gaussian packets/beams as solutions to (8) [3] and (9) [4], following indeed from (27) when setting ψ0 (x) = 1, which amounts to ψ(x, ς) = 1 as well. In this case, the symmetry transformations simply manifest the symmetry of the solutions of Eq. (10) with respect to the shift of the evolution variable ζ [16]. In fact, (26) and (27) b follow directly from the additivity of the evolution operator U(ζ), being b 1 )U(ζ b 2 ) = U(ζ b 1 + ζ2 ). U(ζ As said, the fundamental solution is the result of the evolution of the initial data δ(x); namely, referring to the HE, we can write b S(x, t) = U(t)δ(x),

t > 0.

(28)

x2

Therefore, the Gaussian e− 4ε can be regarded as resulting from δ(x) at the t = ε (apart 1 from the factor √4πε ), and hence 2

x b e− 4ε = U(ε)δ(x).

This implies that 2

− x4ε b b U(ε)δ(x) b v(x, t) = U(t)e 1 ∝ U(t) b + ε)δ(x) = S(x, t + ε), = U(t

just conveying (26) for u(x, t) = 1. In practice, it is as we had translated the origin of the ”time” from t0 = 0 to t0 = −ε. 8

Of course, the same can be seen for (27) with the simple replacement t → iς [16]. Central to the theory of the HE is the well-known Appell transformation [2, 18]. It establishes the possibility to pass from one solution of the HE to another according to the rule x 1 w(x, t) = S(x, t)u( , − ). (29) t t Apart from its role in connection with, for instance, the heat polynomials, the interest in the Appell transformation arises from its significative property, proven in [7], according to which it is essentially the only transformation mapping solutions of the HE into another in the sense that any simmetry transformation of the HE can be seen as composed by Appell transformations and scaling and shift of both coordinates x, t. Appell transformation for the PWE (as well as for the SE) can be considered, of course, as discussed in [19, 20], accordingly yielding the optical analogue of (29) as 2 x 1 1 x 1 iξ φ(x, ς) = W (x, ς)ψ(± , − ) = √ e 4ζ ψ(± , − ). ς ς ς ς 4πiς

(30)

Evidently, in conformity to the formal analogy of the equations of concern, the aforementioned result regarding the role of the Appell transformation in relation to the symmetry transformations of the HE [7] can be reformulated in connection with the optical Appell transformation. In fact, in [21] the latter is seen to be the only symmetry transformation of the PWE in the sense that any symmetry transformation of the PWE can be obtained by composing Appell transformations with scaling and translations of both coordinates.2 We conclude recalling that, as relations (26) and (27) relate to the symmetry operator 1

x2

(for Eq. (10)) e− 2ε K+ = e− 4ε , the Appell transformation (29) and (30) relate to the sym∂2 π 2 π b 1 b metry operator represented by the Fourier trasform Fb = e−i 2 (K+ +K− − 2 ) = e−i 4 (x − ∂x2 −1) [16]. As said, the analysis here presented frames within an investigation aimed at establishing if, at which extent and in which form some properties of the evolution equations - in particular, those exemplified in connection with the just discussed HE, SE and PWE - might be extended to equations, which are not of evolution type or demand to deal with fractional differential operators, like the RHE, KGE,WE and RSE, mentioned in the Introduction. As we will show in the forthcoming section, operator factorization ` a la Dirac can help in this.

3

b

Dirac-like factorization to disentangle root operator functions

As is well known, the algebra of the operators is definitely different from that of c-numbers. Thus, for instance, the identity A2 + B 2 = (A + B)2 = (A + B)(A + B), 9

can not hold if A and B are numbers (real or complex). In contrast, it can hold if A and B are anticommuting operators or matrices, for which so o n b B b =A bB b+B bA b = 0. A, This suggests to write down the square root function form p A2 + B 2 = Aα + Bβ ,

√

A2 + B 2 in the “disentangled”

(31)

where α and β turn out to be “mathematical objects” such that α2 = β 2 = 1, α β + βα = 0,

(32)

in order for the desired equality (31) be satisfied. In principle, A and B can be either numbers or commuting operators; of course, the latter case is of major interest, and it will be considered here. Clearly, α and β cannot simply be numbers; indeed, as a direct consequence of (32), they must be traceless matrices with eigenvalues equal to ±1, and hence of order 2n × 2n, n = 1, 2, .., and determinant equal to (−1)n . Thus, on one side one looses the scalar nature of the original function, which in the stated identity (31) is to be understood as a multiple of the 2n × 2n unit matrix I 2n×2n . In fact, Eq. (31) conveys a matrix identity in a proper 2n-dimensional vector space, whose meaning and ultimate dimensions (following those of α and β) are dictated by the context inherent in the problem at hand. On the other side, one gains a root-free matrix form expression, that could facilitate the solution of the problem although it must be reintenpreted in the light of the gained degree (or, degrees) of freedom (naturally conveyed, as seen, by the procedure). In addition, the method can open new perspectives within the theory of fractional calculus, suggesting alternative formulations to already established treatments and/or definitions. At a basic level, involving up to three addends in the square root, the smallest admissible dimension 2n = 2 is enough to ensure that the desired matrices α and β can be realized. So, on the basis of (32), we may identify them with any two of the Pauli matrices       0 1 0 −i 1 0 σ1 = , σ2 = , σ3 = . (33) 1 0 i 0 0 −1 We recall that σ 2j = I 2×2 , {σ j , σ k } = 2δj,k I 2×2 , [σ j , σ k ] = 2iεjkl σ l , 10

j, k, l = 1, 2, 3,

I 2×2 being the 2 × 2 unit matrix, and εjkl the Levi-Civita tensor:   1 if (j, k, l) is an even permutation of (1, 2, 3), −1 if (j, k, l) is an odd permutation of (1, 2, 3), εjkl =  0 if any two indices are equal . Eventually, we can write q b2 + B b 2 I 2×2 = Aσ b j + Bσ b k , j 6= k A

j, k = 1, 2, 3

or, more in general, q b2 + B b2 + C b 2 I 2×2 = Aσ b j + Bσ b k + Cσ b l , j 6= k 6= l A

j, k, l = 1, 2, 3

(34)

(35)

As said, the correspondence of each of the operators involved in the square root with a specific Pauli matrix is a mere matter of convenience, possibly suggested by the problem under investigation. Therefore, the resulting matrix expression of the original (scalar) operator function is not unique. Anyway, the right hand side in (35) can formally be b B, b C) b and the matrix vector b ≡ (A, understood as the dot product of the operator vector v σ ≡ (σ j , σ k , σ l ), i.e. in symbols q b2 + B b2 + C b 2 I 2×2 = v b · σ. A (36)

3.1

Possible applications: practical and conceptual issues

We have already mentioned the Dirac equation (6) and addressed it as following from
∂2 mc 2 a factorization procedure applied to the Klein-Gordon operator ∇2 − c12 ∂t , in 2 − ~ practice along the lines exemplified above. As said, the procedure can be applied to various (physical and/or mathematical) contexts, and also be variously finalized. Evidently, one of such contexts is that concerned with the solution of evolution equations ruled by root functions of differential operators. As a typical example, we may consider the equation q ∂ζ ψ(ξ, ζ) = − 1 − ∂ξ2 ψ(ξ, ζ), (37) ψ(ξ, 0) = ψ0 (ξ). ~ Setting ζ = i λctc and ξ = λxc , λc = mc being the Compton wavelength of the particle, the above equation would yield the (1+1)D version of the RSE (5), and accordingly ψ(ξ, ζ) would represent the particle wave function. Then, we will refer to (37) as relativistic evolution equation.

11

The Dirac-like “linearization” procedure turns the problem of the solution of (37) into that of the solution of the system of two coupled linear homogeneous partial differential equation of the first order ∂ζ ψ(ξ, ζ) = −(σ j − i∂ξ σ k ) ψ(ξ, ζ), j 6= k,

(38)

ψ(ξ, 0) = ψ 0 (ξ), for the two component vector ψ(ξ, 0). The solution to the above “evolution equation” is indeed immediately written in the form ψ(ξ, ζ) = e−ζ(σj −i∂ξ σk ) ψ 0 (ξ), (39) in full analogy with Eq. (10). Whatever be the specific Pauli matrices chosen in the factorization, on account of that the exponent K = −ζ(σ j − i∂ξ σ k ) is a traceless matrix or equivalently of that σ 2j = I 2×2 , the evolution matrix2 U (ξ, ζ) = e−ζ(σj −i∂ξ σk ) can be given the explicit expression q q sinh[ζ( 1 − ∂ξ2 )] q U (ξ, ζ) = cosh[ζ( 1 − ∂ξ2 )]I 2×2 − (σ j − i∂ξ σ k ), 1 − ∂ξ2

(40)

that we will reconsider in Sec. 4 when further commenting on the relativistic evolution equations. Another possible context of application of the Dirac-like “linearization” procedure is that of the theory of the fractional calculus [23]. As an example, let us consider the operator p b = a + ∂x , O (41) a being an arbitrary constant. For it the following interpretation, 2

We recall that for any 2 × 2 traceless matrix  K=

a c

b −a

 ,

whose square is then a multiple of the unit matrix by − det K = a2 + bc, i.e. K 2 = (− det K)I 2×2 , one has that eK = cosh ω I 2×2 + (sinh ω/ω)K, where ω is any of the two complex roots ω 2 = − det K = a2 + bc [22]. Of course, if ω = 0, amounting to K 2 = 0, one simply has eK = I 2×2 +K. See also footnote 4.

12

b (x) = √a + ∂x f (x) = (a + ∂x ) √1 Of π a + ∂x

Z

∞

dss−1/2 e−as e−s∂x f (x),

0

b −ν , 0, can be worked out, resorting to the integral representation of the operator L as Z ∞ b b −ν = 1 dssν−1 e−sL , 0 (42) L Γ(ν) 0 which reproduces for operators the well-known Laplace-transform identity for c-numbers. Note that the shift operator e−s∂x under the integral yields e−s∂x f (x) = f (x − s). Alternatively, in the light of the above analysis, the scalar operator can be replaced by the operator matrix p p √ a + ∂x → aσ j + ∂x σ k , j 6= k j, k = 1, 2, 3, (43) for any specific choice of the inherent Pauli matrices, thus opening new perspectives within the theory of fractional calculus. The operator nature of the l.h.s. would be conveyed by the matrix nature of r.h.s.; √ indeed, ∂x may be seen as acting on 1, thus giving p 1 ∂x 1 = √ πx according to the Euler definition of fractional derivative ∂xν xµ =

Γ(µ + 1) xµ−ν . Γ(ν − µ + 1)

Thus, the operator (41) can be regarded as acting in a 2D vector space through the matrix ! √ 0 a − √iπx b √ O := , a + √iπx 0 or also through b := O

√ ! −i a , i a − √1πx √1 √πx

b := O

√1 √πx

a

√

a

− √1πx

! ,

each matrix being obtained in correspondence with a specific choice of the Pauli matrices in (43). In our opinioin, this view deserves to be explored.

13

3.2

Estension of the procedure to higher-degree root operator functions

It is quite natural to address the question whether it could be possible to extend the procedure to higher-order root operator functions. In other words, we will try to establish whether it could be possible to write down q m bm + B b m I = Aα b + Bβ b , A (44) or, more in general, with m operators involved in, q m bm I = A b1 α1 + A b2 α2 + .. + A bm αm , bm + .. + A bm + A A m 2 1

(45)

with the αs and β matrices being identified through suitable conditions analogous to (32). 3.2.1

Cube root

Thus, for instance, the factorization b3 + B b 3 ) I = (Aα b + Bβ) b 3, (A allowing for the disentanglement of the cube root operator as q 3 b3 + B b 3 I = Aα b + Bβ b , A

(46)

b B, b and matrices α and β such to satisfy the is possible for commuting operators A, three-term relations 

α, β

2

α3 = β 3 = I, + {β, {α, β}} = 0, β, α2 + {α, {α, β}} = 0,

(47)

(in a sense paralleling the two-term relations (32)), which can equivalently be written as α3 = β 3 = I, α + {β, βαβ} = 0, β + {α, αβα} = 0.

(48)

We see that α and β are traceless matrices, with eigenvalues conveyed by the third roots of unity: µ3 = 1, and hence

√ √ 1 1 µ0 = 1, µ1 = − (1 − i 3), µ2 = − (1 + i 3). 2 2 Therefore, they must be of the order 3n × 3n, n = 1, 2, .., with determinant ∆ = (µ0 µ1 µ2 )n = 1. 14

The matrices



 0 1 0 τ1 =  0 0 1 , 1 0 0



 0 µ1 0 τ 2 =  0 0 µ2  , 1 0 0

(49)

of smallest admissible dimension, provide a suitable pair of matrices satisfying the required conditions. They are seen to span, by repeated commutators, an 8-dimensional Lie algebra. For instance,   0 0 1 √ [τ 1 , τ 2 ] = −i 3τ 3 , τ 3 =  µ1 0 0  = τ > 2. 0 µ2 0 For completeness’  µ2 τ4 =  0 0

sake, we write down the   0 0 µ1 1 0 , τ5 =  0 0 µ1 0

expressions of the other τ -matrices,    0 0 0 0 1 1 0  = τ ∗4 , τ 8 =  1 0 0  0 µ2 0 1 0 (50)









0 µ2 0 0 0 1  µ 0 0  = τ ∗3 = τ > τ 6 =  0 0 µ1  = τ ∗2 = τ > , τ = 2 7 7 6. 1 0 0 0 µ1 0 Each of them is such that τ 3j = I 3×3 , and ∆j = 1, j = 1, .., 8. Also, their commutators are √ [τ j , τ k ] = −i 3fjkl τ l , the relevant structure constants fjkl being f123 =

f134 = f142 = f246 = f268 = f284 = f358 = f382 =

f478 = f483 = f562 = f674 = 1, f156 =

f146 = f175 = f235 = f251 = f346 = f371 = f461 =

f573 = f587 = f685 = f786 = −1. Of course, fjkl = −fkjl . Interestingly, only 4 commutators vanish, i.e. [τ 1 , τ 8 ] = [τ 2 , τ 7 ] = [τ 3 , τ 6 ] = [τ 4 , τ 5 ] = 0, and accordingly, the involved pairs of matrices are the only ones that do not allow for the desired factorization ` a la Dirac of the sum of third-power operators 3 . Therefore, 24 possible pairs of matrices suitable for the disentanglement of the cube root are conveyed by the set of τ -matrices. 3

Note in fact that the relations (47) could not be satisfied by (non null) commuting matrices.

15

If a third term is added in the sum, amounting to the linearization issue q 3 b3 + B b3 + C b 3 I 3×3 = Aα b + Bβ b + Cγ, b A

(51)

a triplet of matrices is needed, such that each and each pair of them satisfy the relations (47), in addition to the further one X (α β γ) = 0, (52) p∈S3

the sum being over all the six possible products of the three matrices obtained from all their permutations p (∈ S3 )4 . One can see that 24 suitable triplets of matrices can be extracted from the set of τ matrices, the choice being a matter of convenience in conformity to the problem under analysis. It is worth noting that the τ -matrices have been already deduced in [23] in connection with the analysis of the fractional Dirac equation. There, the triplets of matrices allowing for (51) are explicitly signalized. Paralleling the analysis, developed in connection with the square root functions of operators, regarding the evolution equation (37), we may consider an evolution equation involving a cube root of the differential operators ∂ξ3 , namely ∂ζ ψ(ξ, ζ) =

3

q

1 + ∂ξ3 ψ(ξ, ζ),

(53)

ψ(ξ, 0) = ψ0 (ξ). By the Dirac-like procedure, it is recast into the system of three coupled linear homogeneous partial differential equation of the first order for the three component vector ψ(ξ, 0), ∂ζ ψ(ξ, ζ) = (τ j + ∂ξ τ k ) ψ(ξ, ζ), j 6= k,

(54)

ψ(ξ, 0) = ψ 0 (ξ), for any choice of the suitable pairs of the afore-introduced τ -matrices. As before, the solution can be formally written as ψ(ξ, ζ) = eζ(τ j +∂ξ τ k ) ψ 0 (ξ).

(55)

However, in order to get an explicit expression for the evolution matrix U (ξ, τ ) = eζ(τ j +∂ξ τ k )

(56)

4 The relations in (47) for each pair of matrices can as well be cast in the form (52) if applied to sets of three matrices in which two of them are the same.

16

one needs to resort to appropriate ordering techniques since the τ -matrices involved in the linearization of the original cube root operators should not commute with each other. Just to fix mind, let us work with the matrices (49). Then, we apply the Zassenhaus formula [24,25] giving the exponential of the sum of two operators as the in general infinite product of operators according to eX+Y = eX eY b

b

b b

∞ Y

e Cj , b

j=1

where the first terms in the product explicitly write as h i b1 = − 1 X, b Yb , C 2 h h ii h h ii 1 b Yb + 1 X, b X, b Yb , b2 = C Yb , X, 3 6 nh h h iii h h h iiio 1 1 h b h b h b b iii b3 = b Yb b X, b Yb C Yb , Yb , X, + Yb , X, − X, X, X, Y . 8 24 Then, in the case of (56) we find that √ 3 2 i b2 = ζ 3 ∂ 2 ( 1 τ 4 − τ 5 ), b1 = ζ ∂ξ τ 3 , C C ξ 2 2 thus enabling us to write for U (ξ, τ ) at the third order in the evolution parameter ζ the expression U (ξ, ζ) = eζτ 1 eζ∂ξ τ 2 e = eζτ 1 eζ∂ξ τ 2 e

√ i 3 2 ζ ∂ξ τ 3 2 √ i 3 2 ζ ∂ξ τ 3 2

eζ

3 ∂ 2 ( 1 τ −τ ) 5 ξ 2 4

1 3 2 ∂ξ τ 4

e2ζ

e−ζ

+ O(ζ 4 )

3∂2τ ξ 5

+ O(ζ 4 ),

the latter expression being allowed by the commutation [τ 4 , τ 5 ] = 0. We conclude by saying that each term in the above product of exponential matrices can be written in a form similar to (40). In fact, as a consequence of that τ 3j = I, the exponential matrix eaτ j turns out to be the sum of three terms; precisely, eaτ j = A0 (a)I 3×3 + A1 (a)τ j + A2 (a)τ 2j .

(57)

The coefficients Aj (a), j = 0, 1, 2 are given by 1 2 a3 , ; ), 3 3 27 2 4 a3 A1 (a) = a 0 F2 (−; , ; ), 3 3 27 a2 4 5 a3 A2 (a) = ), 0 F2 (−; , ; 2 3 3 27 A0 (a) =

0 F2 (−;

17

(58)

where 0 F2 (·) denotes the generalized hypergeometric function, formally represented by the series [17, 26] ∞ Qp X (aj )k z k Qj=1 , (59) p Fq (a1 , .., ap ; b1 , .., bq ; z) = q j=1 (bj )k k! k=0

with (a)k ≡ Γ(a + k)/Γ(a) being the Pochhammer symbol. We recall that p Fq converges for all finite z if p ≤ q. It is worth recalling that the relevant power series expressions of the Aj s, conveyed by (59), have been investigated in [27] as pseudo-hyperbolic functions. It becomes evident that the expression (57) corresponds to (40) (written for a single Pauli matrix) when recalling the link of the hyperbolic functions to the hypergeometric functions as cosh(z) = sinh(z) =

1 z2 , ), 2 4 3 z2 z 0 F1 (−; , ). 2 4 0 F1 (−;

Finally, we note that the set {τ j }j=1,..,8 is closed with respect to the square, since the square of a τ -matrix just equals the matrix in the set with which it commutes; thereby, τ 21 = τ 8 , τ 22 = τ 7 , and so on. This property of the τ -matrices (i.e the cube roots of the unit matrix) perfectly parallels tha of the cube roots of unity; in fact, µ20 = µ0 , µ21 = µ2 , and µ22 = µ1 , so that {µ20 , µ21 , µ22 } = { µ0 , µ1 , µ2 }. Also, as a consequence, we can say that the exponentials eaτ j belong to the algebra spanned (in general, over the complex field C) by the {τ j }j=1,..,8 and the unit matrix. 3.2.2

Quartic root

Likewise, the disentanglement of the quartic root as

4

q b4 + B b 4 I = Aα b + Bβ b , A

(60)

demands for α and β matrices such to satisfy the four-term relations α4 = β 4 = I, 0, {β 2 , {α, β} } = 0,  2 = α , β 2 + {α, β}2 = 0.

{α2 , {α, β} }

(61)

Therefore, we can say that the desired matrices α and β are traceless and with eigenvalues conveyed by the quartic roots of unity: µ4 = 1, i.e. µ0 = 1, µ1 = i, µ2 = −i, µ3 = −1 18

As a consequence, they must be of the order 4n × 4n, n = 1, 2, .., with determinant ∆ = (µ0 µ1 µ2 µ3 )n = (−1)n . The anticommutation relations in the second row of (61) suggest a correspondence of the matrices α2 , {α, β} and β 2 with σ-composed matrices. Thus, working with the smallest admissible dimension, i.e. 4n = 4, we start taking   √ 02×2 −i σ 3 √ ρ1 = , (62) i σ3 02×2 with5

√

 σ3 =

1 0 0 i

 .

Evidently, ρ41 = I. Then, following the conditions (61), the suitable matrix   σ+ σ− ρ2 = , σ− σ+

(63)

is obtained, where 1 σ + = (σ 1 + iσ 2 ) = 2



0 1 0 0



1 , σ − = (σ 1 − iσ 2 ) = 2



0 0 1 0

 .

Note that σ 2+ = σ 2− = 0, and [σ + , σ − ] = σ 3 whereas {σ + , σ − } = I. Of course, ρ1 and ρ2 do not commute with each other (as it must be in order for they to allow for (60); see footnote 3); in fact,   √ σ − −iσ + −i π4 [ρ1 , ρ2 ] = 2ρ3 , ρ3 = e , iσ + −σ − whilst

√ {ρ1 , ρ2 } = i 2ρ3 = i [ρ1 , ρ2 ] .

It can be verified that ρ3 can be a suitable matrix to realize (60) accompanied by either ρ1 or ρ2 . 5 We recall that given a 2 × 2 matrix M with eigenvalues µ1 6= µ2 , the function f (M ) can be evaluated according to [28] µ1 f (µ2 ) − µ2 f (µ1 ) f (µ1 ) − f (µ2 ) f (M ) = I+ M. µ1 − µ2 µ1 − µ 2

19

 By repeated commutators, we span a 15-dimensional Lie algebra of matrices ρj j=1,..,15 such that ρ4j = I, ∀j. Let us explicitly write the other 12 matrices:      √  σ 3 02×2 −σ + σ − 02×2 σ 3 √ ρ4 = , ρ5 = , ρ6 = , 02×2 − σ 3 σ − −σ + σ 3 02×2     σ 3 02×2 −σ + −iσ − ∗ −i π4 ρ7 = ρ3 , ρ8 = , ρ9 = e , (64) 02×2 σ 3 iσ − σ+     02×2 σ 1 −σ − σ + ρ10 = iρ∗6 , ρ11 = , ρ12 = = ρ> 4, σ 1 02×2 σ + −σ −   σ− σ+ ∗ ρ13 = ρ9 , ρ14 = = ρ> ρ15 = iρ∗1 . 2, σ+ σ− Then, with the Lie bracket recast in the form   √ ρj , ρk = 2gjkl ρl , we can specify the structure constants gjkl as g1,7,2 =

g1,14,9 = g3,4,10 = g3,12,15 = g4,9,15 = g6,7,14 = g6,14,3 = g6,13,2

=

g7,15,4 = g9,12,10 = g9,14,6 = g10,12,3 = g10,13,4 = g12,15,9 = −1

g1,2,3 =

g1,13,142 = g2,3,6 = g2,6,9 = g2,9,1 = g3,14,1 = g4,10,9 = g4,15,3

= g7,10,12 = g13,15,12 = 1 g1,4,7 =

g1,9,12 = g2,7,10 = g2,13,15 = g2,15,7 = g4,7,6 = g6,9,4 = g6,12,7

= g7,14,15 = g9,15,14 = g10,142,7 = g12,13,6 = −i g1,3,4 =

g1,12,13 = g2,10,13 = g3,6,12 = g3,10,142 = g3,15,2 = g4,6,13 = g4,13,1

= g7,12,1 = g9,10,2 = g13,14,10 = g14,15,13 = i g1,5,10 =

√ g1,10,11 = g2,5,12 = g2,12,8 = g4,14,8 = g5,14,4 = g6,15,11 = − 2

g2,4,5 = g2,8,4 = g4,5,14 = g4,8,2 = g5,6,15 = g5,10,1 = g5,124,2 = g5,15,6 √ = g8,12,14 = g8,14,12 = g12,14,5 = 2 g1,11,6 = g3,8,7 = g3,9,8 = g7,11,13 = g8,9,13 = g9,11,3 = g9,13,11 = g10,11,15 √ = g10,15,5 = g11,13,7 = g11,15,10 = −i 2 √ g1,6,5 = g3,7,11 = g3,11,9 = g6,11,1 = g7,8,3 = g8,13,9 = i 2, 20

with gjkl = −gkjl . However, not all the matrices (64) satisfy the condition on the determinant being in fact ∆(ρ5 ) = ∆(ρ8 ) = ∆(ρ11 ) = 1; accordingly, such matrices are not suitable for the desired factorization. Besides, they commute  with each other, and each of them commute also with other four matrices in the set ρj j=1,..,15 . In turn, each of the other ρ-matrices commutes only with two elements in the set, one being ρ5 , ρ8 or ρ11 . Of course, this con be deduced from the set of {j, k} indices that do not appear in the above reported table of structure constants gjkl . An accurate analysis reveals that one can rely on 48 possible pairs of matrices allowing for (60); as expected, the matrices ρ5 , ρ8 and ρ11 are not inlcuded in any of the possible pairs. Another property of the ρ-matrices is worth mentioning. It in a sense parallels that concerning the τ -matrices, and suggests ageneralization to the m-th roots of the unit matrix. One can easily verify that the set ρj j=1,..,15 ⊕ I 4×4 is closed under the square and the cube. Interestingly, we find that ρ25 = ρ28 = ρ211 = I 4×4 , as a consequence, indeed, of that they are directly linked to the Pauli matrices. By applying the same terminolgy as for (complex) numbers, we can say that such 4 × 4 matrices are not primitive 4-th roots of the 4 × 4 unit matrix, being in fact also 2-th roots of the 4 × 4 unit matrix6 . This clearly clarifies why such matrices have determinant equal to 1. In contrast, the squares of the other ρ-matrices just equal ρ5 , ρ8 or ρ11 (apart from a minus sign), precisely that with which the matrix of concern commutes. Thus, for instance, ρ21 = ρ8 , ρ22 = ρ11 , ρ23 = ρ5 , and so on. Finally as to the cube, it is obvious that ρ35 = ρ5 , ρ38 = ρ8 , ρ311 = ρ11 , whereas the cube of any other ρ-matrix just equals the matrix (different from ρ5 , ρ8 and ρ11 ) with which the matrix of concern commutes (apart from ±1 or ±i). Thus, for instance, ρ31 = iρ15 , ρ32 = ρ14 , ρ33 = −ρ13 , and so on. aρj Then, as for the τ -matrices, we can say that the exponential matrix belongs to the  e algebra spanned (in general, over the complex field C) by the ρj j=1,..,15 and the unit matrix. In fact, the analog of (57) can be written as eaρj = B0 (a)I 4×4 + B1 (a)ρj + B2 (a)ρ2j + B3 (a)ρ3j , with the coefficients Bk (a), k = 0, 1, 2, 3 being correspondingly given by Bk (a) = 6

ak k+1 k+2 k+3 k+4 a 4 , , , ; ( ) ), 1 F4 (1; k! 4 4 4 4 4

In fact, µ0 and µ3 are not primitive 4-th roots of unity, whilst µ1 and µ2 are.

21

(65)

as an obvious extension of (58) to the matrices (64). Needless to say, for the non primitive 4-th root matrices ρ5 , ρ8 and ρ11 we can write down an expression similar to (40), namely eaρl = cosh(a)I 4×4 + sinh(a)ρl , l = 5, 8, 11. 3.2.3

m-th roots

It is evident that with increasing m in (44) the problem becomes even more complex. However, on the basis of the previous analysis, we can try to draw some basic issues, at least for the two-term case just displayed in (44). The identity (44) yields m + 1 relations involving terms of degree m in the mn × mn matrices α and β. Firstly, the latter come to be the m-th roots of the unit matrix, being αm = β m = I,

(66)

and hence their eigenvalues can be written as 2π

µj = ei m j , j = 0, 2, ...m − 1. It is easy to see that m−1 X

µj = 0,

j=0

thus implying T r(α) = T r(β) = 0. Also, being m−1 Y

µj = (−1)m−1 ,

j=0

we can say that the determinant of the matrices α and β is ∆ = (−1)n(m−1) . As to the other conditions ensuring (44) be satisfied, they can be synthesised in the form X αp1 β q1 αp2 β q2 ....αpα β qβ = 0 (67) p,q

P where the sum is intended to comprise all the powers p and q such that j i j pj = l and P i qi = k, for any choice of integers (l, k) such that l 6= 0, k 6= 0 and l + k = m, thus m! terms of degree m. yielding l!k!

22

We see that the m × m matrices     0 µ0 0 · · · 0 0 µ1 0 · · · 0  0 0 µ0 · · · 0   0 0 µ2 · · · 0       ..   . . . . . . . . ..  , .. .. .. ..  , ν 2 = δ  .. .. .. .. ν1 =  . .       0 0 0 · · · µ0   0 0 0 · · · µm−1  µ0 0 0 · · · 0 µ0 0 0 · · · 0  1 m odd π with δ = , represent a suitable pair of matrices (of smallest allowable orei m m even der) which satisfy (66) and (67). Starting from them, one can span an (m2 −1)-dimensional Lie algebra of matrices {ν j }j=1,..,m2 −1 satisfying (66). On the basis of the previous analysis, we can also state that in general the set {ν}j=1,..,m2 −1 ⊕I m×m is closed under the powers 2, 3, ...m. Consequently, we can write the group element eaν j in the general form e

aν j

= C0 (a)I m×m +

m−1 X

Ck (a)ν kj ,

(68)

k=1

where the coefficients Ck (a), k = 0, 1, 2, m − 1 are Ck (a) =

4

k+1 k+2 k+m a m ak , , ..., ; ( ) ). 1 Fm (1; k! m 4 m m

Relativistic evolution equation: a “direct” solving method

We will reconsider Eq. (37), rewritten here for convenience’s sake, q ∂ζ ψ(ξ, ζ) = − 1 − ∂ξ2 ψ(ξ, ζ),

(69)

ψ(ξ, 0) = ψ0 (ξ), and approach its solution through a more “direct” method. Then, we will show some results obtained through this method along with those obtained from the Dirac-factorization method (Eq. (40)). Equation (37) has been the object of the recent analysis in [29]. Here, we will further elaborate that analysis.

4.1

Evolution operator

In full analogy with Eq. (10), the solution to (69) can be written as −ζ

q

ψ(ξ, ζ) = e

1−∂ξ2

ψ0 (ξ),

(70)

the exponential operator being in fact the evolution operator −ζ b U(ζ) =e

pertaining to Eq. (69). 23

q

1−∂ξ2

,

(71)

4.2

Evolution operator as McDonald transform

b In order to determine the action of U(ζ) on the initial data ψ0 (ξ), we resort to the result [30–32] r Z ∞ π −2√ab −as2 − b2 s e ds = , a, b > 0 (72) e 4a 0 from the integral calculus. Note that the l.h.s. can be interpreted as the Laplace transform b √ of f (s) = e− s / s. Then, assuming that the identity expressed by (72) holds also when the parameters a, or b or both are (commuting) operators, we can write Z ∞ ζ2 2 2 2 ψ(ξ, ζ) = √ dse−s − 4s2 (1−∂ξ ) ψ0 (ξ), (73) π 0 q √ 2 setting in particular a = 1 and b = ζ4 (1 − ∂ξ2 ) so that 2 ab = ζ 1 − ∂ξ2 . One can recognize under the integral the “non-relativistic” evolution operator (14) (at ζ2 “time” 4s 2 ). Accordingly, on account of (15), we can write ψ(ξ, ζ) =

2 πζ

Z

+∞

dξ 0 ψ0 (ξ 0 )

Z

−∞

∞

dsse

  (ξ−ξ0 )2 − −s2 1+ 2 ζ

ζ2 4s2

.

(74)

0

After changing the order of the integrals, and exploiting the integral representation of the modified Bessel function of the second kind (or, McDonald function) Kν , i.e. [17,32,33] Z

∞

s 0

ν−1 −βsp −γs−p

e

2 ds = p

 ν p γ 2p K νp (2 βγ), 0 has also been obtained through the Fourier transform method. The “relativistic” evolution shows a quicker attenuation of the wavefunction amplitude than the “non-relativistic” one. Correspondingly, Fig. 2 displays the (ζ, ξ)-contourplots of the fundamental solution (81) for some value of the “initial time” ζ0 = −ε.

Figure 2: (ζ, ξ)-contourplots exemplifyng the ζ-evolution of the fundamental solution of Eq. (69), i.e. ψD (ξ, ζ), for (a) ε = 0.5, (b) ε = 1, (c) ε = 1.5, and (d) ε = 2.

27

ξ2

In order to have a function, which, just as the Gaussian e− 4 , takes on the unit value at ξ = 0, the factor K1π(ε) is added to (81), so that the function q 2 2 ζ + ε K1 ( (ζ + ε) + ξ ) q ψD (ξ, ζ, ε) = , K1 (ε) (ζ + ε)2 + ξ 2

(82)

pertains to Fig. 2. For completeness’sake, Fig. 3 plots the profiles of the wavefunctions of concern in the same panel at different ζs. We see that, with increasing ε, ψD (ξ, ζ, ε) tends to be even more similar to the ψG (ξ, ζ)|nr .

Figure 3: ξ-profiles of ψG (ξ, ζ)|nr (solid red line), ψG (ξ, ζ)|r (blue dotted line), ψD (ξ, ζ, ε = 1) (green dashed line), ψD (ξ, ζ, ε = 2) (magenta dash-dotted line), and ψD (ξ, ζ, ε = 3) (turquoise solid line) at different ζ, specifically at (a) ζ = 0, (b) ζ = 0.5, and (c) ζ = 1.

Notably, as solutions corresponding to initial monomials can be found for the HE, SE and PWE as recalled in Sec. 2.3, explicit solutions corresponding to initial monomials can be found for Eq. (69) as well. In fact, let us set ψ0 (ξ) = ξ n with n nonnegative integer. The resulting wavefunction is p Z +∞ [n/2] 2 02 2ζ X n! n−2h 0 02h K1 ( ζ + ξ ) p pn (ξ, ζ) = ξ dξ ξ . π h!(n − 2h)! ζ 2 + ξ 02 0 h=0

Since [32] Z 0

+∞

p K (α ζ 2 + s2 ) 2µ Γ(µ + 1) ν dss2µ+1 p = Kν−µ−1 (αz), αµ+1 z ν−µ+1 (ζ 2 + s2 )ν 1 α > 0, − 2 28

we end up with the esplicit expression r pn (ξ, ζ) =

[n/2] 2ζ X n! ζ ( )h ξ n−2h Kh− 1 (ζ). 2 π h!(n − 2h)! 2 h=0

We recall that the McDonald function of half odd integer order means [17, 33] r π −z K− 1 (z) = e , 2 2z r h−1 1 π −z X (h + m − 1)! e , h = 1, 2, ... Kh− 1 (z) = 2 2z m!(h − m − 1)! (2z)m m=0

Therefore, we can say that the pn s are polynomials of degree n. In the asymptotic regime, where the non-relativistic behavior is recovered, the pn s yield the heat polynomials, apart from the rest energy contribution e−ζ . Explicitly, we obtain ζ ζ ξ pn (ξ, ζ) ∼ e−ζ (− )n/2 Hn ( √ ) = e−ζ vn (ξ, ), 2 2 −2ζ vn being the heat polynomials (21). As said, u0 (x) = 1 implies u(x, t) = 1 in (15). Correspondingly, a uniform input for (69), i.e., ψ0 (ξ) = 1, evolves just through the “rest energy” term r 2ζ p0 (ξ, ζ) ∼ K 1 (ζ) = e−ζ . π 2 Interestingly, we find that the first pn s are just the corresponding vn s at ζ2 , apart from the rest energy term e−ζ , namely ζ pn (ξ, ζ) = e−ζ vn (ξ, ), n = 0, 1, 2, 3. 2 Figure 4 shows the plots of the pn s and vn s for n = 4, 5, 6, 7 at ζ = 1. The interest in the pn s stems from the possibility, to be explored, of using them as alternative to the evaluation of the integral transform (76) by exploiting the power series expansion, if any, of the input function. Furthermore, as the heat polynomials realize the power series expansion of the evolved 2 ∂ form (through Eq. (7)) of the eigenstates eχξ of the derivative operator ∂ξ , i.e. eχξ eχ ζ , the relativistic heat polynomials realize the power series expansion of the corresponding “relativistically” evolved form, i.e. χξ −ζ

e e

√

1−χ2

=

∞ X χn n=0

29

n!

pn (ξ, ζ),

(83)

Figure 4: Plots of the pn s and vn s for n = 4, 5, 6, 7 at ζ = 1.

for |χ| ≤ 1. The above relies on the Neumann-type series for the MacDonald function (also referred to as multiplication theorem) [17, 33] 2

2 − 21 ν

(s − τ )

∞ −ν−n X s 1 ( zτ 2 )n Kν+n (zs), Kν [z(s − τ ) ] = n! 2 2

2

1 2

n=0

holding for |τ 2 /s2 | ≤ 1. Equation (83) conveys the generating function relation for the pn s, as (20) conveys that for the vn s. In both cases the generating function is of the type G(ξ, ζ, χ) = A(χ, ζ)eχξ , √ 2 2 with A(χ, ζ) = e−ζ 1−χ , |χ| ≤ 1, for the pn s and A(χ, ζ) = eχ ζ for the vn s, both being amenable for a Taylor series expansion with respect to χ. Therefore, the generating functions pertaining to the pn s as well as the vn s can be ascribed to the App`el family [34]. Also, according to [34], we can express the relativistic heat polynomials in the operator form pn (ξ, ζ) = A(∂ξ , ζ)ξ n , 30

where the derivative operator ∂ξ represents one of the monomial operators of the loweringraising pair Pb = ∂ξ , 0 c = ξ + A (∂ξ , ζ) , M A(∂ξ , ζ)

the prime denoting differentiation with respect to the first argument. They are expected to act on the pn s according to the usual monomial relations Pbpn (ξ, ζ) = npn−1 (ξ, ζ), cpn (ξ, ζ) = pn+1 (ξ, ζ), M which, as can easily be verified, explicitly mean     ∂ξ pn (ξ, ζ) = pn+1 (ξ, ζ). ∂ξ pn (ξ, ζ) = npn−1 (ξ, ζ), ξ +ζq  1 − ∂ξ2 

(84)

(85)

Evidently, relations (84) amount to cPbpn (ξ, ζ) = PbM cpn (ξ, ζ) = npn (ξ, ζ), M signifying the differential equation     ∂ξ2 q ξ∂ + ζ − n pn (ξ, ζ),  ξ  1 − ∂2

pn (ξ, 0) = ξ n .

(86)

ξ

−1/2  Since the operator 1 − ∂ξ2 can be understood as ∞ 1 X Γ( 12 + j) 2j 1 q =√ ∂ξ , j! π 1 − ∂ξ2 j=0

one recovers relations (22) for the vn s as well as the inherent differential equation (23), taking only the 0-th order term in the series. we see that the raising operator for the  Also,  pns is a ξ-differential operator of order 2 n−1 + 1, and accordingly Eq. (86) is of order 2 2 n2 . Finally, it is worth noting that the pn s belong to the family of non-local polynomials, since they obey the integro-differential equation (86) instead of an ordinary differential equation like the vn s. In fact, on the basis of the representations (42) and (75), Eq. (86) amounts to Z ζ +∞ 0 0 ξpn (ξ, ζ) + dξ K0 (ξ − ξ 0 )p00n (ξ, ζ) − npn (ξ, ζ) = 0, pn (ξ, 0) = ξ n . π −∞ Other properties of (76) are under investigation. 31

4.4

Evolution according to the Dirac-factorization approach

In Sect. 3.1 we have worked out the general expression (40) for the solution of Eq. (69) within the context formalism, and hence for the two component  of the Dirac-factorization  ψ1 (ξ, ζ) vector ψ(ξ, ζ) = . ψ2 (ξ, ζ) We will apply such a result to specific initial data, which should allow for a comparison with the results of the previous analysis. Then, let us firstly consider the input vector ! 2 − ξ4 e ψ 0 (ξ) = . (87) 0 Evidently, the definite expression of the evolved vector depends on the specific choice of the Pauli matrices in the square-root factorization. Thus, with H = −(σ 3 − i∂ξ σ 2 ), the evolution of the vector ψ is dictated by ψ(ξ, ζ) =

b − Sb −∂ξ Sb C b − Sb ∂ξ Sb C

! ψ 0 (ξ),

(88)

b and Sb being the cosh- and sinch-operator functions entering (40), i.e C q q sinh[ζ( 1 − ∂ξ2 )] b = cosh[ζ( 1 − ∂ 2 )], q Sb = C . ξ 1 − ∂ξ2 Therefore, with ψ 0 (ξ) given by (87), we obtain ψ(ξ, ζ) =

b − Sb C ∂ξ Sb

!

ξ2

e− 4 ,

(89)

Calculation directly in the space domain (i.e. according to (88)) or equivalently in the Fourier domain, and hence through the expressions r Z ∞   1 2 (C − S) cos(κξ) ψ(ξ, ζ) = 2 dκ e−κ , (90) −κS sin(κξ) π 0 b and S c, namely the functions C and S being the Fourier images of the operators C √ p sinh[ζ( 1 + κ2 )] 2 √ S= , C = cosh[ζ( 1 + κ )], 1 + κ2 yield the same results for the two vector components, which are plotted in Fig. 5. 32

Figure 5: ζ-evolution of the two components (a) ψ1 (ξ, ζ) and (b) ψ2 (ξ, ζ) of the ψ-vector for the input (87), shown at ζ = 0 (solid red line), ζ = 0.3 (blue dotted line), ζ = 0.6 (green dashed line), and ζ = 1 (magenta dash-dotted line).

2

Figure 6: ζ-evolution of the Euclidean norm |ψ| = ψ12 + ψ22 of the vector ψ for the input (87) (see Fig. 5), shown at ζ = 0 (solid red line), ζ = 0.3 (blue dotted line), ζ = 0.6 (green dashed line), and ζ = 1 (magenta dash-dotted line).

33

Figure 6 shows the Euclidean norm |ψ|2 = ψ12 + ψ22 of the vector ψ. A further comparison with the results, previously displayed, is offered by the plots of Fig. 7, which shows the ζ-evolution of the two components of the Dirac vector ψ(ξ, ζ) for the input   ψD (ξ, 0, ε) ψ 0 (ξ) = , (91) 0 with ψD given by (82), and ε = 3. The relevant Euclidean norm |ψ|2 is plotted in Fig. 8.

Figure 7: ζ-evolution of the two components (a) ψ1 (ξ, ζ) and (b) ψ2 (ξ, ζ) of the ψ-vector for the input (91) with ε = 3, shown at ζ = 0 (solid red line), ζ = 0.3 (blue dotted line), ζ = 0.6 (green dashed line), and ζ = 1 (magenta dash-dotted line).

The interpretation and reasonableness of the obtained results should be examined within the context of the specific problem at hand.

5

Concluding notes

We have reviewed the square-root operator factorization method ` a la Dirac along with its extension to higher-degree root operators, as recently suggested in the literature [8, 9]. A deeper analysis of the cube and quartic root operators has been presented along with a precise characterization of the Lie algebras of the pertinent matrices. Then, evolution equations ruled by square root operator functions, referred to as relativistic evolution equations, have been considered, further elaborating the analysis developed in [29]. The presentation here has been finalized to a comparison between the properties of the non-relativistic and relativistic equations, in order to establish which properties of the former can be recovered to the latter. In particular, fundamental solution with its evolution property and solutions arising from initial monomials have been found for the relativistic evolution equation in full analogy with the non relativistic one. 34

2

Figure 8: ζ-evolution of the Euclidean norm |ψ| = ψ12 + ψ22 of the vector ψ for the input (91) with ε = 3 (see Fig. 7), shown at ζ = 0 (solid red line), ζ = 0.3 (blue dotted line), ζ = 0.6 (green dashed line), and ζ = 1 (magenta dash-dotted line).

Appendix In order to prove the limit relation lim V (ξ, ζ) = δ(ξ),

(92)

ζ→0

for the kernel V (ξ, ζ) of the MacDonald transform (76), we start from the integral representation [17] Z ∞ 1 2 2 1/2 dse−x cosh(s) cos(y sinh(s)) cosh(νs), Kν [(x + y ) ] = cos[ν arctan( xy )] 0 which holds for 0. Identifying x ↔ ζand y ↔ ξ, we can write p ζ K1 ( ζ 2 + ξ 2 ) p = π ζ 2 + ξ2 Z

1 p π cos[arctan( ζξ )] ζ 2 + ξ 2

∞

dse−ζ cosh(s) cos(ξ sinh(s)) cosh(s),

0

with the condition 0 being satisfied. Then, letting ζ → 0, the factor preceding the integral becomes ξ2 p → = 1. cos[arctan( ζξ )] ζ 2 + ξ 2 ζ→0 ξ 2 1

35

On the other hand, for the integral we obtain Z Z 1 ∞ 1 ∞ −ζ cosh(s) dse cos(ξ sinh(s)) cosh(s) → ds cos(ξ sinh(s)) cosh(s) = ζ→0 π 0 π 0 Z 1 ∞ ds cos(ξs) = δ(ξ), π 0 thus proving (92).

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