Symmetry Preserving Discretization Schemes through...

Symmetry Pre-serving Dis-

cretiza-tion Schemes

Nelson Faustino

The Scope ofProblemsFunction TheoreticalMethods inNumerical Analysis

Motivation behindthis talk

Lie-algebraicdiscretizationsUmbral CalculusRevisited

Radial algebraapproach

Appell Sets

su(1, 1)symmetries

Discretization ofOperators ofSturm-LiouvilletypeDiscreteElectromagneticSchrodingeroperators

Interplay withBayesian Statistics

Symmetry PreservingDiscretization Schemes through

Hypercomplex Variables

Nelson Faustino

Center of Mathematics, Computation and Cognition, UFABC

nelson.faustino@ufabc.edu.br

ICNAAM 2017, Thessaloniki, Greece, 25–30 September 2017

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Appell Sets

su(1, 1)symmetries

1 The Scope of ProblemsFunction Theoretical Methods in Numerical AnalysisMotivation behind this talk

2 Lie-algebraic discretizationsUmbral Calculus RevisitedRadial algebra approachAppell Setssu(1,1) symmetries

3 Discretization of Operators of Sturm-Liouville typeDiscrete Electromagnetic Schrodinger operatorsInterplay with Bayesian Statistics

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’(. . . ) When Columbus set sail, he was like an appliedmathematician, paid for the search of the solution of a concrete problem:find a way to India. His discovery of the New World was similar to thework of a pure mathematician (. . . )”Vladimir Arnol’d, Notices of AMS, Volume 44, Number 4 (1997)

Figure: From left to right: Discrete Dirac operators on graphs/dualgraphs vs. 7−point representation of the ’discrete’ Laplacian ∆h.

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OverviewWhy should we use Finite Difference Dirac Operators?

Hypercomplex analysis approach:1 Useful to rewrite our main problem in a more compact form

(e.g. Lame/Navier-Stokes/Schrodinger equations);2 Get exact representation formulae to solve vector-field

problems numerically (discrete counterparts);3 The regularity conditions that we need to impose on the

design of convergence schemes are quite lower incomparison with the usual convergence conditionsassociated to standard finite difference schemes.

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OverviewSome references

1 Boundary value problems: Gurlebeck and Sproßig -Quaternionic and Clifford calculus for Engineers andPhysicists (1997).

2 Discrete Fundamental solutions for Difference Diracoperators: Gurlebeck and Hommel, On finite differenceDirac operators and their fundamental solutions, Adv. Appl.Clifford Algebras, 11, 89 – 106 (2003).

3 Numerical implementation using discrete counterparts:Faustino, Gurlebeck, Hommel, and Kahler - DifferencePotentials for the Navier-Stokes equations in unboundeddomains, J. Diff. Eq. & Appl., Journal of Difference Equationsand Applications, 12(6), 577-595.

4 To take a look for further progresses on this direction, attendtomorrow (September 26) the morning talks of the13th Symposium on Clifford Analysis and Applications(ROOM 3).

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Lie algebraic discretization schemesA rigorous way to preserve (continuous) symmetries under discretization.

The roots of Lie algebraic discretization schemes:

Discretization of finite difference operators:A.Dimakis, Muller-Hoissen and T.Striker (1996), Journal ofPhysics A: Mathematical and General, 29(21), 6861

Finite difference operators vs. symplectic solvers:Dattoli, G., Ottaviani, P. L., Torre, A., & Vazquez, L. (1997). LaRivista del Nuovo Cimento (1978-1999), 20(2), 3-133.

Motivation for the approach enclosed on this talk:

Umbral calculus: Finite difference operators as convergentpower series determined in terms of partial derivatives.

Quantum Field Theory: Provides a way to representR−polynomial algebra as the Bose algebra.

Classes of Wigner Quantal Systems: Hypercomplex analysis inits minimal form corresponds to a realization of the Lie superalgebraosp(1|2) (a refinement of the Lie algebra sl(2,R) ∼= su(1, 1)).

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Multivariate polynomialsBasic setting

Monomial over x = (x1, x2, . . . , xn) ∈ Rn: xα = xα11 xα2

2 . . . xαnn

Ring of polynomials R[x ]: Each P(x) ∈ R[x ] is a linearcombination of monomials xα.

Multi-index derivatives for ∂x := (∂x1 , ∂x2 , . . . , ∂xn ):

∂αx := ∂α1x1 ∂

α2x2 . . . ∂

αnxn ∈ End(R[x ]).

Multi-index enumerative notation:

α! = α1!α2! . . . αn!,(βα

)= β!

α!(β−α)!

Binomial formula:

(x + y)β =

|β|∑|α|=0

)xαyβ−α =

|β|∑|α|=0

[∂αx xβ ]x=y

α!xα

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2 . . . xαnn

∂αx := ∂α1x1 ∂

α2x2 . . . ∂

α! = α1!α2! . . . αn!,(βα

)= β!

α!(β−α)!

Binomial formula:

(x + y)β =

|β|∑|α|=0

)xαyβ−α =

|β|∑|α|=0

[∂αx xβ ]x=y

α!xα

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2 . . . xαnn

∂αx := ∂α1x1 ∂

α2x2 . . . ∂

α! = α1!α2! . . . αn!,(βα

)= β!

α!(β−α)!

Binomial formula:

(x + y)β =

|β|∑|α|=0

)xαyβ−α =

|β|∑|α|=0

[∂αx xβ ]x=y

α!xα

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2 . . . xαnn

∂αx := ∂α1x1 ∂

α2x2 . . . ∂

α! = α1!α2! . . . αn!,(βα

)= β!

α!(β−α)!

Binomial formula:

(x + y)β =

|β|∑|α|=0

)xαyβ−α =

|β|∑|α|=0

[∂αx xβ ]x=y

α!xα

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2 . . . xαnn

∂αx := ∂α1x1 ∂

α2x2 . . . ∂

α! = α1!α2! . . . αn!,(βα

)= β!

α!(β−α)!

Binomial formula:

(x + y)β =

|β|∑|α|=0

)xαyβ−α =

|β|∑|α|=0

[∂αx xβ ]x=y

α!xα

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Multivariate polynomialsShift-invariant operators

Taylor series representation in R[x ]: P(x + y) = exp(y · ∂x )P(x).

Shift-invariant operator: Q(∂x ) is shift-invariant iffQ(∂x ) exp(y · ∂x ) = exp(y · ∂x )Q(∂x ).

Theorem (First expansion theorem)

A linear operator Q : R[x ]→ R[x ] is shift-invariant if and only if it can beexpressed (as a convergent series) in the gradient ∂x , that is

Q =∞∑|α|=0

aαα!∂αx ,

where aα = [Qxα]x=0.

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Q =∞∑|α|=0

aαα!∂αx ,

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Q =∞∑|α|=0

aαα!∂αx ,

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Basic polynomial sequencesA way to construct polynomial sequences towards interpolation theory

Definition (Basic polynomial sequence)

A polynomial sequence {Vα}α, where Vα is a multivariate polynomial ofdegree |α| such that

1 V0(x) = 1 (Initial condition);2 Vα(0) = δα,0 (Interpolating property);3 Oxj Vα(x) = αjVα−vj (x).(Delta operator)

is called basic polynomial sequence of the multivariate deltaoperator Ox = (Ox1 ,Ox2 , . . . ,Oxn ).

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Some ResultsResults may be found on my PhD dissertation entitled Discrete Clifford analysis,Universidade de Aveiro (2009)

Theorem (Basic polynomial sequences vs polynomials of binomial type)

{Vα(x)}α is a basic polynomial sequence if and only if is a sequenceof binomial type, i.e.

Vβ(x + y) =

|β|∑|α|=0

)Vα(x)Vβ−α(y).

Theorem (A. Di Bucchianico, 1999)

Let Q = Q(∂x ) be a shift invariant operator. Let Ox be a multivariatedelta operator with basic polynomial sequence {Vα}α. Then

Q =∑|α|≥0

aαα!

Oαx , with aα = [QVα(x)]x=0.

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Theorem (Basic polynomial sequences vs polynomials of binomial type)

{Vα(x)}α is a basic polynomial sequence if and only if is a sequenceof binomial type, i.e.

Vβ(x + y) =

|β|∑|α|=0

)Vα(x)Vβ−α(y).

Theorem (A. Di Bucchianico, 1999)

Let Q = Q(∂x ) be a shift invariant operator. Let Ox be a multivariatedelta operator with basic polynomial sequence {Vα}α. Then

Q =∑|α|≥0

aαα!

Oαx , with aα = [QVα(x)]x=0.

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Theorem (Expansion theorem)

Q = Q(∂x ) is uniquely determined by

Q =∞∑|α|=0

aα(x)Oαx

where the polynomials aα(x) are given by

∞∑|α|=0

aα(x)tα =QV (x , t)V (x , t)

, with V (x , t) =∑∞|α|=0

Vα(x)α!

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Finite difference toolbox

1 Equidistant lattice with mesh width h > 0:

hZn ={

x = (x1, . . . , xn) ∈ Rn :xh∈ Zn

}2 Forward/backward finite difference operators

(∂+jh f)(x) =

f(x + hej )− f(x)

h, (∂−j

h f)(x) =f(x)− f(x − hej )

3 Translation property: ∂+jh and ∂−j

h are interrelated by(T±j

h f)(x) = f(x ± hej ) i.e.

T−jh (∂+j

h f)(x) = (∂−jh f)(x) and T+j

h (∂−jh f)(x) = (∂+j

h f)(x).

4 Product rules for finite difference operators:

∂+jh (g(x)f(x)) = (∂+j

h g)(x)f(x + hej ) + g(x)(∂+jh f )(x)

∂−jh (g(x)f(x)) = (∂−j

h g)(x)f(x − hej ) + g(x)(∂−jh f)(x).

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hZn ={

x = (x1, . . . , xn) ∈ Rn :xh∈ Zn

(∂+jh f)(x) =

f(x + hej )− f(x)

h, (∂−j

h f)(x) =f(x)− f(x − hej )

h f)(x) = f(x ± hej ) i.e.

T−jh (∂+j

h (∂−jh f)(x) = (∂+j

h f)(x).

∂+jh (g(x)f(x)) = (∂+j

h g)(x)f(x + hej ) + g(x)(∂+jh f )(x)

∂−jh (g(x)f(x)) = (∂−j

h g)(x)f(x − hej ) + g(x)(∂−jh f)(x).

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su(1, 1)symmetries

hZn ={

x = (x1, . . . , xn) ∈ Rn :xh∈ Zn

(∂+jh f)(x) =

f(x + hej )− f(x)

h, (∂−j

h f)(x) =f(x)− f(x − hej )

h f)(x) = f(x ± hej ) i.e.

T−jh (∂+j

h (∂−jh f)(x) = (∂+j

h f)(x).

∂+jh (g(x)f(x)) = (∂+j

h g)(x)f(x + hej ) + g(x)(∂+jh f )(x)

∂−jh (g(x)f(x)) = (∂−j

h g)(x)f(x − hej ) + g(x)(∂−jh f)(x).

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su(1, 1)symmetries

hZn ={

x = (x1, . . . , xn) ∈ Rn :xh∈ Zn

(∂+jh f)(x) =

f(x + hej )− f(x)

h, (∂−j

h f)(x) =f(x)− f(x − hej )

h f)(x) = f(x ± hej ) i.e.

T−jh (∂+j

h (∂−jh f)(x) = (∂+j

h f)(x).

∂+jh (g(x)f(x)) = (∂+j

h g)(x)f(x + hej ) + g(x)(∂+jh f )(x)

∂−jh (g(x)f(x)) = (∂−j

h g)(x)f(x − hej ) + g(x)(∂−jh f)(x).

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Radial-type discretizationLie-algebraic formulation

Radial-type approach: Study of finite difference operators belonging tothe algebra

Alg {Lj ,Mj , ej : j = 1, . . . , n},

1 Lj and Mj are position and momentum operators, respectively,satisfying the set of Weyl-Heisenberg algebra relations[Lj , Lk ] = [Mj ,Mk ] = 0 and [Lj ,Mk ] = δjk I

2 e1, e2, . . . , en are the generators of the Clifford algebra of signature(0, n).

Multivector operators: Basic left endomorphisms acting that act onfunctions with values on C`0,n.

Multivector derivative: L =∑n

j=1 ejLj stands the Lie-algebraiccounterpart of the Dirac operator D =

∑nj=1 ej∂xj .

Multivector multiplication: M =∑n

j=1 ejMj stands theLie-algebraic counterpart for the left multiplication of f(x) by aClifford vector X =

∑nj=1 xj ej .

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Alg {Lj ,Mj , ej : j = 1, . . . , n},

∑nj=1 ej∂xj .

∑nj=1 xj ej .

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Alg {Lj ,Mj , ej : j = 1, . . . , n},

∑nj=1 ej∂xj .

∑nj=1 xj ej .

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Alg {Lj ,Mj , ej : j = 1, . . . , n},

∑nj=1 ej∂xj .

∑nj=1 xj ej .

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An IntermezzoQuantum Field Theory (QFT) setting

Basic polynomial sequences (or quasi-monomials) may be constructedexplicitly by a direct application of the Quantum Field Lemma.

Fock space: Vector space (F , 〈·|·〉) such that1 F : Free algebra generated by the elements a−j and a+

j fromthe vacuum vector Φ such that a−j Φ = 0.

2 〈·|·〉: Euclidean inner product in F such that 〈Φ|Φ〉 = 1 and a+j

are adjoint to a−j , i.e. 〈a+j x |y〉 = 〈x |a−j y〉.

Bose algebra: Fock space F whose generators a±j a†j satisfy theHeisenberg-Weyl relations

[a+j , a

+k ] = 0, [a−j , a

−k ] = 0, [a−j , a

+k ] = δjk I.

Standard lemma in QFT: All the basic vectors in F have thefollowing form

ηα :=

n∏j=1

(a†j )αj

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su(1, 1)symmetries

[a+j , a

+k ] = 0, [a−j , a

−k ] = 0, [a−j , a

+k ] = δjk I.

ηα :=

n∏j=1

(a†j )αj

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su(1, 1)symmetries

[a+j , a

+k ] = 0, [a−j , a

−k ] = 0, [a−j , a

+k ] = δjk I.

ηα :=

n∏j=1

(a†j )αj

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su(1, 1)symmetries

[a+j , a

+k ] = 0, [a−j , a

−k ] = 0, [a−j , a

+k ] = δjk I.

ηα :=

n∏j=1

(a†j )αj

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su(1, 1)symmetries

[a+j , a

+k ] = 0, [a−j , a

−k ] = 0, [a−j , a

+k ] = δjk I.

ηα :=

n∏j=1

(a†j )αj

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Canonical discretization of finite differenceoperatorsForward/Backward differences:

First Expansion Theorem: ∂±jh = ± 1

(exp(±h∂xj )− I

Falling/Rising factorials:∏nj=1

(xjT∓jh

)αj1 =

∏nj=1 xj (xj ∓ h) . . . (xj ∓ (αj − 1)h).

Figure: Chromatic polynomial: The falling factorial counts the numberof ways to color the complete graph of order |α| with exactly

∏nj=1 xj

colors.

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Canonical discretization of finite differenceoperatorsForward/Backward differences:

First Expansion Theorem: ∂±jh = ± 1

(exp(±h∂xj )− I

Falling/Rising factorials:∏nj=1

(xjT∓jh

)αj1 =

∏nj=1 xj (xj ∓ h) . . . (xj ∓ (αj − 1)h).

Figure: Chromatic polynomial: The falling factorial counts the numberof ways to color the complete graph of order |α| with exactly

∏nj=1 xj

colors.

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Canonical discretization of finite differenceoperators of hypercomplex typeA list of examples

1 Forward finite differences: The set of operators ∂+jh and

xjT−jh : f(x) 7→ xj f(x − hej ) span the Weyl-Heisenberg algebra of

dimension 2n + 1. Moreover D+h =

∑nj=1 ej∂

+jh and

Xh =∑n

j=1 ejxjT−jh are the corresponding multivector ladder

operators on the lattice hZn.2 Backward finite differences: ∂−j

h and xjT+jh : f(x) 7→ xj f(x + hej )

also span the Weyl-Heisenberg algebra of dimension 2n + 1. Thisturns out D−h =

∑nj=1 ej∂

−jh and X−h =

∑nj=1 ejxjT

+jh as the

corresponding multivector ladder operators on the lattice hZn.3 Discretization of the Hermite operator: D+

h and Xh − D−h areobtained from the set of ladder operators Lj = ∂+j

h andLj = xjT

−jh − ∂

−jh . Moreover Xh − D−h generate hypercomplex

extensions of the Poisson-Charlier polynomials (cf. N.F., Appl.Math. Comp., Vol. 247, pp. 607-622, 2014).

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su(1, 1)symmetries

∑nj=1 ej∂

+jh and

Xh =∑n

∑nj=1 ej∂

−jh and X−h =

∑nj=1 ejxjT

+jh as the

h andLj = xjT

−jh − ∂

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∑nj=1 ej∂

+jh and

Xh =∑n

∑nj=1 ej∂

−jh and X−h =

∑nj=1 ejxjT

+jh as the

h andLj = xjT

−jh − ∂

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Quasi-Monomiality ApproachThe Exponential Generating Function (EGF) approach

Many degrees of freedom for choose discretization operators:(cf. N.F., SIGMA 9 (2013), 065) The set of operators(xj + h

h : f(x) 7→(xj + h

)f(x + hej ) and(

xj − h2

)T−j

h : f(x) 7→(xj − h

)f(x − hej ) satisfy[

∂−jh ,

[∂+j

(xk −

)T−k

]= δjk I

cf. N. F. Appl. Math. Comp., 2014

The EGF of the form

Gh(x , y ;κ) =∏n

h log (1 + hyj )) (1 + hyj )

yield the set of operators Lj = ∂+jh and Mj =

(xj − κ′(∂xj )κ

(∂xj

)−1)

T−jh

as generators of the Weyl-Heisenberg algebra of dimension 2n + 1.Moreover, they are unique.

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Proposition (N.F., Appl. Math. Comp., 2014)

Let κ(t) defined as above and Xh the multiplication operator. If there is amulti-variable function λ(y) (y ∈ Rn) such that

h exp(x · y)

exp(x · y)

n∏j=1

κ(yj )

then the Fourier dual Λh of D+h is given by

Λh = Xh −[log λ

), x].

Quasi-Monomiality formulation: Based on Fock space formalism onecan construct each Clifford-vector-valued polynomial wk (x ; h;λ) of orderk by means of the operational rule

wk (x ; h;λ) = µk (Λh)k a, a ∈ C`0,n.

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Appell set definition: {wk (x ; h;λ) : k ∈ N0} is an Appell setcarrying D+

h if w0(x ; h;λ) = a is a Clifford number and D+h wk (x ; h;λ)

is a Clifford-vector-valued polynomial of degree k − 1 satisfyingD+

h wk (x ; h;λ) = kwk−1(x ; h;λ).

Appell set equivalent formulation: Find for each (x , t) ∈ hZn × Ra EGF Gh(x , t ;λ) satisfying the set of equations

D+h Gh(x , t ;λ) = tGh(x , t ;λ) for (x , t) ∈ hZn × R \ {0}

Gh(x , 0;λ) = a for x ∈ hZn.

Bessel type hypergeometric functions:

Gh(x , t ;λ) = 0F1

;− t2

4(Λh)2

)a + tΛh 0F1

+ 1;− t2

4(Λh)2

= Γ(n

)( tΛh

)− n2 +1 (

J n2−1(tΛh)a + n J n

2(tΛh)

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su(1, 1)symmetries

h wk (x ; h;λ) = kwk−1(x ; h;λ).

Gh(x , t ;λ) = 0F1

;− t2

4(Λh)2

)a + tΛh 0F1

+ 1;− t2

4(Λh)2

= Γ(n

)( tΛh

)− n2 +1 (

2(tΛh)

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su(1, 1)symmetries

h wk (x ; h;λ) = kwk−1(x ; h;λ).

Gh(x , t ;λ) = 0F1

;− t2

4(Λh)2

)a + tΛh 0F1

+ 1;− t2

4(Λh)2

= Γ(n

)( tΛh

)− n2 +1 (

2(tΛh)

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Why one needs su(1,1) based symmetries?The Weyl-Heisenberg symmetry breaking

Main Goal:

For a given polynomial w(t) of degree 1, with µ = ∂+jh w(xj ) = ∂−j

h w(xj ),study the spectra of the coupled eigenvalue problem

E+h f(x) = E−h f(x) = εf(x)

carrying E±h =∑n

j=1 µ−1w

(xj ± h

)∂±j

Drawback: The set of operators∂+j

h , ∂−jh ,W−j

h = µ−1w(xj + h

)T−j

h ,W+jh = µ−1w

(xj + h

h and I,with j = 1, 2, . . . , n, do not endow a canonical realization of anWeyl-Heisenberg type algebra of dimension 4n + 1.

Fill the Weyl-Heisenberg gap: The set of operatorsW−j

h = µ−1w(xj + h

)T−j

h ,W+jh = µ−1w

(xj + h

h andWj = µ−1w (xj ) I generate a Lie algebra isomorphic to sl(2n,R)(N.F., SIGMA 9 (2013), 065– Lemma 1).

20 / 34

Nelson Faustino

Appell Sets

su(1, 1)symmetries

Main Goal:

carrying E±h =∑n

j=1 µ−1w

(xj ± h

)∂±j

h , ∂−jh ,W−j

h = µ−1w(xj + h

)T−j

h ,W+jh = µ−1w

(xj + h

h = µ−1w(xj + h

)T−j

h ,W+jh = µ−1w

(xj + h

20 / 34

Nelson Faustino

Appell Sets

su(1, 1)symmetries

Main Goal:

carrying E±h =∑n

j=1 µ−1w

(xj ± h

)∂±j

h , ∂−jh ,W−j

h = µ−1w(xj + h

)T−j

h ,W+jh = µ−1w

(xj + h

h = µ−1w(xj + h

)T−j

h ,W+jh = µ−1w

(xj + h

20 / 34

Nelson Faustino

Appell Sets

su(1, 1)symmetries

Application to Cauchy problemsN.F. SIGMA, 2013

Homogeneous Cauchy problem in [0,∞)× hZn:

∂tg(t , x) + E+

h g(t , x)− E−h g(t , x) = 0 , t > 0g(0, x) = f(x) , t = 0E+

h g(t , x) = E−h g(t , x) , t ≥ 0.

Semigroup action: The one-parameter representationEh(t) = exp(tE−h − tE+

h ) of the Lie group SU(1, 1) yieldsg(t , x) = Eh(t)f(x) as a polynomial solution of the abovehomogeneous Cauchy problem.

Discrete series connection: One can see that the semigroup(Eh(t))t≥0 gives, in particular, a direct link between the positiveseries representation of SU(1, 1) with the negative ones.

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Nelson Faustino

Appell Sets

su(1, 1)symmetries

∂tg(t , x) + E+

h g(t , x) = E−h g(t , x) , t ≥ 0.

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Appell Sets

su(1, 1)symmetries

∂tg(t , x) + E+

h g(t , x) = E−h g(t , x) , t ≥ 0.

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Nelson Faustino

Appell Sets

su(1, 1)symmetries

Discrete electromagnetic Schrodingeroperators on the lattice

Discrete electromagnetic Schrodinger operators Lh on hZn

Lhf(x) =1

n∑j=1

qhf(x)− ah(xj )f(x + hej )− ah(xj − h)f(x − hej )

)+ q Φh(x)f(x).

µ - mass

q- electric charge

ah(x) =n∑

ejah(xj ) - discrete magnetic potential.

Φh(x)- electric potential.

22 / 34

Nelson Faustino

Appell Sets

su(1, 1)symmetries

Lhf(x) =1

n∑j=1

)+ q Φh(x)f(x).

µ - mass

q- electric charge

ah(x) =n∑

22 / 34

Nelson Faustino

Appell Sets

su(1, 1)symmetries

Lhf(x) =1

n∑j=1

)+ q Φh(x)f(x).

µ - mass

q- electric charge

ah(x) =n∑

22 / 34

Nelson Faustino

Appell Sets

su(1, 1)symmetries

Lhf(x) =1

n∑j=1

)+ q Φh(x)f(x).

µ - mass

q- electric charge

ah(x) =n∑

22 / 34

Nelson Faustino

Appell Sets

su(1, 1)symmetries

Lhf(x) =1

n∑j=1

)+ q Φh(x)f(x).

µ - mass

q- electric charge

ah(x) =n∑

22 / 34

Nelson Faustino

Appell Sets

su(1, 1)symmetries

Asymptotic approximation of a Sturm-Liouvilleproblem

Lhf(x) = − h2

n∑j=1

)∂f∂xj

)f(x) + O

The above asymptotic approximation is satisfied whenever:1 Asymptotic constraint associated to the discrete magnetic

potential:

ah(x) =n∑

)(1 + O (h)) .

2 Asymptotic constraint associated to the discrete magneticpotential:

qΦh(x) +1

n∑j=1

qh− ah(xj )− ah(xj − h)

23 / 34

Nelson Faustino

Appell Sets

su(1, 1)symmetries

Lhf(x) = − h2

n∑j=1

)∂f∂xj

)f(x) + O

potential:

ah(x) =n∑

)(1 + O (h)) .

qΦh(x) +1

n∑j=1

23 / 34

Nelson Faustino

Appell Sets

su(1, 1)symmetries

Lhf(x) = − h2

n∑j=1

)∂f∂xj

)f(x) + O

potential:

ah(x) =n∑

)(1 + O (h)) .

qΦh(x) +1

n∑j=1

23 / 34

Nelson Faustino

Appell Sets

su(1, 1)symmetries

The factorization approachIsospectral relations

Pair of ladder operators

Consider now the pair (A+h ,A

−h ) of ladder operators defined

componentwise by

n∑j=1

ejA+jh with A+j

√qh4µ

(ah(xj )T

+jh −

A−h =n∑

ejA−jh with A−j

√qh4µ

qhI − ah(xj − h)T−j

From the assumption that the vacuum vector ψ0(x ; h) = φ(x ; h)s(s ∈ Pin(n)) annihilated by A+

h , it readily follows that the pair (A+h ,A

−h ) is

isospectral equivalent to the pair (D+h ,Mh), with

D+h f(x) =

n∑j=1

ej∂+jh , Mh =

n∑j=1

(hah(xj − h)2T−j

h −4

q2hI).

24 / 34

Nelson Faustino

Appell Sets

su(1, 1)symmetries

The factorization approachquasi-monomials vs. bound states of the discrete electromagnetic Schrodingeroperators

Moreover, in case where the discrete electric and magnetic potentials aregiven by

Φh(x) =h

n∑j=1

(φ(x ; h)2

φ(x + hej ; h)2 +φ(x − hej ; h)2

φ(x ; h)2

ah(x) =n∑

qhφ(x ; h)

φ(x + hej ; h)

it follows straightforwardly from the factorization propertyLh = 1

2 (A+h A−h + A−h A+

h ) that Lh is isospectral equivalent to theanti-commutator MhD+

h + D+h Mh.

Indeed, the isospectral formula

φ(x ; h)−1Lh(φ(x ; h)f(x) = − q4µh

(MhD+h + D+

yields naturally from the combination of the factorization property withthe aforementioned isospectral relations.

25 / 34

Nelson Faustino

Appell Sets

su(1, 1)symmetries

A sketch of a recent paper of minepublished recently at Applied Mathematics and Computation

We have shown that Lh is asymptotically equivalent to thediscrete harmonic oscillator − 1

2m ∆h + qΦh(x) with mass m ∼ µqh

,whose kinetic term is written in terms of the ’discrete’ Laplacian ∆h.

For the particular choice ah(xj ) =1q

(1h + µ

)it readily

follows that that the asymptotic expansion of Lh reduces to

Lhf(x) = − 12µq

(E+h f(x)− E−h f(x)) + V

)f(x),

with V(x

)= −

n∑j=1

h+ qΦh(x). Hereby E±h corresponds to the

forward/backward counterpart of the radial derivative

E =n∑

xj∂xj , carrying the polynomial w(xj ) = 1 + µxj .

26 / 34

Nelson Faustino

Appell Sets

su(1, 1)symmetries

We have shown that Lh is asymptotically equivalent to thediscrete harmonic oscillator − 1

2m ∆h + qΦh(x) with mass m ∼ µqh

,whose kinetic term is written in terms of the ’discrete’ Laplacian ∆h.

For the particular choice ah(xj ) =1q

(1h + µ

)it readily

follows that that the asymptotic expansion of Lh reduces to

Lhf(x) = − 12µq

(E+h f(x)− E−h f(x)) + V

)f(x),

with V(x

)= −

n∑j=1

h+ qΦh(x). Hereby E±h corresponds to the

forward/backward counterpart of the radial derivative

E =n∑

xj∂xj , carrying the polynomial w(xj ) = 1 + µxj .

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Nelson Faustino

Appell Sets

su(1, 1)symmetries

From the Energy condition associated to the Fock space Fh

endowed by the Clifford module `2(hZn; C`0,n) := `2(hZn)⊗ C`0,n,the vacuum vectors of the form ψ0(x ; h) = φ(x ; h)s (s ∈ Pin(n)), thequantity

n∑j=1

ejXj = x

= hnψ0(x ; h)†ψ0(x ; h)

may be regarded as a discrete quasi-probability law on hZn,carrying a set of independent and identically distributed (i.i.d.)random variables X1,X2, . . . ,Xn.

We have used of the Bayesian probability framework towardsDirac’s insight on quasi-probabilities (they may take negative values)to compute some examples involving the well-known Poisson andhypergeometric distributions, likewise quasi-probability distributionsinvolving the generalized Mittag-Leffler/Wright functions. In some ofthe cases they may take negative values.

27 / 34

Nelson Faustino

Appell Sets

su(1, 1)symmetries

From the Energy condition associated to the Fock space Fh

endowed by the Clifford module `2(hZn; C`0,n) := `2(hZn)⊗ C`0,n,the vacuum vectors of the form ψ0(x ; h) = φ(x ; h)s (s ∈ Pin(n)), thequantity

n∑j=1

ejXj = x

= hnψ0(x ; h)†ψ0(x ; h)

may be regarded as a discrete quasi-probability law on hZn,carrying a set of independent and identically distributed (i.i.d.)random variables X1,X2, . . . ,Xn.

We have used of the Bayesian probability framework towardsDirac’s insight on quasi-probabilities (they may take negative values)to compute some examples involving the well-known Poisson andhypergeometric distributions, likewise quasi-probability distributionsinvolving the generalized Mittag-Leffler/Wright functions. In some ofthe cases they may take negative values.

27 / 34

Nelson Faustino

Appell Sets

su(1, 1)symmetries

ExamplesGeneralization of the so-called Poisson distribution

hnφ(x ; h)2 =

Eα,β(

4q2−αh2

)−1 4xjh q

(2−α)xjh h−

Γ(β + α

) , if x ∈ hZn≥0

0 , otherwise

As a matter of fact, the Mittag-Leffler function

Eα,β(λ) =∑∞

m=0λm

Γ(β + αm)is well defined for Re(α) > 0, Re(β) > 0.

1 Discrete Electric Potential:

Φh(x) =h

n∑j=1

Γ(α + β + α

)Γ(β + α

) +Γ(β + α

)Γ(β − α + α

2 Discrete Magnetic Potential:

ah(x) =n∑

√√√√√ 1qα

Γ(α + β + α

)Γ(β + α

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Nelson Faustino

Appell Sets

su(1, 1)symmetries

ExamplesGeneralization of the so-called Poisson distribution

hnφ(x ; h)2 =

Eα,β(

4q2−αh2

)−1 4xjh q

(2−α)xjh h−

Γ(β + α

) , if x ∈ hZn≥0

0 , otherwise

As a matter of fact, the Mittag-Leffler function

Eα,β(λ) =∑∞

m=0λm

Γ(β + αm)is well defined for Re(α) > 0, Re(β) > 0.

1 Discrete Electric Potential:

Φh(x) =h

n∑j=1

Γ(α + β + α

)Γ(β + α

) +Γ(β + α

)Γ(β − α + α

2 Discrete Magnetic Potential:

ah(x) =n∑

√√√√√ 1qα

Γ(α + β + α

)Γ(β + α

28 / 34

Nelson Faustino

Appell Sets

su(1, 1)symmetries

Appell Set PropertyThe Poisson-Charlier connection

1 Remark: For the special choices q = 2 and λ = 1h2 , the resulting

ladder operator that yields from the Mittag-Leffler distribution

Mh =n∑

((xj +

)T−j

h −1h

corresponds to a finite difference

approximation of the Clifford-Hermite operator

xI − D = − exp

(|x |2

)D exp

(−|x |

2 The resulting quasi-monomials generated from the operationalformula mk (x ; h) = µk (Mh)k s (s ∈ Pin(n)), carrying the constants

µ2m = (−1)m

(k = 2m) and µ2m+1 =

(−1)m

2 + 1)

(k = 2m + 1)

possess the Appell set property D+h mk (x ; h) = kmk−1(x ; h). They

correspond to an hypercomplex extension of the Poisson-Charlierpolynomials in disguise.

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Nelson Faustino

Appell Sets

su(1, 1)symmetries

Appell Set PropertyThe Poisson-Charlier connection

1 Remark: For the special choices q = 2 and λ = 1h2 , the resulting

ladder operator that yields from the Mittag-Leffler distribution

Mh =n∑

((xj +

)T−j

h −1h

corresponds to a finite difference

approximation of the Clifford-Hermite operator

xI − D = − exp

(|x |2

)D exp

(−|x |

2 The resulting quasi-monomials generated from the operationalformula mk (x ; h) = µk (Mh)k s (s ∈ Pin(n)), carrying the constants

µ2m = (−1)m

(k = 2m) and µ2m+1 =

(−1)m

2 + 1)

(k = 2m + 1)

possess the Appell set property D+h mk (x ; h) = kmk−1(x ; h). They

correspond to an hypercomplex extension of the Poisson-Charlierpolynomials in disguise.

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Nelson Faustino

Appell Sets

su(1, 1)symmetries

ExamplesThe Generalized Wright distribution

n∏j=1

[(δ, γ)(β, α)

q1+γ−αh2

Γ(δ + γ

)Γ(β + α

) ααxj

h γ−γxj

h 4xjh q−

(1+γ−α)xjh h−

xjh + 1

) , x ∈ hZn≥0

0 , otherwise

1 Notice that the Wright series 1Ψ1

[(δ, γ)(β, α)

]is absolutely

convergent for |λ| < αα

γγand of |λ| =

γγ, Re(β)− Re(δ) > 1

h2 >γ2γ

4q1+γ−α and of h2 =

4q1+γ−α , Re(β)− Re(δ) > 1

whenever α− γ = −1.

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Nelson Faustino

Appell Sets

su(1, 1)symmetries

Generalized Wright distributionsSome additional remarks

For γ = δ = 1, the aforementioned likelihood function is theMittag-Leffler distribution in disguise. Moreover, if α = Re(α) > 0,

α→ 0+ and h >2q

, the previous distribution simplifies to

hnφ(x ; h)2 =

(1− 4

q−2xjh h−

2xjh , if x ∈ hZn

0 , otherwise

For β = δ, the likelihood function amalgamates the Poissondistribution (α = γ = 1) as well as the orthogonal measure that

gives rise, up to the constant(

1− 4q2h2

)−βn, to the

hypergeometric distribution on hZn≥0, carrying the parameter

λ = 4q2h2 (α→ 0+, γ = 1 and h > 2

31 / 34

Nelson Faustino

Appell Sets

su(1, 1)symmetries

Generalized Wright distributionsSome additional remarks

For γ = δ = 1, the aforementioned likelihood function is theMittag-Leffler distribution in disguise. Moreover, if α = Re(α) > 0,

α→ 0+ and h >2q

, the previous distribution simplifies to

hnφ(x ; h)2 =

(1− 4

q−2xjh h−

2xjh , if x ∈ hZn

0 , otherwise

For β = δ, the likelihood function amalgamates the Poissondistribution (α = γ = 1) as well as the orthogonal measure that

gives rise, up to the constant(

1− 4q2h2

)−βn, to the

hypergeometric distribution on hZn≥0, carrying the parameter

λ = 4q2h2 (α→ 0+, γ = 1 and h > 2

31 / 34

Nelson Faustino

Appell Sets

su(1, 1)symmetries

BibliographyMain references used in my talk

Faustino, Nelson (2013). Special Functions ofHypercomplex Variable on the Lattice Based on SU (1, 1).Symmetry, Integrability and Geometry: Methods andApplications 9, no. 0 : 65-18.

Faustino, N. (2014). Classes of hypercomplex polynomials ofdiscrete variable based on the quasi-monomiality principle.Applied Mathematics and Computation, 247, 607-622.

Faustino, N. (2017). Hypercomplex Fock states for discreteelectromagnetic Schrodinger operators: A Bayesianprobability perspective. Applied Mathematics andComputation, 315, 531-548.

Faustino, Nelson (2017). Symmetry PreservingDiscretization Schemes through Hypercomplex Variables,Conference: 15th International Conference of NumericalAnalysis and Applied Mathematics , DOI:10.13140/RG.2.2.35900.74882

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Nelson Faustino

Appell Sets

su(1, 1)symmetries

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Nelson Faustino

Appell Sets

su(1, 1)symmetries

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Nelson Faustino

Appell Sets

su(1, 1)symmetries

32 / 34

Nelson Faustino

Appell Sets

su(1, 1)symmetries

A final thought

”To first approximation, the human brain is a harmonic oscillator.”

Barry Simon to Charles Fefferman in a private conversation as theywalked around the Princeton campus.

Figure: Barry SimonFigure: Charles Fefferman

SimonFest 2006: Barry Stories–http://math.caltech.edu/SimonFest/stories.html]fefferman

33 / 34

Nelson Faustino

Appell Sets

su(1, 1)symmetries

A final thought

33 / 34

Nelson Faustino

Appell Sets

su(1, 1)symmetries

A final thought

33 / 34

Nelson Faustino

Appell Sets

su(1, 1)symmetries

Thank you for your attention!

The author would like to thank the organizers of the ICNAAM2017 for their kind invitation and for the financial support as well.

Figure: Pictures from my university, located at ABC Paulista (Sao Paulo,Brazil)

34 / 34

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