Generalized Pareto optimum and semi-classical spinors

# Generalized Pareto optimum and semi-classical spinors

## Abstract

In 1971, S.Smale presented a generalization of Pareto optimum he called the critical Pareto set. The underlying motivation was to extend Morse theory to several functions, i.e. to find a Morse theory for differentiable functions defined on a manifold of dimension . We use this framework to take a Hamiltonian to its normal form near a singular point of the Fresnel surface. Namely we say that has the Pareto property if it decomposes, locally, up to a conjugation with regular matrices, as , where has singularities of codimension 1 or 2, and is a regular Hermitian matrix (“integrating factor”). In particular this applies in certain cases to the matrix Hamiltonian of Elasticity theory and its (relative) perturbations of order 3 in momentum at the origin.

## 1 Matrix valued Hamiltonian systems and crossing of modes

Here we recall some well known facts about generic normal forms for a matrix Hamiltonian near a crossing of modes. Consider a real valued symmetric Hamiltonian . We may also replace by a more general symbol (in a suitable class) with asymptotic sum , and consider its semi-classical Weyl quantization

 Hw(x,hDx;h)U(x;h)=(2πh)−d∫∫ei(x−y)η/hH(x+y2,η;h)U(y)dydη

This defines an operator essentially self-adjoint on the space of square-integrable “spinors” . For simplicity we keep denoting by the principal symbol which actually plays the main role in the present analysis.

Let , and be the characteristic variety.

At points of , the polarisation space has positive dimension .

For , is a smooth hypersurface, and modulo an elliptic factor, we can reduce to a scalar symbol with simple characteristics; the corresponding operator, of real principal type, can again be locally conjugated to by an elliptic Fourier integral operator (FIO); this induces well-known polarization properties for solutions of the original system.

Effects which are truly specific for systems occur therefore when ; let be the singular part of .

For generic Hamiltonian , is a stratified set. Splitting off elliptic summands, can be (generically) reduced to a system. For a symmetric system , one should generically move 2 parameters to bring it to a matrix with an eigenvalue of multiplicity 2, and one more to bring this eigenvalue to 0; thus the top (larger) stratum of is of codim 3. On directions transverse to , looks like a quadratic cone. In the next item we review some known results, making these observations more precise.

a) Classical and semi-classical normal forms for generic

We first mention Arnold’s [1] normal forms for (generic) near the stratum of of codim 3: at a conical point (i.e. where ), there are symplectic coordinates such that modulo an elliptic factor, we can bring to one of the following expressions :

 Δ(x,ξ)≡p1q1−p22hyperbolic % conical point Δ(x,ξ)≡p21+q21−p22% elliptic conical point

When Braam and Duistermaat [6] [7] have further brought to its normal form near the stratum of of codim 3 at a conical point : Modulo elliptic summands, we can reduce to the case where is a real symmetric matrix, and in the sense of quadratic forms, i.e. by replacing by , where is elliptic, there are symplectic coordinates such that

 A∗HA=(−p1+p2q1p3q1p3p1+p2) hyperbolic conical point A∗HA=(p1−p2q2p3q2p3p1+p2) elliptic conical point

Moreover these normal forms extend naturally to quantization of Weyl symbols, composing by a suitable FIO microlocally defined near the conical point, and associated with the canonical change of coordinates . Namely, in the hyperbolic case

 U∗(A∗HA)wU=(−hDq1+hDq2q1hDq3q1hDq3hDq1+hDq2)

and similarly in the elliptic case.

Normal forms are useful for studying the Hamiltonian flow of or finding quasi-modes for in terms of classical functions. These normal forms are structurally stable, i.e. they are not affected by small perturbations of a generic in the topology. Even more precise results are available in 1-D, see e.g. [15] in the framework of Born-Oppenheimer approximation, or [8], [3] for Bogoliubov-de Gennes Hamiltonian.

b) Particular cases: , codim 3 singularities

In many physical situations however, the genericity assumption is not fulfilled, and special reductions should be carried out :

Conical refraction in 3-D Recall from [11] in dimension that the normal form is given by the symmetric matrix

 H=(p1+p2p3p3p1−p2) (1)

which is independent of space variables. This is not structurally stable, because the singular part of is involutive.

Graphene Hamiltonian in 2-D. The Hamiltonian is a complex Hermitian matrix [9]:

 H(p)=(0f(p1,p2)¯¯¯f(p1,p2)0)=|λ(p)|(0e−iθ(p)e−iθ(p)) (2)

Here are quasi-momenta, and the eigenvalues are explicitely given by

 λ(p)=±t√3+2cos(√3p1a)+4cos(√3p2a/2)cos(3p2a/2)

Energy vanishes at Dirac points (2 Dirac points per hexagonal cell). The linearization at the Dirac point is of the form

 H(p)=v(0k1−ik2k1+ik20),v=3t/(2a) (3)

c) Particular case: higher order singularities

Here we are concerned in the case where vanishes of order 4 at the conical point (leaving open the case . The physical example is the Hamiltonian quadratic in the momenta from Elasticity Theory [5] in (2+1)-D, that is obtained in the following way.

On the set of maps with Sobolev regularity , we consider the Lagrangian density:

 L((xy),(ϕtψt),(ϕxψx),(ϕyψy))=T−V=12(ϕ2t+ψ2t)−12(ϕ2x+ψ2y)−12A(x,y)(ϕy+ψx)2

Euler-Lagrange equations from the variational principle lead to extremal curves (rays) in the -space, and the set of points connected to the origin in by such a ray is called the “world front”. A matter of interest are the singularities of the world front.

Applying Fourier transformation with respect to

 ϕ↦ˆϕ, ϕx↦ξˆϕ, ϕy↦ηˆϕ, ϕt↦τˆϕ, etc\dots

switches from Lagrangian formulation to Hamiltonian formulation and leads to a Pseudo-differential Hamiltonian system (as a quadratic form in ) with principal symbol

 H=(τ200τ2)−(ξ2+A(x,y)η2A(x,y)ξηA(x,y)ξηA(x,y)ξ2+η2) (4)

Genericity properties for this Hamiltonian imply as above that the singular set is of codimension 3, so that it can be brought to one of Arnold-Braam-Duistermaat normal forms (in 2-D) near . However, genericity breaks down in the case of constant coefficient , which justify a direct approach.

In particular, for , the spatial part has determinant

 Δ(ξ,η)=12(ξ2+η2)2

which vanishes of order 4 at .

Our purpose, precisely in this case, is to provide a normal form for near . This could be achieved by a straightforward diagonalisation of , but our method carries naturally to (relative) perturbations of this Hamiltonian, depending on alone; more naively this example is intended as an illustration of the role played by the generalized Pareto set.

## 2 Generalized Pareto optimum

A central problem in Differential Calculus consists in maximizing a function: Morse theory on a smooth manifold provides a globalization of this problem.

Economists are rather concerned in maximizing simultenaously several “utility functions”, obtaining this way the notion of Pareto optimum in a free exchange economy.

In 1971, S.Smale presented a generalization of Pareto optimum he called the critical Pareto set [17]. The main mathematical motivation is to find a Morse theory for differentiable functions defined on a manifold of dimension . Note that this problem is distinct from this of relative extrema of a single function, which is relevant to Lagrange multipliers.

a) “Classical” Pareto optimum in a free exchange economy

Consider a free exchange economy consisting in consumers , and for each , let represent the (positive) amount of goods , with . We define the “commodity space” as . An unrestricted state of the economy is a point , but we may restrict to the affine space with total ressource of good .

Each consumer is supposed to have an utility function (generally an homogenous function of ). Thus consumer prefers to iff . The level sets , are called “indifference surfaces”.

One considers exchanges in which will increase each on . A state is called “Pareto optimal” iff there is no such that for all , and for some . If is not Pareto, is not economically stable. For Pareto optimum equals the usual notion of a (constrained) maximum of .

Physicists would replace everywhere the words “maximizing” by “minimizing”, and “total resource” by “total energy”.

b) Generalized Pareto optimum in the sense of Smale.

Here we do not only consider (joined) maxima, but also (joined) critical points.

Let be a smooth manifold, dim , and be smooth functions defined by (vector of “utilities”).

Let , where denotes the derivative (Jacobian) of at , and the set of with all .

Definition 1 [17]: We call the Pareto critical set.

Thus the relation means that there is no smooth curve starting at , and such that increases for all ’s (gradient flow dynamics). For , Pareto critical set is just the set of critical points of .

If is a (single) Morse function on , is a discrete set, and the Hessian is defined intrinsically on . For , need not be discrete; but when it consists, as is usual, of a submanifold of dim , then is still intrinsically defined on and valued in the 1-D space .

The open subset of stable points (classical Pareto set), which reduces for to the set of local maxima of , plays a special rôle.

Paradigm of Pareto critical in 2-D: the immersed Klein bottle in .

The paradigm of a Morse function on a compact 2-manifold is the “height function” on the embedded torus, and its (Pareto) critical points are the minimum, 2 saddles, and maximum. Similarly Pareto critical set for 2 functions of 2 variables will be obtained from Klein bottle [Wan], by projecting the immersed bottle in onto a suitable plane, and looking at the “joined extrema” of the coordinates functions on the projection plane. Another, more convenient immersion of Klein bottle in (though not so easy to visualize) is the so called “figure eight” or “bagel” immersion, given by

 x1=(r+cosθ2sinv−sinθ2sin2v)cosθ x2=(r+cosθ2sinv−sinθ2sin2v)sinθ x3=sinθ2sinv+cosθ2sin2v

with a parameter, are the variables. It is obtained by gluing two Möbius bands along their edges. Then the map with typical critical Pareto set is given by .

is a stratified set, consisting in a finite number of segments (codimension 1 strata) where rank, terminating at codimension 2 strata (isolated points) where rank.

We will only consider which leads to the simplest topology.

Elementary Pareto sets: the case of quadratic polynomials in 2-D.

As in the one-dimensional case, quadratic polynomials in provide useful examples of maps with a typical critical Pareto set. For we may take

 u1(x)=12(x21+x22), u2(x)=12(x21+x1x2+x22),Θ={0} u1(x)=x1x2, u2(x)=12(x21−x22),Θ={0} u1(x)=12(x21+x22), u2(x)=12(x21−x22),Θ={x1=0}

More generally, the critical Pareto set of two elliptic or two hyperbolic quadratic linearly independent polynomials reduces to the origin, and to a line for a “mixed” pair.

## 3 Matrix valued Hamiltonian systems with the Pareto property

Since we are interested in 2-spinors, we work with and .

Definition 2: Let be a (real) Hermitian matrix. We say that has the Pareto property iff there exists a smooth map (but with possibly degenerate derivative) , and a (regular) Hermitian matrix such that, locally, and up to conjugation with an elliptic factor of the form , we have:

 H(p)=u′(p)C(p)(u′(p))∗

Remark: A hint on this definition is the following: let such that , then the “pure classical state” , and similarly for when . So any “classical state” is a superposition of “classical pure states”.

These decompositions are local and sometimes can be checked only in the sense of germs. If exact, we say that is “integrable in Pareto sense”. Only in 1-D problems, one can consider general Hamiltonians of the form .

## 4 Pareto property and the quadratic Hamiltonian of Elasticity Theory

Because Hamiltonians depending on alone are not structurally stable, few Hamiltonian systems verify Pareto property. We have :

Theorem 1: For , the quadratic Hamiltonian of Elasticity Theory (4) is integrable in Pareto sense: with , , we have with . Let , we have the “skew-diagonalization”

 12σ2(H(ξ,η)−τ2)σ∗2=u′(ξ,η)(diag(12,0)−(ξ2+η2)−1τ2Id)(u′(ξ,η))∗ (5)

From this we can construct by inverse Fourier transformation semi-classical spinors that verify near “Helmholtz equation” corresponding to (4), standing for the energy parameter. We write for the Fourier transform w.r.t the space variables. There are two linearly independent solutions of the equation

 (diag(12,0)−(ξ2+η2)−1τ2)t(˜ϕ1,˜ψ1)=0

given by

 ϕ1(x,y,τ)=J0(√x2+y2√2τh),ψ1=J0(√x2+y2τh) (6)

and are derived from (5) by convolution integrals. From this it is standard to deduce the spectral properties of .

Relative stability of the Pareto property

Theorem 2: Let

 H(ξ,η)=(ξ2+12η212ξη12ξη12ξ2+η2)+O((ξ,η)3)

Then has the Pareto property (at least in the sense of germs at 0), with , . Moreover there is a skew-diagonalization of type (5), and we can find a set of solutions to the “Helmholtz equation” as in (6).

The proof goes as in Poincaré-Dulac theorem [2]: Namely we seek for a “new” of the form , with , and a “new” of the form . The upper-left matrix element of is given by , with , which has a resonance 2:1. Its lower-right matrix element is given by with . The off-diagonal terms involve the term . It turns out that we can solve (at least perturbatively) this system, the coefficients , , being determined to fulfill the compatibility conditions. The skew-diagonalization of type (5) follows from the fact that is close to a multiple of . We can still construct semi-classical solutions, and their asymptotics in for small , obtained by varying the argument of the Bessel functions.

## 5 Conclusion

We have made an attempt to extend the notion of “non-degenerate critical point” for a scalar Hamiltonian to the notion of “Pareto critical set” for a Hamiltonian system. We focussed to the case where reduces to a point. Our analysis applies to the Hamiltonian quadratic in momenta arising in Elasticity theory, for a particular value of the coefficients. It allows to account for the spectral properties of together with the semi-classical spectral asymptotics of (relative, i.e. depending again only on ) perturbations of near . These Hamiltonians present a codimension 3 singularity of order 4 at . One of the limitations of this approach is due to the fact that the dynamics associated with an matrix valued Hamiltonian depending on alone is of the type of a gradient-flow dynamics, while this fails to be the case for generic Hamiltonians. When the system has the time reversal property, we think that the notion of Nash equilibrium [Sh] associated with more general types of dynamics provided a better alternative to Pareto equilibrium.

Acknowledgements: I thank Ilya Bogaevsky for interesting discussions. This work has been partially supported by the grant PRC CNRS/RFBR 2017-2019 No.1556 “Multi-dimensional semi-classical problems of Condensed Matter Physics and Quantum Dynamics”.

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