Generalized Pareto optimum and semi-classical spinors
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
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  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 :
When Braam and Duistermaat   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
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
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. 
in the framework of Born-Oppenheimer approximation, or ,  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  in dimension that the normal form is given by the symmetric matrix
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 :
Here are quasi-momenta, and the eigenvalues are explicitely given by
Energy vanishes at Dirac points (2 Dirac points per hexagonal cell). The linearization at the Dirac point is of the form
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  in (2+1)-D, that is obtained in the following way.
On the set of maps with Sobolev regularity , we consider the Lagrangian density:
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
switches from Lagrangian formulation to Hamiltonian formulation and leads to a Pseudo-differential Hamiltonian system (as a quadratic form in ) with principal symbol
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
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 .
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 : 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
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
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:
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”
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
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
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 : 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
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.
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”.
- Arnold V 1990 Singularities of Caustics and Wave Fronts, Kluwer, Dordrecht.
- Arnold V 1983 Geometrical methods in the theory of ordinary differential equations. Springer.
- Bensouissi A, M’hadbi N and Rouleux M 2011 Proc. “Days of Diffraction 2011”, Saint-Petersburg. p.39-44. IEEE 101109/DD.2011.6094362
- Bogaevsky I 2016 Int. Conf. Diff. Eq. Dyn. Systems, Suzdal, Russia,
- Bogaevsky I 2003 Moscow Math. J. Vol.3 807
- Braam P J and Duistermaat J J 1993 Indag. Math. N.S. Vol.4 407
- Braam P J and Duistermaat J J 1995 Panorama of Math. Banach Center Publ., Vol 34, 29
- Duncan K P and Györffy B L 2002 Annals of. Phys. Vol.298, 273.
- Fefferman C and Weinstein M 2012 J. American Math. Soc. 25(4), 1169
- Fermanian Kammerer Cl and Mehats F 2016 arXiv: 1605.02451v1
- Ivrii V Ya 1979 Siberian Math. J. Vol.20, 390
- Kaidi N and Rouleux M 1993 Comptes Rendus Acad. Sci. Paris, t.317, Série I, 359
- Mnëv N 1985 Russ. Math. Surv. Vol.40, 137
- Ralston J 1992 Astérisque 210, 263
- Rouleux M 1999 Adv. Part. Diff. Eq. 16, (M. Demuth, E. Schrohe, B. W. Schulze, ed.) Wiley-Vch, 123
- Shen W Non-cooperative and semi-cooperative differential games. Preprint.
- Smale S 1973 Dynamical Systems, M.Peixoto ed. Acad. Press, 531
- Wan H Y 1975 Topology Vol.14. Pergamon Press, 217