Critical graph of a polynomial quadratic differential related to a Schrödinger equation with quartic potential

Critical graph of a polynomial quadratic differential related to a Schrödinger equation with quartic potential

Mondher Chouikhi and Faouzi Thabet
University of Gabès, Tunisia
Abstract

In this paper, we study the weak asymptotic in the plane of some wave functions resulting from the WKB techniques applied to a Shrodinger equation with quartic oscillator and having some boundary condition. In first step, we make transformations of our problem to obtain a Heun equation satisfied by the polynomial part of the WKB wave functions .Especially , we investigate the properties of the Cauchy transform of the root counting measure of a re-scaled solutions of the Schrodinger equation, to obtain a quadratic algebraic equation of the form , where are also polynomials. In second step, we discuss the existence of solutions (as Cauchy transform of a signed measures) of this algebraic equation.This problem remains to describe the critical graph of a related 4-degree polynomial quadratic differential . In particular, we discuss the existence(and their number) of finite critical trajectories of this quadratic differential.


2010 Mathematics subject classification: 30C15, 31A35, 34E05.

Keywords and phrases: Quantum theory. WKB analysis. Quadratic differentials. Cauchy transform.

1
Introduction

Quadratic differentials appear in different mathematical and mathematical-physical domains, such as, orthogonal polynomials, potential theory, ordinary differential equations, quantum theory, moduli spaces …

Recently, quadratic differentials have provided an important tool in the asymptotic study of some solutions of algebraic equations.

In quantum theory, trajectories of some quadratic differentials have crucial role in WKB analysis, more precisely, consider the time-independent Schrödinger equation on the complex plane

(1)

where denotes the constant of Planck, the local coordinate, , the wave function, the potential, and , the energy of a system of mass The determination of an explicit solution of equation 1 is difficult in general case but a general solution can be written as a linear combination of particular solutions.The series expansion in of ) provide a principal part which defines a meromorphic quadratic differential in the complex plane . The critical graph of (the closure of the union of critical trajectories of ) , called also Stokes graph, is important for the WKB analysis which gives an important method to determine particular solution of Schrödinger equation. General study is more detailed in [1] and [2], where they considered the ”classical” case The classical Schrödinger equation is

where

is the hamiltonian. It is obvious that is an eigenfunction of the operator . In quantum mechanics, is generally supposed to be hermitian; in fact this condition guarantees a crucial property of quantum theory : the fact that the energy levels are real. Clearly these energy levels are the eigenvalues of It is shown that the Hermeticity condition of is sufficient and not necessary; see [3]. In fact, under PT-symmetry condition ( space-time reflection ) Bender has shown in [4] that the hamiltonian (which is not hermitian) has a real spectre. PT-symmetry condition allows to study many of hamiltonian that were discarded before.

The PT-symmetry condition was at the origin of many works, especially the numerical and asymptotic studies of the spectrum of operators when is polynomial. Bender and Boettcher conjectured in [5] that when the eigenvalues of are all real and positive. Many works supported this conjecture, [6], [7], etc…

The first rigourous proof of reality and positivity of the eigenvalues of some non-self-adjoint hamiltonian was given by Dorey, Dunning and Tateo in [8]. However, there are some PT-symmetric hamiltonian with polynomial potentials producing non real eigenvalues, see [9]. This PT-symmetric condition of the hamiltonian with polynomial potentials can be expressed by

(2)

In this paper, we focus on a case of quartic ( is a polynomial of degree ) PT-symmetric hamiltonian and we study the weak asymptotic of its spectrum. Starting from a Schrödinger differential equation and using the Cauchy transform defined in (12) we get algebraic equation (6), which gives rise to a particular quadratic differential.

More precisely, we consider the eigenvalue problem

(3)

where is real, is integer and we choose see [3],[10],[11]. This problem is a quasi-exactly solvable : for each there are eigenvalues , with elementary eigenfunctions

and are polynomials of degree at most .

Under condition (2), the potential

is PT-symmetric. With the rotation we obtain the following system

(4)

with and

For the sick of simplicity, we denote by and by . Substituting in (4), we obtain

(5)

It’s clear that are also the eigenfunctions of the operator

associated to the eigenvalue

Let us denote by the cauchy transform of where is the polynomial part of the eigenfunction The eigenvalue problem (3) has infinitely eigenvalues tending to infinity, so, in order to study the asymptotic of the sequence we shall consider a re-scaled polynomial of and

Substituting in (5), we get that

and

In particular, for we get

It was shown in [10], that is bounded. By the Helly selection theorem, we may assume that

and then, there exists a compactly supported positive measure such that

Finally, we obtain the algebraic equation :

(6)

where we are looking for those solutions as Cauchy transform of some compactly-supported Borelian positive measure in the complex plane ; any connected curve of the support of such a measure (if exists) coincides with a finite critical trajectory of a quadratic differential on the Riemann sphere of the form where is polynomial of degree (Subsection 2.2). The general possible configurations of the critical graph of this kind of quadratic differential are will known (see ,,), but the main goal of this work (besides the precise description of the critical graph) is the investigation of the number of finite critical trajectories when is a real polynomial, (Subsection 2.1).

We notice that if we start with the eigenvalue problem

then we analyse the case when The case is studied in [11].

2 Quadratic differentials


Below, we describe the critical graphs of the family of quadratic differentials on the Riemann sphere

where is a monic real polynomial of degree

We begin our investigation by some immediate observations from the theory of quadratic differentials. For more details, we refer the reader to [25],[26]

Recall that finite critical points of the polynomial quadratic differential are its zeros and simple poles; poles of order or greater then called infinite critical points. All other points of are called regular points.

With the parametrization , we get

thus, infinity is an infinite critical point of as a pole of order and respectively. Horizontal trajectories (or just trajectories) of the quadratic differential are the zero loci of the equation

(7)

or equivalently

If is a horizontal trajectory, then the function

is monotone.

The vertical (or, orthogonal) trajectories are obtained by replacing by in equation (7). The horizontal and vertical trajectories of the quadratic differential produce two pairwise orthogonal foliations of the Riemann sphere .

A trajectory passing through a critical point of is called critical trajectory. In particular, if it starts and ends at a finite critical point, it is called finite critical trajectory or short trajectory, otherwise, we call it an infinite critical trajectory. A short trajectory is called unbrowken if it does not pass through any finite critical points except its two endpoints. The closure the set of finite and infinite critical trajectories, that we denote by is called the critical graph.

The local structure of the trajectories is as follow :

  • At any regular point, horizontal (resp. vertical) trajectories look locally as simple analytic arcs passing through this point, and through every regular point of passes a uniquely determined horizontal (resp. vertical) trajectory of these horizontal and vertical trajectories are locally orthogonal at this point.

  • From each zero with multiplicity of there emanate critical trajectories spacing under equal angles .

  • At the pole , there are asymptotic directions (called critical directions) spacing under equal angle , and a neighborhood of this pole, such that each trajectory entering stays in and tends to following one of the critical directions. These critical directions are ;

    See Figure 1.

Figure 1: Structure of the trajectories near a simple zero (left),double zero (center), and pole of order 8 (right)

We have the following observations :

  • If is a horizontal trajectory of , then, is monotone on


  • If two different trajectories are not disjoint, then their intersection must be a zero of the quadratic differential.

  • A necessary condition for the existence of a short trajectory connecting finite critical points of is the existence of a Jordan arc connecting them, such that

    (8)

    However, we will see that this condition is not sufficient.

  • Since the quadratic differential has only two poles, Jenkins Three-pole Theorem (see [25, Theorem 15.2]) asserts that the situation of the so-called recurrent trajectory (whose closure might be dense in some domain in ) cannot happen.

  • Any critical trajectory which is not finite diverges to following one of the directions described above.


A very helpful tool that will be used in our investigation is the Teichmüller lemma (see [25, Theorem 14.1]).

Definition 1

A domain in bounded only by segments of horizontal and/or vertical trajectories of (and their endpoints) is called -polygon.

Lemma 2 (Teichműller)

Let be a -polygon, and let be the critical points on the boundary of and let be the corresponding interior angles with vertices at respectively. Then

(9)

where are the multiplicities of and is the number of zeros of inside

2.1
Critical graph of

We focus on the case when is a real monic polynomial having two simple real zeros and two conjugate zeros. By a linear change of variables, we may assume that

where

Lemma 3

For any condition (8) is satisfied for the pairs of zeros and .

Lemma 4
  • The segment is always a short trajectory connecting

  • Two critical trajectories emanating from the same zero of cannot diverge to in the same direction.

  • No critical trajectory emanating from diverges to in the direction

  • Exactly one trajectory emanating from diverges to in the upper half plane, and it follows the direction

  • There are at most three short trajectories.


Let us consider the set

(10)

Then we have the following observations :

  • the choice of the square root does not play any role in the integral defining (10);

  • since is an real polynomial, it can be shown easily that set is symmetric with respect to the real and imaginary axis;

  • direct calculations show that the points

The following

Lemma 5

Let be the unique real defined by :

(11)

Then, we have

where we denote by

Remark 6

Numeric calculus gives the approximation

Lemma 7

The set is formed by Jordan arcs :

  • two curves emerging from and diverging respectively to infinity in

  • two curves emerging from and diverging respectively to infinity in

From Lemma 7, splits into 4 connected domains :

  • limited by and containing

  • limited by and containing

  • limited by and

  • limited by and See Figure 2.

Figure 2: Approximate plot of the curve

Then main result of this paper is the following :

Proposition 8

For any complex number the segment is a short trajectory of the quadratic differential Moreover,

  • For has an unbrowken short trajectory connecting and . splits into six half-plane domains and one strip domain; see Figure 3.

  • For has two short trajectories connecting to and to splits into six half-plane domains; see Figure 4.

  • For there is no short trajectory connecting and . splits into six half-plane domains and two strip domains; see Figure 5.

Figure 3: Structure of the trajectories near a simple zero (left),double zero (center), and pole of order 8 (right)
Figure 4: (center), and pole of order 8 (right)
Figure 5: Structure of the trajectories near a simple zero (left),double zero (center), and pole of order 8 (right)

Remark 9

  1. If and has no short trajectory, then splits into six half-plane domains and three strip domains. See Figure LABEL:10.

  2. If then the critical graph is symmetric with respect to the real axis. It is obvious that when with simple real zeros : then the segments and are two short trajectories of . See Figure 6.

Figure 6: Structure of the trajectories near a simple zero (left),double zero (center), and pole of order 8 (right)

2.2
The algebraic equation

The Cauchy transform of a compactly supported Borelian complex measure is the analytic function defined in supp by :

(12)

It satisfies:

(13)

and the inversion formula (which should be understood in the distributions sense):

In particular, the normalized root-counting measure of a complex polynomial of a complex polynomial of degree is defined for any compact set in by :

the Cauchy transform of is :

Let us consider the following algebraic equation :

(14)

In this subsection, we give the connection between investigate the existence of a compactly supported Borelian signed measure whose Cauchy transform satisfies almost everywhere in the complex plane where and are two polynomials with degree and at most

The measure (if exists) is called real mother-body measure of the equation (14).

The following proposition gives the connection between ..

Proposition 10

If equation (14) admits a real mother-body measure , then:

  • any connected curve in the support of coincides with a short trajectory of the quadratic differential

    (15)

    where is the discriminant of the quadratic equation (14).

3 Proofs

Proof of Lemma 3. Since is a real polynomial, then and get, after the change of variable in second integral :

Let us give a necessary condition to get two short trajectories joining two different pairs of zeros of in the general case. We write

and consider two disjoint oriented Jordan arcs and joining two different pairs zeros. We define the single-valued function in with condition

From the Laurent expansion at of

we deduce the residue

For we denote by and the limits from the and sides, respectively. (As usual, the side of an oriented curve lies to the left and the side lies to the right, if one traverses the curve according to its orientation.) Let

Since

we have

where is a closed contours encircling the and . After the contour deformation we pick up residue at We get, for any choice of the square roots

We conclude the necessary condition :

 

Proof of Lemma 4.

  • It is straightforward that for we have

  • Suppose that and are two such trajectories emanating from the same zero , spacing with angle Consider the -polygon with edges and and vertices and infinity. The right side of (9) can take only the values or while the left side is at list ; a contradiction.

  • If a critical trajectory emanating from diverges to in the direction then, also, a critical trajectory emanating from must diverge to in the direction these two critical trajectories will form with the segment and an -polygon and, again, this violates Lemma 2.

  • The result follows by combining (ii) and (iii), and using Lemma 2.

  • The case of four short trajectories can be discarded immediately by Lemma 2.

It is an immediate consequence from 2 that it cannot happen that two short trajectories connect two zeros of a holomorphic quadratic differential on the Riemann sphere.   

Proof of Lemma 7. The first observation is that :

(16)

indeed, if for some then

However, if we choose the argument in then for any we have

which gives a contradiction.

In order to prove that is a curve, and since we know that with the observation (16), it is sufficient to prove the result in the set

and the real functions and defined for in by:

In other, the set is defined by :

Observe that is differentiable in and is the only singular point of

We have

By the other hand, with the change of variable we get :

We have for any :

It follows that

We conclude that the set is a regular curve in by applying the Implicit Function Theorem to the function .   

Proof of Lemma 5.

From the power series expansion

we get for

It follows that

which gives the local structure of near

Writing with in the integral defining we get with the change of variable :

(17)

Taking the limits when equality (17) becomes

With the change of variable where

we get

It follows that

and then

which gives equation (11).

To prove the existence and uniqueness of solution of equation (11) in we need to study the function

It order to show that it is sufficient to show that

which is straightforward by studying the rational function

 

Proof of Proposition 8.

  1. Let us begin our proof by the first observation, that, if for some diverges to in the direction then there is no short trajectory connecting and .????

    Suppose that for some with there is no short trajectory connecting and Obviously, all critical trajectories emanating from stay in the upper half plane. Let us denote by the only critical trajectory emanating from and diverging to in the upper half plane. From Lemma 4, follows a certain direction . Then, we claim that otherwise, if then, at most two critical trajectories emanating from must diverge to in the same direction, which contradicts Lemma 4 again.

    We denote by and consider the set

    Since for any diverges to in the direction by continuity of the trajectories (in the Hausdorff metric), we conclude that (otherwise, the segment will be a short trajectory, which contradicts (8).then, for some for s set

  2. If there is no short trajectory connecting and for some then there is a critical trajectory that emanates from and diverges to in the same direction of From the behaviour of orthogonal trajectories at we take an orthogonal trajectory that hits and respectively in two points and (there are infinitely many such orthogonal trajectories ) We consider a path connecting and , formed by the part of from to , the part of from to and the part of from to the critical trajectory intersecting from to Then

    which contradicts the fact that

  3. vv

 


Proof of Proposition 10. Solutions of the quadratic equation (14) are

with some choice of the square root. If is a real mother-body measure of (14), then, condition (13) implies that

we conclude that

From [31, Theorem 1], and since the Cauchy transform of the positive measure coincides a.e. in with an algebraic solution of the quadratic equation (14), it follows that the support of this measure is a finite union of semi-analytic curves and isolated points (see [35]) . Let be a connected curve in the support of For we have

From Plemelj-Sokhotsky’s formula, we have

and then we get

which shows that is a horizontal trajectory of the quadratic differential on the Riemann sphere. For more details, we refer the reader to [28],[30],[32],[27].   

In the end, this analysis can be done in the case where is a real polynomial without real zeros: Suppose that with Then the quadratic differential has at list one short trajectory. Indeed, if no one of the trajectories emanating from is short, then, they must diverge all to in the different asymptotic directions in . It follows from Lemma 2 that at most two trajectories emanating from can diverge to in and then, with consideration of symmetry, the remaining one is a short trajectory that connects and See Figure 7.

Figure 7: Structure of the trajectories near a simple zero (left),double zero (center), and pole of order 8 (right)
Acknowledgement 11

Part of this work was carried out during a visit of F.T to the university of Stockholm, Sweden. We would to thank Professor Boris Shapiro for many helpful discussions.

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Mondher Chouikhi(chouikhi.mondher@gmail.com)

Faouzi Thabet(faouzithabet@yahoo.fr)

Institut supérieur des sciences appliquées et de technologie

de Gabès. Rue Omar Ibn Al khattab 6072. Gabès. Tunisia.

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