On spectral asymptotic of quasi-exactly solvable quartic potential

Motivated by the earlier results of Masoero and De Benedetti (Nonlinearity 23:2501, 2010) and Shapiro et al. (Commun Math Phys 311(2):277–300, 2012), we discuss below the asymptotic of the solvable part of the spectrum for the quasi-exactly solvable quartic oscillator. In particular, we formulate a conjecture on the coincidence of the asymptotic shape of the level crossings of the latter oscillator with the asymptotic shape of zeros of the Yablonskii–Vorob’ev polynomials. Further we present a numerical study of the spectral monodromy for the oscillator in question.


Introduction
A quasi-exactly solvable quartic oscillator was introduced by C. M. Bender and S. Boettcher in [1] and (in its restricted form) is a Schrödinger-type eigenvalue problem of the form L J (w) = w − (x 4 /4 − ax 2 /2 − J x)w = λw (1.1) If J = n + 1 is a positive integer, then L n+1 (w) maps the linear space of quasiexactly solutions of the form { pe −x 3 /6+ax/2 : deg p ≤ n} to itself where p belongs to the linear space of univariate polynomials of degree at most n. The restriction of the operator L n+1 (w) to the latter linear space is a finite-dimensional linear operator whose spectrum and eigenfunctions can be found explicitly using methods of linear algebra. This part of the spectrum and eigenfunctions of (1.1) is usually referred to as solvable. (Observe that the operator L J (w) given by (1.1) has a negative spectrum while physicists usually prefer to work with −L J (w) whose spectrum is positive.) One can easily show that polynomial factors p in the quasi-exactly solutions w = pe h of (1.1) coincide with the polynomial solutions of the degenerate Heun equation where a ∈ C has the same meaning as above and (α, λ) are the spectral variables. 0 0 · · · 3 0 (n − 1)a n(n − 1) 0 0 · · · 0 2 0 na 0 0 · · · 0 0 1 0 (1.4) We will call the bivariate characteristic polynomial Sp n (a, λ) := det(M (a) n − λI ) the n-th spectral polynomial of (1.2). The degree of Sp n (a, λ) equals n + 1 which is also its degree with respect to the variable λ. (The maximal degree of the variable a in Sp n (a, λ) equals n+1 2 ). Additionally observe that for a ∈ R, Sp n (a, λ) is a real polynomial in λ and therefore the spectrum of M (a) n is symmetric with respect to the real axis.
The main goal of this paper is to study the asymptotic of the spectrum of M (a n ) n in two different regimes. In the first regime we require that lim n→∞ a n n 2/3 = 0 while in the second regime we require that lim n→∞ a n n 2/3 = A = 0. Remark. In case of a non-degenerate Heun equation detailed study of similar asymptotic was carried out in [23] and [26].

Theorem 1
(i) If lim n→∞ a n n 2/3 = 0, the maximal absolute value r n (a n ) of the eigenvalues of M (a n ) n grows as 3 4 n 4/3 , i.e., r n (a n ) = 3 4 n 4/3 (1 + o (1)). (ii) If lim n→∞ a n n 2/3 = 0, the sequence {μ (a n ) n } of eigenvalue-counting measures for the spectra of the sequence of matrices { 1 n 4/3 M (a n ) n } weakly converges to the measure ν 0 supported on the union of three straight intervals connecting the origin with three cubic roots of 27 64 , see Fig. 1.
More information about ν 0 can be found in Sect. 2. Our next result is obtained on the physics level of rigor, i.e., modulo three convergence assumptions which are explicated in the proof of Proposition 2, see Sect. 3. To formulate it, take the family of equations which we consider as quadratic equations in the variable C depending on the main variable x and additional parameters A and β.

Proposition 2
If lim n→∞ a n n 2/3 = A = 0, then (under three additional convergence assumptions given in the proof) the sequence {μ (a n ) n } of eigenvalue-counting measures for the sequence of matrices { 1 n 4/3 M (a n ) n } weakly converges to a special compactly supported probability measure ν A .
In particular, the support of ν A consists of all values of the spectral parameter β for which there exists a compactly supported in the x-plane probability measure κ whose Cauchy transform C κ (x) satisfies (1.5) in this plane almost everywhere.

Remark 1
Numerical experiments and some theoretical considerations indicate that if lim n→∞ a n n 2/3 does not exist then there is no non-trivial limiting behavior of the spectrum. We think that for lim n→∞ |a n | n 2/3 = +∞, one can rigorously prove the latter observation. However we are not trying to pursue this aspect in the present paper.

Remark 2
To make the paper self-contained, let us recall that for a complex-valued measure μ compactly supported in C, its logarithmic potential is defined as and its Cauchy transform is defined as

Remark 3
The existence of a signed (real, but not necessarily positive) measure μ whose Cauchy transform C μ (x) satisfies (1.5) almost everywhere in C is closely related to the properties of the family of quadratic differentials depending on parameters A and β. We will present some information about this connection in Sect. 3. (For an accessible introduction to quadratic differentials, see e.g. [25]).

Asymptotic distribution of the branching points of Sp n (a, ) and Yablonskii-Vorob'ev polynomials
To finish the introduction, let us formulate a number of conjectures supported by extensive numerical experiments. For a given positive integer n, denote by n the set of all branching points of the projection of the algebraic curve n (a) : {Sp n (a, λ) = 0} to the a-axis. In other words, n is the set of all values of the complex parameter a for which the matrix M (a) n has a multiple eigenvalue, i.e., Sp n (a, λ) has a multiple root. In physics literature such points are called level crossings. Obviously, one can describe n as the zero locus of the univariate discriminant polynomial D n (a) which is the resultant of Sp n (a, λ) and ∂ Sp n (a,λ) ∂λ with respect to λ. One can show that the degree of D n (a) equals n+1 2 . Further recall that Yablonskii-Vorob'ev polynomials {Q n } are defined as follows, see e.g. [28,29]. Set Q 0 = 1, Q 1 = t, and for n ≥ 1, define Although the latter expression a priori determines a rational function, Q n is in fact a polynomial of degree n+1 2 , see e.g. [27]. The importance of Yablonskii-Vorob'ev polynomials is explained by the fact that all rational solutions of the second Painlevé equation Denote by Z n the zero locus of Q n . (Good exposition of the properties of Q n together with several pictures of Z n can be found in [11]). Our conjectures below reveal an unexpected connection of n and Z n .

Remark 4
One can show that the maximal absolute value of points in Z n grows as For illustration of Conjecture 2 see Fig. 3. We hope that considerations similar to that in [17,18] can help to settle it. The set 40 of the branching points for Sp 40 (a, λ) together with the zero locus −Z 40 after scaling sending the three corners to the cubic roots of 1. The blue dots are the scaled roots of −Z 40 while the red ones are the scaled branching points of Sp 40 (a, λ). The bottom picture shows a fragment of the top one with substantial magnification. (Observe the surprising closeness; the distinction is hardly visible by a naked eye!)

Conjecture 3 When n → ∞, the points in the sequences
3n 2/3 Z n } asymptotically fill the same curvilinear triangular shape F, see Fig. 3.
The interior of F consists of all values a ∈ C for which the support of measure ν a introduced in Proposition 2 is a tripod, i.e., consists of three smooth segments with a common point, see Fig. 8 (left).
The complement of F consists of all values a ∈ C for which the support of ν a is a single smooth segment, see Fig. 8 (down). The boundary of F consists of those a ∈ C for which the support of ν a is a single curve with a singular point, i.e., a belongs to the boundary between the domain where this support is a tripod and the domain where the support is a smooth single curve, see Fig. 8

(right).
Conjecture 3 complements Conjecture 2. For the sequence {Z n } parts of the latter conjecture have been settled in [9], see also [3].

Remark 5
Data sharing not applicable to this article as no datasets were generated or analysed during the current study.
2 Case lim n→∞ a n n 2/3 = 0. Proof of Theorem 1 Let us start our considerations of the spectral asymptotic of sequences {M (a n ) n } with the simplest case a n = 0, n = 1, 2, 3, . . . .

Proposition 3
(i) The sequence Sp n (0, λ) splits into the following three subsequences: (1) for n+1 = 3 , the polynomial Sp n (0, λ) contains only the powers of λ divisible by 3; i.e., we have Sp n (0, λ) = u ( ) (λ 3 ); (2) for n + 1 = 3 + 1, one has Sp n (0, λ) = λv ( )   To prove Proposition 3 following the ideas of [16], we need to introduce a double indexed polynomial sequence containing our original sequence of characteristic polynomials {Sp n (0, λ)}. The most natural way to do this is to consider the principal minors of M (0) n − λI given by: Namely, denote by (k) n (λ) the k-th principal minor of (2.1). Then Since the latter matrix is 4-diagonal with one subdiagonal and two superdiagonals, then, by a general result of [21], its principal minors satisfy a 4-term linear recurrence relation. Simple explicit calculation gives where n is fixed and k runs from 1 to n + 1, with the standard initial conditions Notice that recurrence (2.2) has variable coefficients depending both on k and n. Another form of (2.2) which is a bit easier to study is as follows. To simplify our manipulations with the signs, let us instead of M with the initial conditions n , recurrence (2.3) can be rewritten as the system: We need to show that for any positive integer n, each polynomial in the recurrence (2.4) has positive coefficients and real negative roots. Moreover the roots of any two consecutive polynomials in each of the three sequences {U j=0 are strictly interlacing and are therefore simple. Positivity of the coefficients of polynomials is straightforward from the positivity of coefficients in (2.4) and the initial conditions. Let us settle the negativity of all roots by using induction on j. For j = 1 and n ≥ 3, it is trivial to check that the negative root of the U (1) n is larger than that of V (1) n which is larger than that of W (1) n . Let us consider the case j = 2. Note that U (2) 2 , which implies that the two roots of ξ W (1) n are non-positive and the only root of U (1) n is located strictly between them for n ≥ 3. Hence, then using Lemma 1.10 of [14] or conducting elementary calculations, we can conclude that U (2) n has simple negative roots for n ≥ 3. Using similar elementary calculations we obtain similar results for the pair U (2) n , V (1) n implying that V (2) n has simple negative roots and for the pair V (2) n , W (1) n implying that W (2) n has simple negative roots. Assume now that our hypothesis holds for a given positive integer j. Note that, in each of the three equations in (2.4), the degree of the first polynomial in the righthand side is bigger than the degree of the second polynomial by one. This indicates that the largest root belongs to the polynomial with the larger degree. Furthermore by Corollary 1.30 of [14], we can derive the following results: Here the arrow " ← " indicates that the corresponding pair of polynomials have simple interlacing roots with the largest root belonging to the polynomial at which the arrow points.
(a) Consider the recurrence We can rewrite ξ W n by induction hypothesis and Corollary 1.30 in [14]. Using (2.2), we can conclude that ξ V n , by Lemma 1.31 in [14]. Therefore U ( j+1) n has real and simple roots, by Lemma 1.10 in [14].
n only have negative and simple roots. Proof is concluded by the same argument as in part (a). (c) Consider the recurrence n by part (a), Corollary 1.30 in [14] and the fact that both U Let us now settle Theorem 1 in the special case a n = 0; n = 1, 2, 3, . . . .

Proof
In order to apply the approach of [16], we need the variable recurrence coefficients to stabilize when k n → τ , for any fixed τ ∈ [0, 1]. To get such stabilization, consider the rescaled matrix (2.9) obtained from (2.2) by substituting β = n −4/3 λ. In other words, (2.9) is satisfied by the characteristic polynomials of the principal minors of the matrix 1 n . Then when k n → τ, the latter recurrence transforms into the recurrence: with constant coefficients. The polynomial (2.11) Following [16] and using Proposition 3, we conclude that the Cauchy transform of the asymptotic root-counting measure for the polynomial sequence is obtained by averaging the Cauchy transforms of the asymptotic distributions of (2.10) over τ ∈ [0, 1]. In fact, in the case under consideration even the density of the former distribution can be obtained by averaging the densities of the latter family of distributions which we can confirm as follows.
Recurrence (2.10) is similar to the one considered in the last section of [4] and has very nice asymptotic distribution of its roots, see Fig. 4. Observe that for any τ ∈ [0, 1], the initial conditions for (2.10) are given by Since (2.10) has constant coefficients, the support of the asymptotic root-counting measure of its solution is described by the well-known result of Beraha-Cahane-Weiss [2]. Namely, this support coincides with the set of all β ∈ C such that among three solutions of the characteristic equation with respect to the variable two have the same modulus which is bigger or equal to the modulus of the third solution of (2.12). From considerations of [4] one can easily derive that, for any fixed τ ∈ [0, 1], this support is the union of three intervals starting at the origin and ending at the branching points of (2.12), i.e., those values of β and τ for which (2.12) has a multiple root with respect to . The latter branching points are given by the equation: and their location for three values of τ is shown in Fig. 4. Observe that if for τ ∈ [0, 1], we denote the branching point lying on the positive half-axis by β + (τ ), then it attains its maximal value when τ = 1/2 and this maximum equals 3 4 . As we mentioned before, in the case under consideration for any τ ∈ [0, 1], the roots of all polynomials generated by (2.10) lie on three fixed rays through the origin. Therefore, the density of the asymptotic root-counting measure of the polynomial sequence is obtained by averaging the densities of (2.10) over τ ∈ [0, 1], cf. [12] and [16]. Therefore, the support of the asymptotic root distribution for {Sp n (0, βn 4/3 )} in the β-plane is the union of three intervals of length 3 4 . Proof of Theorem 1 in case when lim n→∞ a n n 2/3 = 0, We need to show that Proposition 3 holds asymptotically for any sequence {a n } of complex numbers satisfying the condition lim n→∞ a n n 2/3 = 0.
To obtain a converging sequence of root-counting measures one has to consider the scaled matrix 1 n 4/3 M (a n ) n − β I given by: (2.16) It is obtained by dividing (2.14) by n 4/3 and setting β = λ/n 4/3 . Its principal minors satisfy the recurrence n (a n , β) − (k − 1)(n − k + 2)a n n 8/3 (k−2) n (a n , β) (a n , β). (2.17) Since lim n→∞ a n n 2/3 = 0, then for k n → τ, the family (2.17) of recurrence relations converges to the earlier family (2.10) corresponding to the case a n = 0, n = 1, 2, 3, . . . . Therefore the measure obtained by averaging the family of root-counting measures for recurrence relations with constant coefficients is exactly the same as in the previous case a n = 0, i.e., it coincides with ν 0 . Additionally observe that by a general result of [16], the support of the asymptotic root-counting measure of the sequence { (k) n (a n , β)} can only be smaller than that of ν 0 and their Cauchy transforms must coincide outside the support of ν 0 . Since the support of ν 0 is the union of three straight intervals through the origin the resulting asymptotic root-counting measure for the sequence {Sp n (a n , βn 4/3 )} in case lim n→∞ a n n 2/3 = 0 coincides with ν 0 as well.

Remark 6
Since the support of the limiting measure ν 0 consists of three segments through the origin it is in principle possible to find integral formulas for the density and the Cauchy transform of ν 0 similar to those presented in [16], [12] and [23]. In particular, in the complement to the support of ν 0 , its Cauchy transform is given by where˜ is the unique solution of (2.12) satisfying lim β→∞˜ β = −1. However it seems difficult to find either a somewhat explicit expression for C ν 0 (β) or a linear differential operator with polynomial coefficients annihilating C ν 0 (β). (Observe that such an operator always exists since C ν 0 (β) belongs to the Nilsson class, see [20]).

"Proof" of Proposition 2
Similarly to the previous section, let (k) n (a n , λ) be the characteristic polynomial of the k-th principal minors of M (a n ) n , see (1.4). As in the previous section let us start with a special sequence {a n = An 2/3 } for some fixed A = 0. Next consider the matrix and denote its k-th principal minor by (k) n (An 2/3 , λ). This sequence of minors satisfies the recurrence relation of length 4 of the form: To get stabilization similar to that of Sect. 2, introduce the scaled matrix Its principal minors (which we denote by (k) n (An 2/3 , β)) satisfy the relation (k) Assuming that k n → τ, we obtain that (3.3) tends to the following relation with constant coefficients: (3.4) whose characteristic equation is given by The polynomial For a generic complex A, the union over all τ ∈ [0, 1] of the supports of the asymptotic root-counting measures μ   [16] corresponded to the first author by A. Kuijlaars, see below). The latter circumstance implies that the measure is obtained as the balayage of measure ν A . (The above mentioned complex version of the main result of [16] is worked out in details in Sect. 6.) However, the next Lemma shows that in case A ≥ 3 3 √ 4 (which fits the situation covered by the main result of [12],) these supports coincide and one can obtain the density of ν A by averaging the densities of (3.4).

Lemma 5 If
, then for any τ ∈ [0, 1], the support of the asymptotic rootcounting measure given by the polynomial sequence defined by (3.4

) is a real interval. This interval connects two branching points defined by (3.7).
Proof Using the standard expression for the discriminant, one can check that all three branching points of (3.5) with respect to satisfy the equation:  , and any τ ∈ [0, 1], all three solutions of (3.7) are real, we calculate the discriminant of (3.7) with respect to β. Again using symbolic manipulations, we get that this discriminant is given by: For τ ∈ [0, 1], the graph of Dsc in the real (A, τ )-plane is presented in Fig. 7. One can easily check that the maximal value of A on this graph is obtained when τ = 1/2 (for example, for A = 3 shown in Fig. 6) we can conclude that all three roots of (3.7) are real for any τ ∈ [0, 1]. The latter circumstance together with the reality of the situation imply that supports of the asymptotic root counting measure is real for any τ ∈ [0, 1]. corresponds to the real corner of the asymptotic limiting domain F appearing in Conjecture 2.

Remark 8 One can additionally show that for
, the union of the supports (which is the union of all real intervals described in Lemma 5 coincides with the interval bounded by the two rightmost branching points given by (3.11), see Fig. 6.

Remark 9
Similarly to the case A = 0, for any A ≥ 3 3 √ 4 , one can represent the Cauchy transform of ν A as in the complement to the support of ν A (which is an interval explicitly given in Lemma 5). Here˜ is the unique solution of (3.5) satisfying the condition lim β→∞˜ β = −1.
(In an appropriate domain in C such presentation for the Cauchy transform of ν A is valid for any complex A.) Let us now discuss Proposition 2.

"Proof" of Proposition 2 under additional convergence assumptions.
To obtain the support of the limiting measure ν A whose existence is claimed in the Proposition, we argue as follows. Assume that we have a (sub)sequence β j n ,n of the eigenvalues of the sequence of matrices {n −4/3 M (An 2/3 ) n } (one eigenvalue for each n) converging to some finite limit which we denote by . Denote by { p n } the corresponding (sub)sequence of eigenpolynomials of (the sequence of) differential operators {T n }, see (1.3) in the Introduction. For each T n the value of its parameter a n equals An 2/3 . Then each eigenpolynomial p n satisfies its own differential equation of the form p n − (x 2 − An 2/3 ) p n + (nx − β j n ,n n 4/3 ) p n = 0. (3.8) First assumption. We assume that if the subsequence {β j n ,n } has a finite limit , then, after appropriate scaling of x, the sequence {μ n } of the root-counting measures of { p n } converges in the weak sense to some limiting measure κ a, whose support consists of finitely many compact curves and points. This assumption implies that the sequence of appropriately scaled Cauchy transforms of {μ n } converges to the Cauchy transform of the limiting measure κ a, . The appropriate scaling of x which might provide such a convergence can be easy guessed from (3.8). Namely, substituting x = n 1/3 and dividing the above equation by n 4/3 p n , we get the relation Second assumption. Assuming that the sequence {μ n } of the root-counting measures of { p n ( n 1/3 )} converges to κ a, , we additionally assume that the sequences of the root-counting measures of its first and second derivatives converge to the same measure κ a, .
(Apparently this assumption can be rigorously proved by using the same arguments as presented in [4].) Under the above two main assumptions and using (3.9), we get that the Cauchy transform C A, of κ A, satisfies the quadratic equation: almost everywhere in C. Up to the variable change x ↔ and ↔ β the latter equation coincides with (1.5).
Third assumption. So far we presented a physics-style argument showing that if a (sub)sequence {β j n ,n } of the eigenvalues of the sequence of matrices {n −4/3 M (An 2/3 ) n } converges to some limit , then there exists a probability measure κ A, whose Cauchy transform satisfies (3.10) almost everywhere in the -plane. Our final assumption is that the converse to the latter assumption is true as well, i.e., for each with the above properties there exists an appropriate subsequence {β j n ,n } of eigenvalues of matrices {n −4/3 M (An 2/3 ) n } converging to . Simultaneous application of the above three assumptions settles Proposition 2 in the case of the special sequence a n = An 2/3 , n = 1, 2, 3, . . . . The argument for the general case lim n→∞ a n = An 2/3 repeats verbatim the one used in the proof of Theorem 1.

Quadratic equations with polynomial coefficients and quadratic differentials
To finish this section let us present some additional results about quadratic differentials and signed measures. The next result is a special case of Proposition 9 and Theorem 12 of [6].

Proposition 6
There exists a signed measure μ A, whose Cauchy transform satisfies We know that the support of μ A, should include all the branching points of (3.10) and consists of the critical horizontal trajectories of (1.6).

Lemma 7
The set of the critical values of the polynomial P( , ) = (( 2 − A) 2 − 4( − )), i.e., the set of all for which P( , ) has a double root with respect to is given by the equation: Proof Straight-forward calculations.

Corollary 3
The endpoints of the support of ν A are contained among the three roots of equation (3.7) when τ = 1/2, see Fig. 8. This equation coincides with (3.11) where is substituted by β.

On branching points and monodromy of the spectrum
Observe that, for any positive integer n and generic values of parameter a, the roots of Sp n (a, λ) with respect to λ are simple. The latter roots are called the quasy-exactly solvable spectrum of the quartic oscillator under consideration. Moreover, for any given n, and any sufficiently large positive a, these roots are real and distinct. The set n ⊂ C of branching points of Sp n (a, λ), i.e., the set of all values of a for which two eigenvalues coalesce, has cardinality n+1 2 . When plotted these branching points form a regular pattern in the complex plane shown in Figs. 3 and 9. In this subsection we present our (mostly) numerical results and conjectures about the monodromy of the roots of Sp n (a, λ) when a runs along different closed paths in the complement to n in the a-plane. We start with the following statement. where Sp n (a, β) is the characteristic polynomial of the tridiagonal matrix In other words, we are comparing the roots of Sp n (a, β) divided by √ a with that of Sp n (a, β) divided by and (k) where γ = β/ √ a and both recurrences have the standard boundary conditions:   Now one can easily check by induction that bivariate polynomial Sp n (a, β) is quasihomogeneous with weight 1 for variable β and weight 2 for variable a. When  Fig. 11 The system of standard paths and the monodromy (transpositions) of the spectrum which they produce for n = 4 a is positive, then Sp n (a, β) is a real-rooted polynomial in β. Since multiplication of λ by e πit and multiplication of a by e 2πit multiplies the whole Sp n (a, β) by a constant, then for any fixed a, the roots of Sp n (a, β) with respect to β lie on the line through the origin whose slope is half of the slope of a. Now consider the roots of Sp n (a, β) with respect to β. Observe that if a = 1, then the spectrum of (4.1) is (−1, −1 + 2 n , −1 + 4 n , . . . , 1 − 4 n , 1 − 2 n , 1). The same argument as above gives that for any a = 0, the roots of Sp n (a, β) with respect to β will be equally spaced on the interval [− √ a, √ a]. Since choosing |a| sufficiently large, we can achieve that all roots of S p To describe (our conjecture on) the monodromy of the spectrum, let us introduce a system of standard paths connecting a base point chosen as a sufficiently large positive number with every branching point, see Fig. 11. Based on our numerical experiments, we see that n form a triangular shape with points regularly arranged into columns and rows in C. There are n columns (enumerated from left to right) where the j-th column consists of n − j branching points with approximately the same real part and there are n rows (enumerated from bottom to top) where the i-th row consists of points with approximately the same imaginary part. We denote the branching points σ i, j ∈ n where i = 1, . . . , 2n − 1 is the row number and j = 1, . . . , n is the column number.
Fixing a base point B as a sufficiently large positive number, connect B with every σ i, j by a "vertical hook" P i, j , i.e., move from B vertically to the imaginary part of σ i, j , then move horizontally to the left untill you almost hit σ i, j , then circumgo σ i, j counterclockwise along a small circle centered at σ i, j and return back to B along the same path. Conjecturally, along such a path one will never hit any other branching points unless σ i, j lies on the real axis. In other words, the imaginary parts of all branching points except for the real ones are all distinct. In case when σ i, j is real one can slightly deform the suggested path (which is a real interval) in an arbitrary way to move it away from the real axis. The resulting monodromy will (conjecturally) be independent of any such small deformation, see below. Finally we can state our surprisingly simple guess. This conjecture has been numerically checked for all n ≤ 10. Observe that since the system of standard paths gives a basis of the fundamental group π 1 (C \ n ), then knowing the monodromy for the standard paths, one can calculate the monodromy along any loop in C \ n based at B.

Appendix I: Estimates for largest roots of Yablonskii-Vorob'ev polynomials
In this section we present both estimates from above and from below as well as the asymptotic behaviour for roots of maximal modulus of Yablonskii-Vorob'ev polynomials. The Yablonskii-Vorob'ev polynomials satisfy the differential-difference relation with Q 0 (t) = 1 and Q 1 (t) = t. The roots of Q n approximately cover a triangular shape invariant under rotation by 2π/3 whose edges are curves instead of straight lines. The whole pattern is invariant under the dihedral group of symmetries of an equilateral triangle. The roots of maximal modulus lie on a circle with centre the origin. These and other properties are listed in [11].
Our strategy is to connect roots and coefficients of Yablonskii-Vorob'ev polynomials. Similar approach was used in [15]. Our results are sharper and we also derive the asymptotic formula. As the first step we form rational functions ρ n (t) = (log(Q n (t))) − t/4.
Another relation between ρ n and Q n is

The general form of this sequence is
Besides, the factor 3 √ −1 shows the direction in which the three roots of maximal modulus move when n → ∞.
Numerical results show the rate of convergence to the asymptotic value, cf. The main result of this section is Proposition 13 mentioned in Sect. 3 which undoubtedly has independent interest. For the sake of completeness and for the convenience of our readers we include its proof below. Here we adopt the notation used in paper [16] by A. Kuijlaars , ∀z ∈ ρ(T ).
On the other hand, recall that for a bounded operator B it holds where D B = {z ∈ C | |z| ≤ B }. Let us remark that σ (B) ⊂ D B . Let {a n } n≥1 , {b n } n≥0 ⊂ C and a n = 0, ∀n ≥ 1. Further J n denotes the (n + 1) × (n + 1) Jacobi matrix of the form . . . . . . a n−1 b n−1 a n a n b n Recall also that the polynomial sequence { p n } n≥0 determined by recurrence a n p n−1 (x) + b n p n (x) + a n+1 p n+1 (x) = x p n (x), n ≥ 0, and initial conditions p −1 = 0 and p 0 = 1, is related with J n by formula On the other hand, one has ∀z ∈ C, |z| > 3M.
Proof By simple linear algebra, (note that e n stands for the (n + 1)-th vector of the standard basis of C n+1 ). Consequently, One the other hand, Observe that if R n ∈ C n+1,n+1 is the permutation matrix determined by equations R n e k = e n−k , ∀k ∈ {0, 1, . . . , n}, then b n a n a n b n−1 a n−1 a n−1 b n−2 a n−2 . . . . . . . . .
If convenient, we identify matrix R n with the operator R n ⊕ 0 acting on 2 (N 0 ) In what follows, {a n,N | n, N ∈ N} ⊂ C, {b n,N | n ∈ N 0 , N ∈ N} ⊂ C, and stands for the semi-infinite (complex) Jacobi matrix. Finally, P n ∈ B( 2 (N 0 )) denotes the orthogonal projection on span{e 0 , . . . , e n }. Proof Recall an easily verifiable statement: Let X be a Banach space, B ∈ B(X ) and B n a uniformly bounded sequence of operators acting on X . If B n ϕ → Bϕ, as n → ∞, for all ϕ from a dense subset of X , then s-lim n→∞ B n = B. Take arbitrary {n j }, {N j } ⊂ N, N j → ∞ and n j /N j → t, for j → ∞. Let us denote temporarilyĴ j = R n j J (N j )R n j Assumption (6.2) guarantees the existence of j 0 ∈ N such that sup j≥ j 0 P n j J (N j )P n j = sup j≥ j 0 Ĵ j < ∞.

Proposition 10 Let
Hence, taking into account the above statement, it suffices to verify that lim j→∞Ĵ j e n = J (A, B)e n , ∀n ∈ N 0 . Take n ∈ N 0 and j > n, then Ĵ j e n − J (A, B)e n 2 = |a n j −n+1,N j − A| 2 + |b n j −n,N j − B| 2 + |a n j −n,N j − A| 2 → 0, for j → ∞, by assumption (6.1). If n = 0, set a n j +1,N j = 0 in the above equation. All in all, the claim is verified.
It is again quite easy to see that for any B n , B ∈ B(X ) such that sup n B n ≤ M and s-lim n→∞ B n = B, one has s-lim n→∞ (B n − z) −1 = (B − z) −1 , ∀z / ∈ D M and the convergence is local uniform in z. Indeed, since from which the strong convergence of resolvents follows. The local uniformness follows, for example, from the Mantel's theorem. The next statement is in fact a corollary of Proposition 10, however, it is also a complex generalization of [16, Thm. 2.1]. Therefore we formulate it as a proposition. However, in the general case of complex sequences a n,N and b n,N , one can get only the relation (6.3) outside a ball, |z| > M, and not for almost all z ∈ C. This however does not imply the weak convergence (6.4). To our best knowledge nor the existence of the weak limit is guaranteed (only a subsequence, by Helly's theorem). Thus, one can not expect the validity of Theorem 12 in the complex setting. However, one can get at least the following.

Remark 11
Note the functions a and b are assumed to be continuous in 0 (from the right). This additional condition simplifies the proof considerably and we do not aim here to achieve a full generality.
Proof Note that coefficients a n,N and b n,N are uniformly bounded if n/N is restricted to a compact subsets of [0, ∞), as it follows from the assumptions. Take t > 0 and 0 < < t, then sup{|b n,N | | |n/N − t| ≤ } + 2 sup{|a n,N | | |n/N − t| ≤ } < ∞.
for |z| > 3M. The function in the last integral is known to coincide with the logarithmic potential of ω [b(s)−2a(s),b(s)+2a(s)] at z. All in all, we obtained for |z| > 3M. By the harmonicity of logarithmic potentials U μ outside the support μ and Identity principle for harmonic functions the last equality can be extended to all z / ∈ (suppν ∪ suppσ (t)).