# Off-Spectral Analysis of Bergman Kernels

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## Abstract

The asymptotic analysis of Bergman kernels with respect to exponentially varying measures near emergent interfaces has attracted recent attention. Such interfaces typically occur when the associated limiting Bergman density function vanishes on a portion of the plane, *the off-spectral region*. This type of behavior is observed when the metric is negatively curved somewhere, or when we study partial Bergman kernels in the context of positively curved metrics. In this work, we cover these two situations in a unified way, for exponentially varying weights on the complex plane. We obtain a uniform asymptotic expansion of the *coherent state of depth**n* rooted at an off-spectral point, which we also refer to as the *root function* at the point in question. The expansion is valid in the entire off-spectral component containing the root point, and protrudes into the spectrum as well. This allows us to obtain error function transition behavior of the density of states along the smooth interface. Previous work on asymptotic expansions of Bergman kernels is typically local, and valid only in the bulk region of the spectrum, which contrasts with our non-local expansions.

## 1 Introduction

### 1.1 Bergman kernels and emergent interfaces

This article is a companion to our recent work [15] on the structure of planar orthogonal polynomials. We will make frequent use of methods developed there, and recommend that the reader keep that article available for ease of reference.

The study of Bergman kernel asymptotics has by now a sizeable literature. The majority of the contributions have the flavor of local asymptotics near a given point \(w_0\), under a positive curvature condition. However, in the study of partial Bergman kernels for the subspace of all functions vanishing to a given order at the point \(w_0\), the assumption of vanishing has the effect of introducing a negative point mass for the curvature form at \(w_0\). In addition to the negative curvature which comes from considering partial Bergman kernels defined by vanishing, we allow for the direct effect of patches of negatively curved geometry. Around the set of negative curvature, a *forbidden region* (or *off-spectral set*) emerges. This forbidden region is typically larger than the set of actual negative curvature, and may consist of several connectivity components. Recently, the asymptotic behavior of Bergman kernels near the interface at the edge of the forbidden region has attracted considerable attention. In this work, we intend to investigate this in the fairly general setting of exponentially varying weights in the complex plane. The restriction of the Bergman kernel to the diagonal gives us the density of states, which drops steeply at the interface. Indeed, in the forbidden region the density of states vanishes asymptotically, with exponential decay. One of our main results is that the density of states across the interface converges to the error function in a blow-up, provided the interface is smooth.

The key to obtaining the above-mentioned result is in fact our main result. It concerns the *expansion of the coherent state*\(\mathrm {k}_n(z,w_0)\)*of depth n rooted at a given off-spectral point*\(w_0\). This is the normalized reproducing kernel function at the point \(w_0\) for the Bergman space defined by vanishing to order *n* at \(w_0\). When \(n=0\), the coherent state of depth 0 is just the normalized Bergman kernel \(K(w_0,w_0)^{-\frac{1}{2}}K(z,w_0)\). An important feature of the asymptotic expansion is that *z* and \(w_0\) are allowed to be macroscopically separated. Such truly off-diagonal expansions have, to the best of our knowledge, not appeared elsewhere. It is easy to see that the coherent state of order \(n=0,1,2,\ldots \), at \(w_0\) form an orthonormal basis for the Bergman space, which gives an expansion of the density of states, which leads to the error function asymptotics. Similarly, if we want to handle partial Bergman kernels given by vanishing to order \(n_0\) at \(w_0\), we expand in the basis given by the coherent states of depth \(n\ge n_0\).

### 1.2 Coherent states and elementary potential theory

*m*denote a large positive real parameter and \(Q:{{\mathbb {C}}}\rightarrow {{\mathbb {R}}}\) a continuous potential, and consider the Bergman space \(A^2_{mQ}\) defined as the collection of all entire functions

*f*in the Lebesgue Hilbert space \(L^2_{mQ}\) with finite weighted norm

*suppose that*\(A^2_{mQ}\)

*is nontrivial*, so that \(K_m\) is nontrivial as well. We consider a point \(w_0\in {{\mathbb {C}}}\), and note that \(K_m(w_0,w_0)\ge 0\) automatically. We claim that in fact \(K_m(w_0,w_0)>0\). Indeed, if \(K_m(w_0,w_0)=0\), we would have that \(f(w_0)=0\) for each \(f\in A^2_{mQ}\). But if \(f\ne 0\), we may factor \(f(z)=(z-w_0)^Ng(z)\), where \(N\ge 1\) and

*g*is entire with \(g(w_0)\ne 0\). Clearly, \(g\in A^2_{mQ}\), since division by \((z-w_0)^N\) makes the function smaller outside a disk of radius 1 about \(w_0\). Finally, the fact that \(g(w_0)\ne 0\) forces \(K_m(w_0,w_0)>0\), a contradiction. For nontrivial \(A^2_m\), we consider the

*coherent state*(normalized Bergman kernel)

*n*and a point \(w_0\in {{\mathbb {C}}}\), we consider the subspace \(A^2_{mQ,n,w_0}\) of \(A^2_{mQ}\), consisting of those functions that vanish to order at least

*n*at \(w_0\). It may happen for some

*n*that this space is trivial, for instance when the potential

*Q*has logarithmic growth only, because then the space \(A^2_{mQ}\) consists of polynomials of a bounded degree. We denote its reproducing kernel by \(K_{m,n,w_0}\), and observe that \(K_{m,0,w_0}=K_m\). We define the

*coherent state of depth*

*n*

*at*\(w_0\) (or the

*root function*of order

*n*), denoted \(\mathrm {k}_{m,n,w_0}\), as the unique solution to the optimization problem

*n*at \(w_0\) is connected with the reproducing kernel \(K_{m,n,w_0}\):

*n*-th derivative at \(w_0\). The root function \(\mathrm {k}_{m,n,w_0}\) will play a key role in our analysis, similar to that of the orthogonal polynomials in the context of polynomial Bergman kernels. The root functions \(\mathrm {k}_{m,n,w_0}\) all have norm equal to 1 in \(A^2_{mQ}\), except when they are trivial and have norm 0. As a result of the relation (1.2.2), we may alternatively call the root function \(\mathrm {k}_{m,n,w_0}\) a

*normalized partial Bergman kernel*.

*spectrum*\(\mathcal {S}\), also called the

*spectral droplet*. This is the closed set defined in terms of the following obstacle problem. Let \(\mathrm {SH}({{\mathbb {C}}})\) denote the cone of all subharmonic functions on the plane \({{\mathbb {C}}}\), and consider the function

*Q*does not have a subharmonic minorant, then the problem degenerates: \({\hat{Q}}=-\infty \) identically. However, if

*Q*is \(C^{1,1}\)-smooth and has some modest growth at infinity, it is known that \({\hat{Q}}\in C^{1,1}\) as well, and it is a matter of definition that \({\hat{Q}}\le Q\) pointwise (see, e.g., [12]). Here, \(C^{1,1}\) denotes the standard smoothness class of differentiable functions with Lipschitz continuous first order partial derivatives. We define the

*spectrum*(or the

*spectral droplet*) as the contact set

*forbidden region*is the complement \(\mathcal {S}^c={{\mathbb {C}}}{\setminus } \mathcal {S}\). The spectral droplet \(\mathcal {S}\) is associated with the Bergman kernels \(K_m\). There is also the corresponding notion of a spectral droplet associated with the partial Bergman kernels \(K_{m,n,w_0}\) as defined in Sect. 2.1 below, which we now briefly outline. For \(0\le \tau <+\infty \), let \(\mathrm {SH}_{\tau ,w_0}({{\mathbb {C}}})\) denote the convex set

*partial Bergman density*

*Bergman density*. In the limit as \(m,n\rightarrow +\infty \) with \(n=m\tau \), it is known that

*bulk*of the spectral droplet \(\mathcal {S}_{\tau ,w_0}\) is the set

### 1.3 Further background on Bergman kernel expansions

Our motivation for the above setup originates with the theory of random matrices, specifically the *random normal matrix ensembles*. We should mention that an analogous situation occurs in the study of complex manifolds. The Bergman kernel then appears in the study of spaces of \(L^2\)-integrable global holomorphic sections of \(L^m\), where \(L^m\) is a high tensor power of a holomorphic line bundle *L* over the manifold, endowed with an hermitian fiber metric *h*. If \(\{U_i\}_i\) is a coordinate system on a manifold, then a holomorphic section *s* to a line bundle *L* can be written in local coordinates as \(s=s_i e_i\), where \(e_i\) are local basis elements for *L* and \(s_i\) are holomorphic functions. The pointwise norm of a section *s* of \(L^m\) may be then be written as \(|s|_{h^m}^2=|s_i|^2\mathrm e^{-m\phi _i}\) on \(U_i\), for some smooth real-valued functions \(\phi _i\). Along with a volume form on the base manifold, this defines an \(L^2\) space which shares many characteristics with the spaces considered here.

The asymptotic behavior of Bergman kernels has been the subject of intense investigation. However, the understanding has largely been limited to the analysis of the kernel inside the bulk of the spectrum, in which case the kernel enjoys a full *local* asymptotic expansion. The pioneering work on Bergman kernel asymptotics begins with the efforts by Hörmander [16] and Fefferman [10]. Developing further the microlocal approach of Hörmander, Boutet de Monvel and Sjöstrand [7] obtain a near-diagonal expansion of the Bergman kernel close to the boundary of the given domain. Later, in the context of Kähler geometry, the influential *peak section method* was introduced by Tian [22]. His results were refined further by Catlin and Zelditch [8, 23], while the connection with microlocal analysis was greatly simplified in the more recent work by Berman, Berndtsson, and Sjöstrand [6]. A key element of all these methods is that the kernel is determined by the local geometry around the given point. This feature is absent when we consider the kernel near an off-spectral point or near a boundary point of the spectral droplet.

In the recent work [15], we analyze the boundary behavior of polynomial Bergman kernels, for which the corresponding spectral droplet is compact, connected, and has a smooth Jordan curve as boundary. The analysis takes the path via a full asymptotic expansion of the orthogonal polynomials, valid off a sequence of increasing compacts which eventually fill the droplet. By expanding the polynomial kernel in the orthonormal basis provided by the orthogonal polynomials, the error function asymptotics emerges along smooth spectral boundaries. This compares with the earlier work [3], which captures the boundary effects at the level of mean field asymptotics, and builds on the methods developed in [2]. Prior to the work [15], little was known about the general behavior of planar orthogonal polynomials for exponentially varying weights. However, in specific situations, such as the two-parameter weight family considered in [5], very detailed analysis was available.

The appearance of an interface for partial Bergman kernels in higher dimensional settings and in the context of complex manifolds has been observed more than once, notably in the work by Shiffman and Zelditch [21] and by Pokorny and Singer [17]. That the error function governs the transition behavior across the interface was observed later in several contexts. For instance, in [18], Ross and Singer investigate the partial Bergman kernels associated to spaces of holomorphic sections vanishing along a divisor, and obtain error function transition behavior under the assumption that the setup is invariant under a holomorphic \(S^1\)-action. This result was later extended by Zelditch and Zhou [24], in the context of \(S^1\)-symmetry. More recently, Zelditch and Zhou [25] obtain the same transition for so-called *spectral partial Bergman kernels*, defined in terms of the Toeplitz quantization of a general smooth Hamiltonian.

Our methods for obtaining asymptotic expansions of coherent states centered at off-spectral points do not easily extend to the higher complex dimensional setting. It appears plausible, however, that our analysis would apply in the context of sections of positive holomorphic line bundles on compact Riemann surfaces.

### 1.4 Off-spectral and off-diagonal asymptotics of coherent states

*Q*(

*z*) an

*admissible potential*, by which we mean the following:

- (i)\(Q:{{\mathbb {C}}}\rightarrow {{\mathbb {R}}}\) is \(C^2\)-smooth, and has sufficient growth at infinity:$$\begin{aligned} \tau _Q:=\liminf _{|z|\rightarrow +\infty }\frac{Q(z)}{\log |z|}>0. \end{aligned}$$
- (ii)
*Q*is real-analytically smooth and strictly subharmonic in a neighborhood of \(\partial \mathcal {S}\), where \(\mathcal {S}\) is the contact set of (1.2.3), - (iii)
there exists a bounded component \(\Omega \) of the complement \(\mathcal {S}^c ={{\mathbb {C}}}{\setminus }\mathcal {S}\) which is simply connected, and has real-analytically smooth Jordan curve boundary.

*off-spectral component*we mean a connectivity component of the complement \(\mathcal {S}^c\). This situation occurs, e.g., if the potential is strictly superharmonic in a portion of the plane, as is illustrated in Fig. 1. In terms of the metric, this means that there is a region where the curvature is negative.

A word on notation. To formulate our first main result, we need the function \({{\mathcal {Q}}}_{w_0}\), which is bounded and holomorphic in the off-spectral component \(\Omega \) and whose real part equals *Q* along the boundary \(\partial \Omega \). To fix the imaginary part, we require that \({{\mathcal {Q}}}_{w_0}(w_0)\in {{\mathbb {R}}}\). In addition, we need the conformal mapping \(\varphi _{w_0}\) which takes \(\Omega \) onto the unit disk \({{\mathbb {D}}}\) with \(\varphi _{w_0}(w_0)=0\) and \(\varphi _{w_0}'(w_0)>0\). Since the boundary \(\partial \Omega \) is assumed to be a real-analytically smooth Jordan curve, both the function \({{\mathcal {Q}}}_{w_0}\) and the conformal mapping \(\varphi _{w_0}\) extend analytically across \(\partial \Omega \) to a fixed larger domain. By possibly considering a smaller fixed larger domain, we may assume that the extended function \(\varphi _{w_0}\) is conformal on the larger region. These observations are essential for our first main result, since we want the asymptotics to hold across the interface \(\partial \Omega \).

### Theorem 1.4.1

*Q*is an admissible potential, we have the following. Given a positive integer \(\kappa \) and a positive real

*A*, there exist a neighborhood \(\Omega ^{\circledast }\) of the closure of \(\Omega \) and bounded holomorphic functions \({{\mathcal {B}}}_{j,w_0}\) on \(\Omega ^{\circledast }\) for \(j=0,\ldots ,\kappa \), as well as domains \(\Omega _{m}=\Omega _{m,A}\) with \(\Omega \subset \Omega _{m}\subset \Omega ^{\circledast }\) which meet

*A*is big enough, then

### Remark 1.4.2

Using an approach based on Laplace’s method, the functions \({{\mathcal {B}}}_{j,w_0}\) may be obtained algorithmically, for \(j=1,2,3,\ldots \), see Theorem 3.2.2 below. The details of the algorithm are analogous with the case of the orthogonal polynomials presented in [15].

Commentary to the theorem. The point \(w_0\) may be chosen as any fixed point in \(\Omega \), while the point *z* is allowed to vary anywhere inside the set \(\Omega _m\), which contains the entire off-spectral component \(\Omega \) as well as a shrinking neighborhood of the boundary \(\partial \Omega \). Hence, we obtain *macroscopically off-diagonal asymptotics*, where the points *z* and \(w_0\) are either both off-spectral, or where *z* is spectral near-boundary and \(w_0\) off-spectral, respectively. To our knowledge, this is the first instance of such asymptotics. Indeed, earlier work covers either diagonal or near-diagonal asymptotics inside the bulk of the spectrum (and to some extent near the spectral boundary). The analysis of Bergman kernel asymptotics for two macroscopically separated points in \(\mathrm {bulk}(\mathcal {S}):=\{z\in \mathrm {int}\,\mathcal {S}:\varDelta Q(z)>0\}\) appears difficult if we ask for high precision, and the same can be said for the case when \(w_0\in \Omega \) is off-spectral and \(z\in \mathrm {int}\,\mathcal {S}\) is a bulk point. To address the latter issue, we may ask what happens in Theorem 1.4.1 if *z* is not in the set \(\Omega _m\). Since \(\Omega _m\) captures most of the \(L^2\)-mass of the root function in view of Theorem 1.4.1, we see that the coherent state is minuscule outside \(\Omega _m\) in the \(L^2\)-sense. If we want corresponding pointwise control, we may appeal to e.g. the subharmonicity estimate of Lemma 3.2 of [1].

### 1.5 Expansion of partial Bergman kernels in terms of root functions

Here we make minimal assumptions on the potential \(Q:{{\mathbb {C}}}\rightarrow {{\mathbb {R}}}\): it should be continuous. As mentioned earlier, we let the root function \(\mathrm {k}_{m,n,w_0}\) be as in (1.2.2) whenever the associated optimization problem has a positive maximum, while otherwise \(\mathrm {k}_{m,n,w_0}=0\).

### Theorem 1.5.1

*Q*, we have that

### Proof

*f*is nontrivial, there exists an integer \(N\ge n\) such that \(f(z)=c\,(z-w_0)^N +\mathrm {O}(|z-w_0|^{N+1})\) near \(w_0\), where \(c\ne 0\) is complex. At the same time, the existence of such nontrivial

*f*entails that the corresponding root functions \(\mathrm {k}_{m,N,w_0}\) is nontrivial as well, and that \(K_{m,N,w_0}(\zeta ,\zeta )\asymp |\zeta -w_0|^{2N}\) for \(\zeta \) near \(w_0\). On the other hand, the orthogonality between

*f*and \(\mathrm {k}_{m,N,w_0}\) gives us that

### 1.6 Off-spectral asymptotics of partial Bergman kernels

Given a point \(w_0\in {{\mathbb {C}}}\) we recall the partial Bergman spaces \(A^2_{mQ,n,w_0}\), and the associated spectral droplets \(\mathcal {S}_{\tau ,w_0}\) (see (1.2.5)), where we keep \(n=\tau m\). Before we proceed with the formulation of the second result, let us fix some terminology.

### Definition 1.6.1

*Q*is said to be \((\tau ,w_0)\)-

*admissible*if the following conditions hold:

- (i)\(Q:{{\mathbb {C}}}\rightarrow {{\mathbb {R}}}\) is \(C^2\)-smooth and has sufficient growth at infinity:$$\begin{aligned} \tau _Q:=\liminf _{|z|\rightarrow +\infty }\frac{Q(z)}{\log |z|}>0. \end{aligned}$$
- (ii)
*Q*is real-analytically smooth and strictly subharmonic in a neighborhood of \(\partial \mathcal {S}_{\tau ,w_0}\). - (iii)
The point \(w_0\) is an off-spectral point, i.e., \(w_0\notin \mathcal {S}_{\tau ,w_0}\), and the component \(\Omega _{\tau ,w_0}\) of the complement \(\mathcal {S}_{\tau ,w_0}^c\) containing the point \(w_0\) is bounded and simply connected, with real-analytically smooth Jordan curve boundary.

*Q*is \((\tau ,w_0)\)-admissible for each \(\tau \in I\) and \(\{\Omega _{\tau ,w_0}\}_{\tau \in I}\) is a smooth flow of domains, then

*Q*is said to be \((I,w_0)\)-admissible.

Generally speaking, off-spectral components may be unbounded. It is for reasons of simplicity that we focus on bounded off-spectral components in the above definition.

*Q*is \((I,w_0)\)-admissible for some non-trivial compact interval \(I=I_{0}\). For an illustration of the situation, see Fig. 3.

*Q*on \(\partial \Omega _{\tau ,w_0}\) and is real-valued at \(w_0\). It is tacitly assumed to extend holomorphically across the boundary \(\partial \Omega _{\tau ,w_0}\). We now turn to our second main result.

### Theorem 1.6.2

*Q*is \((I_0,w_0)\)-admissible, where the interval \(I_0\) is compact. Given a positive integer \(\kappa \) and a positive real

*A*, there exists a neighborhood \(\Omega ^{\circledast }_{\tau ,w_0}\) of the closure of \(\Omega _{\tau ,w_0}\) and bounded holomorphic functions \({{\mathcal {B}}}_{j,\tau ,w_0}\) on \(\Omega ^{\circledast }_{\tau ,w_0}\), as well as domains \(\Omega _{\tau ,w_0,m}=\Omega _{\tau ,w_0,m,A}\) with \(\Omega _{\tau ,w_0}\subset \Omega _{\tau ,w_0,m}\subset \Omega ^{\circledast }_{\tau ,w_0}\) which meet

*n*at \(w_0\) enjoys the expansion

*A*big enough, we have

### Remark 1.6.3

As in the case of the normalized Bergman kernels, the expressions \({{\mathcal {B}}}_{j,\tau , w_0}\) may be obtained algorithmically, for \(j=1,2,3,\ldots \) (see Theorem 3.2.2 below).

Commentary to the theorem.(a) As in Theorem 1.4.1, the point \(w_0\) may be chosen as any fixed point in \(\Omega \), while the point *z* is allowed to vary anywhere inside the set \(\Omega _{\tau ,w_0,m}\). Arguing in a fashion analogous to what we did in the commentary following Theorem 1.4.1, we may conclude that \(\mathrm {k}_{m,n,w_0}\) is small, both pointwise and in the \(L^2\)-sense, away from the set \(\Omega _{\tau ,w_0,m}\).

(b) Theorems 1.4.1 and 1.6.2 seemingly cover different instances of the root function asymptotics. In fact, in a certain sense, they are equivalent. Indeed, modulo the necessary technicalities we may obtain Theorem 1.6.2 from Theorem 1.4.1 by replacing the potential *Q*(*z*) with the singular potential \({\tilde{Q}}_{\tau ,w_0}(z)=Q(z)-\tau \log |z-w_0|\). On the other hand, the former theorem is the limit case \(\tau \rightarrow 0\) of the latter.

(c) Theorem 1.6.2 should be compared with Theorem 4.1 in the work of Shiffman and Zelditch [21]. There, an asymptotic expansion of the diagonal restriction \(K_{NP}(z,z)\) of the Bergman kernel associated to a family of scaled Newton polytopes *NP* is obtained, as \(N\rightarrow +\infty \). This expansion is valid deep inside the corresponding forbidden region. In one complex dimension, this is equivalent to studying a certain weighted partial Bergman density for a polynomial Bergman space. Hence there is some overlap with the present work as well as with [15]. The genuinely off-diagonal and off-spectral asymptotic expansion of the coherent states obtained here seem not to have any analogue elsewhere.

### 1.7 Interface transition of the density of states

*Q*is \((I,w_0)\)-admissible for some (short) open interval containing the point \(\tau \) if \(\tau >0\), while if \(\tau =0\),

*Q*is admissible. Fix a point \(z_0\in \partial \Omega \), and denote by \(\nu \in {{\mathbb {T}}}\) the inward unit normal to \(\partial \Omega \) at \(z_0\). We define the rescaled density \(\varrho _m=\varrho _{m,\tau ,w_0,z_0}\) by

### Corollary 1.7.1

### 1.8 Comments on the exposition and a guide to the proofs

In Sect. 2, we explain the proofs of the main results, which are Theorems 1.4.1 and 1.6.2 together with Corollary 1.7.1. Our approach is analogous to that of [15], and we make an effort to explain exactly what needs to be modified for the techniques to apply in the present context. As in [15], the proofs involve the construction of a flow of loops near the spectral boundary, called the *orthogonal foliation flow*. The flow is constructed in an iterative procedure, which produces both the terms in the asymptotic expansion of the coherent state as well as the successive terms in the expansion of the flow. The flow \(\{\gamma _t\}_t\) is constructed to have the following property. If the normal velocity is denoted by \(\upsilon _{\mathrm {n}}\), we want the Szegő kernel for the point \(w_0\) with respect to the interior of the curve \(\gamma _t\) and the induced weighted arc-length measure \(\mathrm e^{-2mQ}\upsilon _{\mathrm {n}}{\mathrm d}\sigma \) on \(\gamma _t\) to be stationary in *t*. In turn, this allows us to glue together the Szegő kernels to form the Bergman kernel. In order to obtain the coefficient functions of the expansions in a more straightforward fashion, we apply the method developed in [15] which is based on Laplace’s method. Another important ingredient, which allows localization to a neighborhood of each off-spectral component, is Hörmander’s \({\bar{\partial }}\)-estimate, suitably modified to the given needs. Since this is the method which allows us to change the geometry drastically, we sometimes refer to it as \({\bar{\partial }}\)-*surgery*.

In Sect. 3, we develop a more general version of the foliation flow lemma, which allows us to introduce a conformal factor in the area form. To avoid unnecessary repetition, both the orthogonal foliation flow and the algorithm for computing the coefficient functions in the asymptotic expansions are explained only in this more general setting. We supply a couple of applications of this extension, including a stability result for the root functions and the orthogonal polynomials under a \(\frac{1}{m}\)-perturbation of the potential *Q* (Theorems 3.2.1 and 3.4.2).

## 2 Off-Spectral Expansions of Normalized Kernels

### 2.1 A family of obstacle problems and evolution of the spectrum

*Q*. To see this, we consider the perturbed potential

The following proposition summarizes some basic properties of the function \(\hat{Q}_{\tau ,w_0}\) given by (1.2.4). We refer to [12] for the necessary details.

### Proposition 2.1.1

The evolution of the free boundaries \(\partial \mathcal {S}_{\tau ,w_0}\), which is of fundamental importance for our understanding of the properties of the normalized reproducing kernels, is summarized in the following.

### Proposition 2.1.2

*h*on \(\Omega _{\tau ,w_0}\), we have that

### Proof

*h*is harmonic on \(\Omega _{\tau ,w_0}\) and \(C^2\)-smooth up to the boundary, and apply Green’s formula to obtain

The second part follows along the lines of [15, Lemma 2.3.1]. \(\quad \square \)

The study of Laplacian growth (also called Hele-Shaw flow) has a long history, see e.g. [11]. In the context of curved surfaces, it was considered by Hedenmalm and Shimorin [14], and further developed in [13] and [19]. Next, we turn to an off-spectral growth bound for weighted holomorphic functions.

### Proposition 2.1.3

*Q*is admissible and denote by \({{\mathcal {K}}}_{\tau ,w_0}\) a closed subset of the interior of \(\mathcal {S}_{\tau ,w_0}\). Then there exist constants \(c_0\) and \(C_0\) such that for any \(f\in A^2_{mQ,n,w_0}({{\mathbb {C}}})\) it holds that

### 2.2 Some auxilliary functions

*Q*that will be useful in the sequel. We denote by \({{\mathcal {Q}}}_{\tau ,w_0}\) the bounded holomorphic function on \(\Omega _{\tau ,w_0}\) whose real part on the boundary curve \(\partial \Omega _{\tau ,w_0}\) equals

*Q*, uniquely determined by the requirement that \({\text {Im}}{{\mathcal {Q}}}_{\tau ,w_0}(w_0)=0\). We also need the function \(\breve{Q}_{\tau ,w_0}\), which denotes the harmonic extension of \(\hat{Q}_{\tau ,w_0}\) across the boundary of the off-spectral component \(\Omega _{\tau ,w_0}\). These two functions are connected via

*Q*, the off-spectral component \(\Omega _{\tau ,w_0}\) is a bounded simply connected domain with real-analytically smooth Jordan curve boundary. Without loss of generality, we may hence assume that \({{\mathcal {Q}}}_{\tau ,w_0}\), \(\breve{Q}_{\tau ,w_0}\) as well as the conformal mapping \(\varphi _{\tau ,w_0}\) extend to a common domain \(\Omega _0\), containing the closure \(\bar{\Omega }_{\tau ,w_0}\). By possibly shrinking the interval \(I_0\), we may moreover choose the set \(\Omega _0\) to be independent of the parameter \(\tau \in I_0\).

### 2.3 Canonical positioning

An elementary but important observation for the main result of [15] is that we may ignore a compact subset of the interior of the compact spectral droplet \(\mathcal {S}_\tau \) associated to polynomial Bergman kernels when we study the asymptotic expansions of the orthogonal polynomials \(P_{m,n}\) (with \(\tau =\frac{n}{m}\)). Indeed, only the behavior in a small neighborhood of the complement \(\mathcal {S}_\tau ^{\mathrm {c}}\) is of interest, and the \({\bar{\partial }}\)-surgery method allow us to disregard the rest. The physical intuition behind this is the interpretation of the probability density \(|P_{m,n}|^2\mathrm e^{-2mQ}\) as the net effect of adding one more particle to the system, and since the positions in the interior of the droplet are already occupied we would expect the net effect to occur near the boundary. The fact that we may restrict our attention to a simply connected proper subset of the Riemann sphere \({\hat{{{\mathbb {C}}}}}\) breaks up the rigidity and allows us to apply a conformal mapping to place ourselves in an appropriate model situation.

*canonical positioning operator*for the point \(w_0\) with respect to the off-spectral component \(\Omega _{\tau ,w_0}\) be given by

As for the coherent states, we will analyze them in terms of the canonical positioning operator \(\mathbf {\Lambda }_{m,n,w_0}\). This simplifies the geometry of \(\Omega _{\tau ,w_0}\) by mapping it to the unit disk \({{\mathbb {D}}}\), and simplifies the weight. Indeed, the function \(R_{\tau ,w_0}\) is flat to order 2 at the unit circle, and consequently the weight \(\mathrm e^{-2mR_{\tau ,w_0}}\) behaves like a Gaussian ridge.

We summarize the properties of the operator \(\mathbf {\Lambda }_{m,n,w_0}\) in the following proposition. For a potential *V* and a domain \(\Omega \) with \(w_0\in \Omega \), we denote by \(A^2_{mV,n,w_0}(\Omega )\) the space of holomorphic functions on \(\Omega \) which vanish to order *n* at \(w_0\in \Omega \), endowed with the topology of \(L^2(\mathrm {e}^{-2mV},\Omega )\). In case \(n=0\) we simply denote the space by \(A^2_{mV}(\Omega )\).

### Proposition 2.3.1

*Q*be a \((\tau ,w_0)\)-admissible potential, and let \(\Omega _{\tau ,w_0}\) denote the corresponding off-spectral component. Moreover, let \(R_{\tau , w_0}\) be given by (2.3.2). Then, for \(\eta >1\) sufficiently close to 1, the operator \(\mathbf {\Lambda }_{m,n,w_0}\) defines an invertible isometry

### Proof

The conclusion is immediate by the defining normalizations of the conformal mapping \(\varphi _{\tau ,w_0}\). \(\quad \square \)

The following definition is an analogue of Definition 3.1.2 in [15]. We denote by \(\Omega _1\) a domain containing the closure of the off-spectral component \(\Omega _{\tau ,w_0}\), and let \(\chi _{0,\tau }\) denote a \(C^\infty \)-smooth cut-off function which vanishes off \(\Omega _1\), and equals 1 in a neighborhood of the closure of \(\Omega _{\tau ,w_0}\).

### Definition 2.3.2

Let \(\kappa \) be a positive integer. A sequence \(\{F_{m,n,w_0}\}_{m,n}\) of holomorphic functions on \(\Omega _0\) is called a *sequence of approximate root functions of order**n**at*\(w_0\)*of accuracy*\(\kappa \) for the space \(A^2_{mQ,n,w_0}\) if the following conditions are met as \(m\rightarrow +\infty \) while \(\tau =\frac{n}{m}\in {I}_{w_0}\):

We remark that the exponents in the above error terms are chosen for reasons of convenience, related to the correction scheme of Sect. 2.5.

### 2.4 The orthogonal foliation flow

The (approximate) orthogonal foliation flow \(\{\gamma _{m,n,t}\}_t\) is a smooth flow of closed curves near the unit circle \({{\mathbb {T}}}\), originally formulated in [15] in the context of orthogonal polynomials. The defining property in that context is that the orthogonal polynomials \(P_{m,n}\) should be (approximately) orthogonal to the lower degree polynomials along the curves \(\Gamma _{m,n,t}=\phi _\tau ^{-1}(\gamma _{m,n,t})\) with respect to the induced measure \(\mathrm e^{-2mQ}\upsilon _{\mathrm {n}}\mathrm {ds}\), where \(\upsilon _{\mathrm {n}}\) denotes the normal velocity of the flow \(\{\Gamma _{m,n,t}\}_t\) and \(\mathrm {ds}\) denotes normalized arc length measure.

Smoothness classes and polarization of smooth functions. We fix the smoothness class of the weights under consideration, and adapt Definition 4.2.1 in [15] to the the present setting. First, we need the notion of polarization, which applies to real-analytically smooth functions. If *R*(*z*) is real-analytic, there exists a function of two complex variables, denoted by *R*(*z*, *w*), which is holomorphic in \((z,{{\bar{w}}})\) in a neighborhood of the diagonal, with diagonal restriction \(R(z,z)=R(z)\). The function *R*(*z*, *w*) is referred to as the *polarization* of *R*(*z*), and it is uniquely determined by its diagonal restriction *R*(*z*). If *R*(*z*, *w*) is such a polarization of a function *R*(*z*) which is real-analytically smooth near the circle \({{\mathbb {T}}}\) and quadratically flat there, then \(R(z)=(1-|z|^2)^2R_0(z)\) and in polarized form \(R(z,w)=(1-z{{\bar{w}}})^2R_0(z,w)\), where \(R_0(z,w)\) is holomorphic in \((z,{{\bar{w}}})\) in a neighborhood of the diagonal where both variables are near \({{\mathbb {T}}}\). The function \(R_0(z,w)\) is then the polarization of \(R_0(z)\).

### Definition 2.4.1

*R*on \({{\mathbb {D}}}(0,\eta )\) such that

*R*is quadratically flat on \({{\mathbb {T}}}\) with \(\varDelta R|_{{\mathbb {T}}}>0\) and satisfies

*R*is real-analytically smooth and has a polarization

*R*(

*z*,

*w*) which is holomorphic in \((z,{{\bar{w}}})\) on the \(2\sigma \)-fattened diagonal annulus

*uniform family*, provided that for each \(R\in S\), the corresponding \(R_0(z,w)\) is uniformly bounded and bounded away from 0 on \(\hat{{\mathbb {A}}}(\eta ,\sigma )\) while the constant \(\alpha (R)\) is uniformly bounded away from 0.

The point with above definition is that it lets us encode uniformity properties of the potentials \(R_{\tau ,w_0}\) with respect to the parameter \(\tau \) and the point \(w_0\).

For a polarized function *f*(*z*, *w*), we let \(f_{{\mathbb {T}}}(z)\) denote the restriction of *f*(*z*, *z*) for \(z\in {{\mathbb {T}}}\), wherever it is well-defined. We recall from Proposition 4.2.2 of [15] that if *f*(*z*, *w*) is holomorphic in \((z,{{\bar{w}}})\) on the \(2\sigma \)-fattened diagonal annulus \(\hat{{\mathbb {A}}}(\eta ,\sigma )\) and if the parameters meet \(1<\eta \le \sqrt{1+\sigma ^2}+\sigma \), then it follows that the function \(f_{{{\mathbb {T}}}}\) extends holomorphically to the annulus \({\mathbb {A}}(\eta ^{-1},\eta )\). We may need to restrict the numbers \(\eta ,\sigma \) further. Indeed, it turns out that we need that the functions \(\log \varDelta R\), \(\hat{R}=\sqrt{R}\) (chosen to be positive inside the unit circle and negative outside) as well as \(\log (-z\partial _z\hat{R})\) have polarizations which are holomorphic in \((z,{\bar{w}})\) for \((z,w)\in \hat{{\mathbb {A}}}(\eta ,\sigma )\) and uniformly bounded there as well. If *R* belongs to a uniform family of \({\mathfrak {W}}(\eta _0,\sigma _0)\), then there exist \((\eta _1,\sigma _1)\) such that these properties hold for the polarizations with \(\eta =\eta _1\) and \(\sigma =\sigma _1\) (See Proposition 4.2.3 of [15]), where we moreover require that \(1<\eta _1\le \sqrt{1+\sigma _1^2}+\sigma _1\).

### Lemma 2.4.2

Let \({{\mathcal {K}}}\) be a compact subset of each of the domains \(\Omega _{\tau ,w_0}\), where \(\tau \in I_0\). Then there exist constants \(\eta ,\sigma \) with \(\eta >1\) and \(\sigma >0\), such that the collection of weights \(R_{\tau ,w_0}\) with \(w_0\subset {{\mathcal {K}}}\) and \(\tau \in I_0\) is a uniform family in \({\mathfrak {W}}(\eta ,\sigma )\).

This is completely analogous to the corresponding claim in of [15], which was expressed in the context of an exterior conformal mapping.

The orthogonal foliation flow near the unit circle. The existence of the orthogonal foliation flow around the circle \({{\mathbb {T}}}\) and the asymptotic expansion of the root functions after canonical positioning are stated in the following lemma (compare with Lemma 4.1.2 in [15]). For the proof, we refer to the sketched proof of Lemma 3.3.2 below, as well as the complete proof of Lemma 4.1.2 in [15], for the case of orthogonal polynomials.

### Lemma 2.4.3

*s*,

*t*both close to 0, the domains \(\psi _{s,t}\big ({{\mathbb {D}}}\big )\) grow with

*t*, while they remain contained in \({{\mathbb {D}}}(0,\eta _1)\). Moreover, for \(\zeta \in {{\mathbb {T}}}\) we have that

*s*, when

*t*varies in the interval \([-\beta _s,\beta _s]\) with \(\beta _s:=s^{1/2}\log \frac{1}{s}\), the flow of loops \(\{\psi _{s,t}({{\mathbb {T}}})\}_t\) cover a neighborhood of the circle \({{\mathbb {T}}}\) of width proportional to \(\beta _s\) smoothly. In addition, the first term \(B_0\) is zero-free, positive at the origin, and has modulus \(|B_0|=\pi ^{-\frac{1}{4}}(\varDelta R)^{\frac{1}{4}}\) on \({{\mathbb {T}}}\). The other terms \(B_j\) are all real-valued at the origin. The implied constant in (2.4.1) is uniformly bounded, provided that

*R*is confined to a uniform family of \({\mathfrak {W}}(\eta _0,\sigma _0)\).

### 2.5 \({\bar{\partial }}\)-corrections and asymptotic expansions of root functions

In this section, we supply a proof of the main result, Theorem 1.6.2. The proof consists of two parts. First, we construct a family of approximate root function of a given order and accuracy, after which we apply Hörmander-type \({\bar{\partial }}\)-estimates to correct these approximate kernels to entire functions. The precise result needed for the correction scheme runs as follows.

### Proposition 2.5.1

*n*: \(|u(z)|=\mathrm {O}(|z-w_0|^n)\) around \(w_0\). Then

*u*meets the bound

This is an immediate consequence of Corollary 2.4.2 in [15], and essentially amounts to Hörmander’s classical bound for the \({\bar{\partial }}\)-equation in the given setting.

We turn to the proof of Theorem 1.6.2.

### Sketch of proof of Theorem 1.6.2

As the proof is analogous to that of Theorems 1.3.3 and 1.3.4 in [15], we supply only an outline of the proof.

*n*at \(w_0\) with the stated uniform error bound, and finally to show that it is close to the true normalized reproducing kernel.

*m*is large enough, as the main term is bounded away from 0 in modulus, and consecutive terms are much smaller. Also, for large enough

*m*, it holds that \(\chi _1=1\) on \({{\mathcal {D}}}_{m,n,w_0}\). We now introduce the function

*t*, the composition \(q_{m,n}\circ \psi _{m,n,t}\) is holomorphic, so that we may apply the mean value property:

*g*vanishes to order \(n+1\) or higher at \(w_0\), then \(\chi _0\mathrm {k}_{m,n,w_0}^{\langle \kappa \rangle }\) and

*g*are approximately orthogonal in \(L^2_{mQ}\).

For further details regarding the above computations, we refer to Subsection 4.8 of [15].

*u*vanishes to order

*n*at the root point \(w_0\). As a consequence, the function

*n*at the root point \(w_0\). Moreover, the small perturbations \(u_{m,n,w_0}\) and \({{\mathbf {P}}}_{m,n+1,w_0}\mathrm {k}^\star _{m,n,w_0}\) vanish at least to order

*n*at \(w_0\). It follows that \(\tilde{\mathrm {k}}_{m,n,w_0}\) vanishes precisely to the correct order, that is to say,

*C*is close to being positive real, since the \({\bar{\partial }}\)-correction \(u_{m,n,w_0}\) is small. Indeed, we have \(C=(1+\mathrm {O}(\mathrm e^{-\alpha _1m}))\,C_1\) where the constant \(C_1>0\) may depend on all the parameters but the parameter \(\alpha _1>0\) is a uniform constant. Since the function \(\tilde{\mathrm {k}}_{m,n,w_0}\) is automatically orthogonal to \(A^2_{mQ,n+1,w_0}\), it follows that \(\tilde{\mathrm {k}}_{m,n,w_0}\) equals a scalar multiple of the true root function \(\mathrm {k}_{m,n,w_0}\):

*D*depending only on

*Q*. The only remaining issue is that the error terms are slightly worse than claimed. However, by replacing \(\kappa \) with an integer larger than \(\kappa +2+A^2D\) and by deriving the expansion with the indicated higher accuracy, we conclude that the desired error terms may be obtained as well.

*m*is big enough. In view of (2.5.7) the first integral on the right-hand side equals \(1+\mathrm {O}(m^{-\kappa -\frac{1}{3}})\). For any \(z\in \mathrm {supp}(\chi _0){\setminus }\Omega _{\tau ,w_0,m}\) we have the bound \(2m(Q-\breve{Q}_{\tau ,w_0})(z)\ge A^2D\log m\) where

*D*is the positive constant encountered previously. Consequently, we have the estimate

*A*is chosen large enough, it follows that

### Proof sketch of Theorem 1.4.1

The proof of Theorem 1.4.1 is entirely analogous to the above proof of Theorem 1.6.2, essentially amounting to putting \(\tau =0\) in the latter context. In the setting of Theorem 1.4.1, there exists already a forbidden region around the point \(w_0\), and hence permits us to consider \(\tau =0\). Indeed, the reason why we required that \(\tau >0\) in the context of Theorem 1.6.2 was to allow for the instance when the off-spectral component \(\Omega _{\tau ,w_0}\) shrinks down to the point \(\{w_0\}\) as \(\tau \rightarrow 0\). \(\quad \square \)

### 2.6 Interface asymptotics of the Bergman density

In this section we show how to obtain the error function transition behavior of Bergman densities at interfaces, where the interface may occurs as a result of a region of negative curvature (understood as where \(\varDelta Q<0\) holds in terms of the potential *Q*) or as a consequence of dealing with partial Bergman kernels. Here, we focus on the the partial Bergman kernel analysis. In fact, we may think of the first instance of the full Bergman kernel as a special case and maintain that it is covered by the presented material.

The following Corollary of the main theorem summarizes the asymptotics of normalized off-spectral partial Bergman kernels in a suitable form. The domains \(\Omega _{\tau ,w_0,m}\) are as in Theorem 1.6.2, for a given positive parameter *A* chosen suitably large.

### Corollary 2.6.1

### Proof

In view of the decomposition (2.2.1), this is just the assertion of Theorem 1.6.2 with accuracy \(\kappa =1\). \(\quad \square \)

We proceed with a sketch of the error function asymptotics at interfaces, in particular we point out why we may carry on exactly as in the proof of Theorem 1.4.1 of [15].

### Proof sketch of Corollary 1.7.1

## 3 The Foliation Flow for More General Area Forms

### 3.1 More general area forms

*V*is a positive \(C^2\)-smooth function which is real-analytic in a neighborhood of the fixed smooth spectral interface of interest, which meet the polynomial growth bound

*V*, we factor \(V\mathrm {dA}=V_{{\mathbb {S}}}\mathrm {dA}_{{\mathbb {S}}}\), where \(V_{{\mathbb {S}}}(z)=(1+|z|^2)^2V(z)\), and see that our weighted measure is

*n*with respect to the \(L^2\)-space with measure \(\mathrm e^{-2mQ}V_{{\mathbb {S}}}\,\mathrm {dA}_{{\mathbb {S}}}\) becomes isometrically isomorphic to the \(L^2\)-space of rational functions on the sphere \({\mathbb {S}}\) with a pole of order at most

*n*at the origin, with respect to the \(L^2\)-space with measure \(\mathrm e^{-2mQ\circ \lambda }V_{{\mathbb {S}}}\circ \lambda \, \mathrm {dA}_{{\mathbb {S}}}\). This provides an extension of the scale of root functions to zeros of negative order (i.e. poles), and the apparent similarities between orthogonal polynomials and root functions may be viewed in this light. This analogy goes even deeper than that. Assuming that 0 is an off-spectral point for the weighted \(L^2\)-space with measure \(\mathrm e^{-2mQ\circ \lambda }V_{{\mathbb {S}}}\circ \lambda \,\mathrm {dA}_{{\mathbb {S}}}\), we may multiply by a suitable power of the conformal mapping from the off-spectral region to the unit disk \({{\mathbb {D}}}\), which preserves the origin, to obtain a space of functions holomorphic in a neighborhood of the off-spectral region. Hörmander-type estimates for the \({\bar{\partial }}\)-equation then permit us to correct the functions so that they are entire, with small cost in norm.

*Q*. Indeed, if we consider \({{\tilde{Q}}}=Q-m^{-1}h\) for some smooth function

*h*of modest growth, we have that

### 3.2 The asymptotics of root functions and orthogonal polynomials for more general area forms

*n*or higher. These are closed subspaces of \(A^2_{mQ,V}\) which get smaller as

*n*increases: \(A^2_{mQ,V,n+1,w_0}\subset A^2_{mQ,V,n,w_0}\). The successive difference spaces \(A^2_{mQ,V,n,w_0}\ominus A^2_{mQ,V,n+1,w_0}\) have dimension at most 1. If the dimension equals 1, we single out an element \(\mathrm {k}_{m,n,w_0,V}\in A^2_{mQ,V,n,w_0}\ominus A^2_{mQ,V,n+1,w_0}\) of norm 1, which has positive derivative of order

*n*at \(w_0\). In the remaining case when the dimension equals 0 we put \(\mathrm {k}_{m,n,w_0,V}=0\). As before, we call \(\mathrm {k}_{m,n,w_0,V}\)

*root functions*, and observe that these are the same objects we defined earlier for \(V=1\) in terms of an extremal problem.

### Theorem 3.2.1

*V*, with respect to the interface \(\partial \Omega _{\tau ,w_0}\), we have, using the notation of the same theorem, for fixed accuracy and a given positive real

*A*, the asymptotic expansion of the root function

*A*, where \(\tau =\frac{n}{m}\), and the implied constant is uniform. Here, the main term \({{\mathcal {B}}}_{0,\tau ,w_0}\) is zero-free and smooth up to the boundary on \(\Omega _{\tau ,w_0}\), positive at \(w_0\), with prescribed modulus

The proof of this theorem is analogous to that of Theorem 1.6.2, given that we have explained how to modify the orthogonal foliation flow with respect to the general area form in Lemma 3.3.2. The lemma is applied with \(s=m^{-1}\). We omit the necessary details.

*Q*in the canonical positioning procedure, and we put analogously

*f*in the Hardy space \(H^2\) that vanish at the origin.

### Theorem 3.2.2

*l*, which is allowed to assume only non-negative values. However, the same definition works also for \(l< 0\), which is necessary for the present application. In terms of the operators \(\mathbf {M}_k\) and \({{\mathbf {L}}}_k\), we have

### 3.3 The flow modified by a conformal factor

We proceed first to modify the book-keeping slightly by formulating an analogue of Definition 2.4.1, which applies to weights after canonical positioning.

### Definition 3.3.1

*R*,

*W*) of non-negative \(C^2\)-smooth weights defined on \({{\mathbb {D}}}(0,\eta )\) is said to belong to the class \({\mathfrak {W}}_\circledast (\eta ,\sigma )\) if \(R\in {\mathfrak {W}}(\eta ,\sigma )\) and if the weight

*W*meets the following conditions:

- (i)
*W*is real-analytic and zero-free in the neighborhood \({\mathbb {A}}(\eta ^{-1},\eta )\) of the unit circle \({{\mathbb {T}}}\), - (ii)
The polarization

*W*(*z*,*w*) of*W*extends to a bounded holomorphic function of \((z,{{\bar{w}}})\) on the \(2\sigma \)-fattened diagonal annulus \(\hat{{\mathbb {A}}}(\sigma , \eta )\), which is also bounded away from 0.

*S*of pairs (

*R*,

*W*) is said to be a uniform family in \({\mathfrak {W}}_\circledast (\eta ,\sigma )\) if the weights

*R*with \((R,W)\in S\) are confined to a uniform family in \({\mathfrak {W}}(\eta ,\sigma )\), while

*W*(

*z*,

*w*) is uniformly bounded and bounded away from 0 in \(\hat{{\mathbb {A}}}(\eta ,\sigma )\).

*f*(

*z*,

*w*) is holomorphic in \((z,{{\bar{w}}})\) on the set \(\hat{{\mathbb {A}}}(\sigma _1, \eta _1)\), then the function \(f_{{{\mathbb {T}}}}(z)=f(z,{\bar{z}}^{-1})\) may be continued holomorphically to the annulus \({\mathbb {A}}(\eta _1^{-1},\eta _1)\).

We proceed with the main result of this section.

### Lemma 3.3.2

*s*,

*t*small enough it holds that the domains \(\psi _{s,t}\big ({{\mathbb {D}}}\big )\) increase with

*t*, while they remain contained in \({{\mathbb {D}}}(0,\eta _1)\). Moreover, for \(\zeta \in {{\mathbb {T}}}\), we have

*s*, when

*t*varies in the interval \([-\beta _s,\beta _s]\) with \(\beta _s:=s^{1/2}\log \frac{1}{s}\), the flow of loops \(\{\psi _{s,t}({{\mathbb {T}}})\}_t\) cover a neighborhood of the circle \({{\mathbb {T}}}\) of width proportional to \(\beta _s\) smoothly. In addition, the main term \(B_{0}\) is zero-free, positive at the origin, and has modulus \(|B_0|=\pi ^{-\frac{1}{4}}(\varDelta R)^{\frac{1}{4}}W^{-\frac{1}{2}}\) on \({{\mathbb {T}}}\), and the other terms \(B_{j}\) are all real-valued at the origin. The implied constant in (3.3.1) is uniformly bounded, provided that (

*R*,

*W*) is confined to a uniform family of \({\mathfrak {W}}_\circledast (\eta _0,\sigma _0)\).

In order to obtain this lemma, we need to modify the algorithm which gives the original result. We proceed to sketch an outline of this modification. The omitted details are available in [15], and we try to guide the reader for easy reading.

*n*, we introduce

### Proof of Lemma 3.3.2

*R*, and from this we may obtain the coefficients \(\hat{\psi }_{0,l}\). Indeed, if we take logarithms of both sides of the equation and multiply by

*s*we obtain

*t*in the interval \([-\beta _s,\beta _s]\), so that \(J_{\Psi _s}\) gets defined on the annulus \({\mathbb {A}}(1-\beta _s,1+\beta _s)\), provided that the coefficient functions which define \(\psi _{s,t}\) can be found. Assuming some reasonable stability with respect to the variable

*s*as \(s\rightarrow 0^+\) in (3.3.4), we obtain in the limit that

*t*and locally near \({{\mathbb {T}}}\). For \(t>0\), we choose the curve outside the unit circle, while for \(t<0\) we choose the other one. We normalize the mapping \(\psi _{0,t}\) so that it preserves the origin and has positive derivative there. In this fashion, the coefficients \(\hat{\psi }_{0,l}\) get determined uniquely by the level set condition. We note that the smoothness of the level curves was worked out in some detail in Proposition 4.2.5 in [15]. Moreover, the coefficient functions \(\hat{\psi }_{0,l}\) are given in terms of Herglotz integrals as in Proposition 4.6.1 of [15], with the obvious modifications.

*s*and

*t*, by the multivariate Taylor formula, this is equivalent to having the system of equations

*R*,

*W*) remains confined to a uniform family of \({\mathfrak {W}}_\circledast (\eta _0,\sigma _0)\), the result follows.

*s*and

*t*are given as follows. When \(j\ge 0\) and \(l\ge 1\), we have

*R*,

*W*) remains in a uniform family in \({\mathfrak {W}}_\circledast (\eta _0,\sigma _0)\), and may be explicitly written down using the multivariate Faà di Bruno’s formula. The crucial point for us is the dependence structure of the functions \(\mathfrak {T}_{j,l,W}\), which remains the same as in the algorithm for the orthogonal polynomials:

- (\(\mathfrak {T}\)-i)
*For*\(l\ge 1\),*the function*\(\mathfrak {T}_{j,l,W}\)*is an expression in terms of the functions*\(b_0,\ldots ,b_j\)*as well*\(\hat{\psi }_{p,q}\)*for indices*Open image in new window*with*\((p,q)\prec _{\mathrm {L}}(j+1,l-1)\),*and also involves**R**and**W*, whereas- (\(\mathfrak {T}\)-ii)
*For*\(l=0\),*the function*\(\mathfrak {T}_{j,0,W}\)*is an expression in terms of*\(b_0,\ldots ,b_{j-1}\)*and*\(\hat{\psi }_{p,q}\)*for indices*Open image in new window*with*\((p,q)\prec _{\mathrm {L}}(j+1,0)\),*and also involves**R**and**W*.

*W*is strictly positive in a fixed neighborhood of the unit circle \({{\mathbb {T}}}\), so that \(\log W\) is a real-analytic function in the same region. A natural approach to the computations is to write

*t*close to 0. However, for \(t\ne 0\) there are two such loops, one which is inside the circle \({{\mathbb {T}}}\), and one which is outside. For \(t>0\) we choose the outside loop, whereas for \(t<0\) we instead choose the inside loop. This way, the domain enclosed by \(\Gamma _t\) grows with

*t*.

*Q*and the conformal factor

*V*as well as the regularity of the conformal mapping \(\varphi _{\tau ,w_0}\), the function

*W*is real-analytic and positive in a neighborhood of \({{\mathbb {T}}}\), and moreover the pair (

*R*,

*W*) meets the regularity requirements of Definition 3.3.1. This means that essentially we are in the same setting as explained in [15]. For instance, we may conclude that the function \(b_{0}\) given by (3.3.13) extends as a bounded holomorphic function on a disk \({{\mathbb {D}}}(0,\eta _1)\) with radius \(\eta _1>1\).

We proceed to Step 3 with \(j_0=1\). We note that Open image in new window entails that \(0\le j\le \kappa \).

Step 3. To begin with, we have an integer \(1\le j_0\le \kappa \) for which we have already successfully determined the coefficient functions \(b_j\) for \(0\le j\le j_0-1\) as well as \(\hat{\psi }_{j,l}\) for all Open image in new window with \((j,l)\prec _{\mathrm {L}}(j_0,0)\). Note that is known to be so for \(j_0=1\), by Steps 1 and 2. In this step, we intend to determine all the coefficient functions \(\hat{\psi }_{j,l}\) with Open image in new window and \((j,l)\prec _{\mathrm {L}}(j_0+1,0)\), and keep track of the Eq. (3.3.9) with Open image in new window that get solved along the way. The induction hypothesis includes the assumption that the system of Eq. (3.3.9) holds for all Open image in new window with \(l\ge 1\) and \((j,l)\prec _{\mathrm {L}}(j_0-1,1)\) (this is vacuous for \(j_0=1\)), as well as for (*j*, *l*) with \(0<j<j_0\) and \(l=0\).

*l*, starting with \(l=0\). Assume for the moment that it has been carried out for all \(l=0,\ldots ,l_0-1\). The coefficient \(\hat{\psi }_{j_0,l_0}\) which we are looking for appears as the leading term in the Eq. (3.3.10) corresponding to \((j,l)=(j_0-1,l_0+1)\), which reads

*j*,

*l*) with \(0<j<j_0\) and \(l=0\). This completes Step 3.

*j*,

*l*) with \(l=0\) and \(0<j\le j_0\), as well as for all Open image in new window with \(l\ge 1\) and \((j,l)\prec _{\mathrm {L}}(j_0,1)\). This completes Step 4, and we have extended the set of known data so that we may proceed to Step 3 with \(j_0\) replaced with \(j_0+1\). Here, we should insert a word on smoothness. If

*f*is real-analytically smooth along \({{\mathbb {T}}}\), then \({{\mathbf {H}}}_{{\mathbb {D}}}[f]\) gets to be holomorphic in a larger disk \({{\mathbb {D}}}(0,\eta )\) for some \(\eta >1\). Hence real-analytic smoothness carries over to the next step in the iterative procedure.

The above algorithm continues until all the unknowns have been determined, up to the point where the whole index set Open image in new window has been exhausted. In the process, we have in fact solved the system of Eq. (3.3.9) for all indices Open image in new window. This means that if we form \(h_s\) and \(\psi _{s,t}\) in terms of the functions \(b_j\) and \(\hat{\psi }_{j,l}\) obtained with the above algorithm, and put \(f_s=\exp (h_s)\), we find by exponentiating the Taylor series expansion of \(\varpi _{s,t}\) in the parameters *s* and *t* that the flow Eq. (3.3.1) holds. This completes the sketch of the proof of the lemma. \(\quad \square \)

### 3.4 Orthogonal polynomial asymptotics for a general area form

The results concerning root functions for more general area forms apply also to the setting of orthogonal polynomials. Although many things are pretty much the same, we make an effort to explain what the precise result is in this context.

*Q*which applies to the setting of orthogonal polynomials. We recall that the spectral droplet \(\mathcal {S}_\tau \) is the contact set

### Definition 3.4.1

*Q*is \(\tau \)-admissible if the following conditions are met:

- (i)
\(Q:{{\mathbb {C}}}\rightarrow {{\mathbb {R}}}\) is \(C^2\)-smooth,

- (ii)
*Q*meets the growth bound$$\begin{aligned} \tau _Q:=\liminf _{|z|\rightarrow +\infty }\frac{Q(z)}{\log |z|}>\tau >0, \end{aligned}$$ - (iii)
The unbounded component \(\Omega _\tau \) of the complement of the spectral droplet \(\mathcal {S}_\tau \) is simply connected on the Riemann sphere \(\hat{{{\mathbb {C}}}}\), with real-analytic Jordan curve boundary,

- (iv)
*Q*is strictly subharmonic and real-analytically smooth in a neighborhood of the boundary \(\partial \Omega _\tau \).

*V*is assumed to be nonnegative, positive near the curve \(\partial \Omega _\tau \), and real-analytically smooth in a neighborhood of \(\partial \Omega _\tau \) with at most polynomial growth or decay at infinity (3.1.1). We denote by \(\phi _\tau \) the surjective conformal mapping

*Q*on the boundary \(\partial \Omega _\tau \), and whose imaginary part vanishes at infinity.

*n*, positive leading coefficient, and unit norm in \(A^2_{mQ,V}\). They have the additional property that

### Theorem 3.4.2

*Q*is \(\tau \)-admissible for \(\tau \in I_0\), where \(I_0\) is a compact interval of the positive half-axis. Suppose in addition that

*V*meets the above regularity requirements. Given a positive integer \(\kappa \) and a positive real

*A*, there exists a neighborhood \(\Omega ^{\circledast }_{\tau }\) of the closure of \(\Omega _{\tau }\) and bounded holomorphic functions \({{\mathcal {B}}}_{j,\tau ,V}\) on \(\Omega ^{\circledast }_{\tau }\), as well as domains \(\Omega _{\tau ,m}=\Omega _{\tau ,m,\kappa ,A}\) with \(\Omega _{\tau }\subset \Omega _{\tau ,m}\subset \Omega ^{\circledast }_{\tau }\) which meet

In view of Lemma 3.3.2, the construction of approximately orthogonal quasi-polynomials may be carried out in the same way as in the case when \(V=1\).

### 3.5 \({\bar{\partial }}\)-surgery in the context of more general area forms

*V*, we need to mention some modifications that are required. We recall that

*V*meets the conditions (3.1.1) and (3.1.2) and put

*m*is large enough. Here, \(\Omega _\tau \) is the unbounded component of the off-spectral set \(\mathcal {S}_\tau ^c\). Moreover, it follows from the growth and decay bounds (3.1.1) that \(Q_m\) meets the required growth bound for large enough

*m*. We consider the solution \(\hat{Q}_{\tau ,m}\) to the obstacle problem

*Q*and the conformal factor

*V*, the coincidence set

*m*. Indeed, by the main theorem of [20], the free boundary moves with a smooth normal velocity under a smooth perturbation, and it follows that \(\partial \mathcal {S}_{\tau ,m}\) is contained in a \(\mathrm {O}(m^{-1})\)-neighborhood of \(\partial \mathcal {S}_\tau \). We note that \(\hat{Q}_{\tau ,m}\) is automatically harmonic off \(\mathcal {S}_{\tau ,m}\) and that \(\varDelta \hat{Q}_{\tau ,m}\ge 0\) holds generally.

*u*to the problem

*u*is holomorphic off the compact set \(\mathcal {S}_{\tau ,m}\), this estimate implies a polynomial growth bound \(u(z)=\mathrm {O}(|z|^{n-1})\) as \(|z|\rightarrow +\infty \).

After these modifications, the \({\bar{\partial }}\)-surgery may be performed as in the earlier context of the planar area measure \(\mathrm {dA}\). For the details, we refer to Section 4.9 in [15] and the comments in Sect. 2.5 above.

## Notes

### Acknowledgements

Open access funding provided by Royal Institute of Technology. We wish to thank Steve Zelditch, Bo Berndtsson, and Robert Berman for their interest in this work. In addition we should thank the anonymous referee for his or her helpful efforts. Funding was provided by Vetenskapsrådet (Grant No. 2016-04912).

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