### 数学代写|数论作业代写number theory代考|Asymptotics of Random Resonances

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## 数学代写|数论作业代写number theory代考|Delta-Interactions

This paper is written in Memory of Erik Balslev. Erik’s pioneering works have been very influential in many areas of analysis, mathematical physics, and analytical $\mathrm{~ n u m b e ̀ r ~ t h e o o r y . ~ S o ̄ m e ~ o f ~ h i s ~ r e s e a r a r c h ~ t o ̄ p i c s ~ a a r e ̄ ~ c l o s e l l y ~ c o n n n e ̀ c t e}$ paper. In fact, one of Erik’s major contributions into mathematical physics is the clarification of the very concept of resonance $[11-14,16]$. Moreover, the technique of complex dilations that he developed in cooperation with Jean-Michel Combes [15] has played an important role in making accessible tools of analytic perturbation theory in problems involving resonances and eigenvalues embedded in continuous spectra. The connection of spectral theory with problems of analytic number theory, that Erik discovered and masterly developed in a series of important papers [1720], has inspired us to the study of the interplay between resonances, exponential polynomials, and point interactions.

In this paper we introduce a 3-D continuous model for random resonances using Hamiltonians of a generalized Schrödinger type involving point interactions. To make the Hamiltonian random, we assume that the interactions are generated by a point process with suitable properties. The main goal of this paper is to introduce main notions and problems for related random resonances and consider some of their asymptotic properties on relatively simple examples of binomial point processes.
While Schrödinger operators with random point interactions have been introduced in mathematical papers and their self-adjoint spectra have been investigated (see [3, $28,33,35,38]$ and references therein), it seems that the resonances for such models are not yet adequately mathematically studied.

For other models of random resonances, the mathematical theory has been attracting an increasing attention during recent years. It is worth to mention the monograph [51] and the paper [39]. One of the problems suggested in the introduction to [51] as a promising direction of future research concerns the connection between Weyl asymptotics and asymptotics of random resonances. We would like to note that Sect. 3 of the present paper addresses a somewhat connected problem in the context

of Schrödinger operators with random point interactions. Namely, we prove that the Weyl-type asymptotics, which has been recently introduced for deterministic point interactions in [44], takes place almost surely (a.s.) for our stochastic example.
From a more general perspective, random resonance effects were intensively studied in Physics (see, e.g., the literature in [39]). One of the first mentioning in the mathematical context of resonances of random Schrödinger operators known to us is in the paper [32], where the question of estimation of the support of distribution of random resonances served as one of the motivations for a resonance optimization problem (see also [37] for a more recent discussion of this interplay).

We shall present now in detail the model of random resonances that will be studied in this paper. Let $\Upsilon$ be a point process on $\mathbb{R}^{3}$, let $(\Omega, \mathcal{F}, \mathbb{P})$ be the underlying complete probability space, and let $\eta_{\Upsilon}$ be the random (counting) measure associated with $\Upsilon$. Throughout the paper we assume that
(A0) the point process $\Upsilon$ is simple and finite.

## 数学代写|数论作业代写number theory代考|Point Process of Random Resonances

Let $\left{a_{n}\right}_{n \in \mathbb{N}{0}}$ be a sequence of $\mathbb{C}$-valued random variables on the complete probability space $(\Omega, \mathcal{F}, \mathbb{P})$. Then the random power series $F(z)=\sum{n=0}^{\infty} a_{n} z^{n}$ is said to be a random entire function $(o n \mathbb{C}$ ) if the radius of convergence of $F$ is a.s. equal to $\infty$. The random entire function $F$ is said to be a random polynomial if a.s. $a_{n} \neq 0$ only for a finite number of indices $n \in \mathbb{N}_{0}$ (we refer to $[9,10,23]$ for basic facts of the theories of random analytic functions and random polynomials).

In this section, it will be shown that the multiset $\Sigma\left(H_{\Upsilon}\right)$ of resonances of the random operator $H_{\Upsilon}$ is a.s. the multiset of zeros of a random entire function. Then it is easy to see that $\Sigma\left(H_{\Upsilon}\right)$ is a point process in $\mathbb{C}$.

Definition 2.1 Consider a finite simple point process $\beta=\left{\beta_{j}\right}_{j=1}^{# \rho}$ on $\mathbb{C}$. Let $\left{p_{j}\right}_{j=1}^{\infty}$ be a sequence of (complex valued) random polynomials. Then any random function of the form
$$\sum_{j=1}^{# \beta} e^{\beta_{j} z} p_{j}(z)$$
is a.s. defined on the whole complex plane $\mathbb{C}$ and is said to be a random exponential polynomial.

It is easy to check that a random exponential polynomial is a random entire function (concerning the theory of deterministic exponential polynomials we refer to $[21,22])$.

## 数学代写|数论作业代写number theory代考|Asymptotic Density and Weyl-Type Asymptotics

A substantial part of the mathematical studies of deterministic resonances is devoted to the asymptotics of the their counting function
$$\mathfrak{N}{H{Y}}(R)=#\left{k \in \Sigma\left(H_{Y}\right):|k| \leq R\right}$$
In $[44]$, the asymptotics $\mathfrak{N}{H{Y}}(R)=\frac{C}{\pi} R+O(1)$ as $R \rightarrow \infty$ with a certain constant $C \geq 0$ was established for deterministic Hamiltonians $H_{Y}$ with $# Y=N \in \mathbb{N}$ point interactions and it was proved that $C \leq V(Y):=\max {\sigma \in S{\psi Y}} \sum_{j=1}^{# Y}\left|Y_{j}-Y_{\sigma(j)}\right| .$ The number $V(Y)$ was called in [44] the size of the set $Y$. In the case $C=V(Y)$, it was said (slightly changing the wording in [44]) that the Weyl-type asymptotics of $\mathfrak{N}{H{Y}}(R)$ takes place.

We use in the present paper the terminology of [7] and say that $\operatorname{Ad}\left(H_{Y}\right):=C / \pi$ is the total asymptotic density of resonances of $H_{Y}$. This is motivated by the equality
$$\operatorname{Ad}\left(H_{Y}\right)=\lim {R \rightarrow \infty} \frac{\mathfrak{N}{H_{Y}}(R)}{R} \quad \text { (see (3.1) and the line following it). }$$

By Theorem 2.2, the total asymptotic density of random resonances $\operatorname{Ad}\left(H_{\Upsilon}\right)$ is an $[0,+\infty]$-valued random variable for any point process $\Upsilon$ satisfying $(\mathrm{A} 0)$. Combining this with the deterministic result of [44] one sees that $\operatorname{Ad}\left(H_{\Upsilon}\right)$ is a $[0,+\infty)$-valued random variable and that
$$\operatorname{Ad}\left(H_{\Upsilon}\right) \leq \frac{V(\Upsilon)}{\pi} \text { a.s. }$$
The main result of this section says, roughly speaking, that for point processes $\Upsilon$ with good enough ‘diffuse’ sampling distributions the Weyl-type asymptotics for random resonances of $H_{\Upsilon}$ holds with the probability 1 . For the sake of simplicity, we prove this result only for the uniform binomial processes $\Theta\left(m, \mathbb{B}_{r}\right)$ in $\mathbb{R}^{3}$-balls (see Sect. 1).

## 数学代写|数论作业代写number theory代考|Delta-Interactions

(A0) 点过程Υ是简单和有限的。

## 数学代写|数论作业代写number theory代考|Point Process of Random Resonances

\ sum_ {j = 1} ^ {# \ beta} e ^ {\ beta_ {j} z} p_ {j} (z)\ sum_ {j = 1} ^ {# \ beta} e ^ {\ beta_ {j} z} p_ {j} (z)

## 数学代写|数论作业代写number theory代考|Asymptotic Density and Weyl-Type Asymptotics

\mathfrak{N}{H{Y}}(R)=#\left{k \in \Sigma\left(H_{Y}\right):|k| \leq R\right}\mathfrak{N}{H{Y}}(R)=#\left{k \in \Sigma\left(H_{Y}\right):|k| \leq R\right}

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