| Russian Mathematical Surveys |
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This article is cited in 2 scientific papers (total in 2 papers) Spectral inequality for Schrödinger's equation with multipoint potential P. G. Grinevichabc, R. G. Novikovde a Steklov Mathematical Institute of Russian Academy of Sciences, Moscow b Landau Institute for Theoretical Physics of Russian Academy of Sciences c Lomonosov Moscow State University d CMAP, CNRS, École Polytechnique, Institut Polytechnique de Paris, Palaiseau, France e Institute of Earthquake Prediction Theory and Mathematical Geophysics, RAS
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     To Iskander Asanovich Taimanov on his 60th birthday 1. IntroductionThe important role of exactly solvable models in quantum mechanics is well known. For equations of Schrödinger type there are many important potentials for which, in dimension $d=1$ this equation is exactly solvable for all energies, while in dimension $d=2$ it is exactly solvable for a single particular value of the energy. In the construction and investigation of such potentials methods of soliton theory are efficient, for instance, transformations of Darboux–Moutard type. The literature devoted to this line of research is quite extended. Without attempting to present a complete list, we note, for example, [1]–[8] and the references in these works. On the other hand, in dimension $d=2$ and higher, apart from equations with highly symmetric potential, the stock of cases that are exactly solvable for all energies is quite limited. One such case is Schrödinger’s equation with potential equal to a sum of point potentials of Bethe–Peierls type. For the first time such a point potential was introduced in [9] for $d=3$, to describe the interaction between neutrons and protons. Subsequently, point and multipoint scatterers were considered by many authors, including Thomas [10], Fermi [11], Zel’dovich [12], Berezin and Faddeev [13]; also see the monograph [14]. Also nowadays such potentials are under extensive investigation (for instance, see [3] and [15]–[25], as well as the references there). In this paper we consider the stationary Schrödinger equation (in dimensionless variables)
$$
\begin{equation}
-\Delta\psi+V(x)\psi=E\psi, \quad V(x)=v_0(x)+v(x), \quad x\in\mathbb{R}^d, \quad d=1,2,3,
\end{equation}
\tag{1}
$$
where $v_0(x)$ is a sufficiently regular real function on $\mathbb{R}^d$ which increases at infinity sufficiently rapidly, and $v(x)$ is a sum of $n$ point scatterers which can formally be expressed as It is well known that point scatterers can only be defined in dimensions $d=1,2,3$ (for instance, see [3]). If $d=1$, then a point scatterer is just a standard Dirac delta function with an arbitrary coefficient. For $d=2$ or $d=3$, $\varepsilon\delta(x)$ denotes a ‘renormalized’ delta function depending on the real parameter $\varepsilon=\varepsilon(\alpha)$. More precisely, a complex function $\psi$ is a solution of (1) if the following conditions are satisfied: 1) away from the points $y_j$ the function $\psi$ satisfies Schrödinger’s equation (1) with potential $v_0(x)$; 2) the following limit relations hold in a neighbourhood of the points $y_j$:
Remark 1. In the definition of a point scatterer we can start from a potential well of any shape and make scaling; for a generic well the limit does not depend on its initial shape. However, we can select a special shape of the well so as to construct another point potential, which can be regarded as a ‘renormalized’ $\delta'$-function (for $d=3$, see [26]). Remark 2. One motivation for introducing point potentials for $d=3$ in [9] was the absence of any kind of complete theory of strong interactions at that time. However, the simple idea that if the object has size much less than the wavelength, then we can treat it as a point, turned out to be efficient. The use of point potentials made it possible in [11], in particular, to explain why slow neutrons interact stronger with nuclei than fast ones [27]. The reason is that in wave problems, when the length of the scattered wave (for instance, the de Broglie wave of the scattered particle in quantum mechanics or the acoustic wave in acoustics problems) is much larger than the geometric size of the scatterer, then the effective scattering cross-section can be much larger than the geometric cross-section of the scatterer: it can have the order of the square of the wavelength. On the other hand, if the size of the scatterer is much larger than the wavelength, then the scattering cross-section is determined just by the size of the scatterer. In this paper we continue to develop the spectral theory of equation (1). More precisely, we consider a homogeneous boundary condition for this equation, including the Dirichlet, Neumann, and Robin cases, and the case of transmission eigenvalues. For such spectral problems we show, in particular, that if the energy $E$ is an eigenvalue with multiplicity $m>n$ for equation (1), where $V(x)\equiv v_0(x)$, then, in the case when $V(x)=v_0(x)+v(x)$, where $v(x)$ is the $n$-point potential defined above, this energy is also an eigenvalue with multiplicity at least $m-n$. This paper is a continuation of [22]. In particular, for $d=2,3$, when $v_0(x)\equiv 0$ all energies are transmission eigenvalues with infinite multiplicity. Hence this property also holds for all $n$-point potentials, as discovered in [22]. 2. Dirichlet, Neumann, and Robin problemsConsider equation (1) with Robin boundary condition
$$
\begin{equation}
a(x)\psi(x)+b(x)\,\frac{\partial\psi(x)}{\partial\nu}\bigg|_{\partial D}=0
\end{equation}
\tag{8}
$$
in a bounded domain $D$ with regular boundary $\partial D$, where $\partial/\partial\nu$ is the derivative in the direction of the outward normal $\nu$ to the boundary of the domain. Condition (8) covers the Dirichlet and Neumann boundary conditions as special cases. We also assume that $\operatorname{supp}v(x)$ lies in the interior of $D$. Theorem 1. Let $E$ be an eigenvalue with multiplicity $m>n$ for problem (1), (8), where $V(x)\equiv v_0(x)$. Then $E$ is also an eigenvalue with multiplicity at least $m-n$ for problem (1), (8) where $V(x)=v_0(x)+v(x)$ and $v(x)$ is the potential from (2). Remark 3. To our knowledge the result of Theorem 1 is new even for $v_0\equiv 0$. Remark 4. A nice illustration to Theorem 1 is the case when $D$ is a ball in $\mathbb{R}^3$, $v_0\equiv 0$, and the Robin condition reduces to the Diriclet or Neumann one. The reason is that there are eigenvalues with an arbitrarily high multiplicity in this problem. Theorem 1 follows from the lemma below. Lemma 1. Let $\psi(x)$ satisfy equation (1) for $V(x)\equiv v_0(x)$. Also let $\psi(y_j)=0$, $j=1,\dots,n$, where the points $y_j$ are as in (2). Then $\psi(x)$ also satisfies equation (1) for $V(x)=v_0(x)+v(x)$. Lemma 1 follows from the definition of a solution of (1) for $V(x) =v_0(x)+v(x)$, which involves formulae (3)–(7). Proof of Theorem 1. Let $\psi_l(x)$, $l=1,\dots,m$, be linearly independent eigenfunctions of spectral problem (1), (8) for $V(x)\equiv v_0(x)$. Consider the system of $n$ linear equations with respect to the $m$ unknowns $z_l$: Its space of solutions is at least $(m-n)$-dimensional. Now, for each set of $z_l$ satisfying (9) the function solves (1) for $V(x)\equiv v_0(x)$, satisfies the boundary condition (8), and vanishes at the points $y_j$. Hence, by Lemma 1 the function $\Psi(x)$ satisfies (1) for $V(x)\equiv v_0(x)+v(x)$ and satisfies the boundary condition (8). Moreover, as the solutions $\psi_l(x)$ are linearly independent, $\Psi(x)$ is distinct from identical zero if at least one coefficient $z_l$ is distinct from zero. Hence the space of functions $\Psi(x)$ constructed in accordance with (9) and (10) has dimension at least $m-n$. $\Box$
3. Transmission eigenvaluesThe problem of transmission eigenvalues is currently a subject of significant interest: see, for instance, [28]–[33] and [22]. An energy $E$ is called an interior transmission eigenvalue for equation (1) in a domain $D$ if there exists a non-trivial pair of functions $\phi(x)$, $\psi(x)$ such that and
$$
\begin{equation}
\psi(x)\equiv \phi(x), \quad \frac{\partial}{\partial\nu}\psi(x)\equiv \frac{\partial}{\partial\nu}\phi(x) \quad\text{for all} \ x\in\partial D.
\end{equation}
\tag{13}
$$
The dimension of the space of such pairs is called the multiplicity of the interior transmission eigenvalue. Note that for $V(x)\equiv 0$ and $d>1$ this multiplicity is infinite. As in § 2, we assume that $\operatorname{supp}v(x)$ lies in the interior of $D$. Theorem 2. Let $E$ be an interior transmission eigenvalue with multiplicity $m>n$ in the sense of (11)–(13) for equation (1) in the domain $D$, where $V(x)\equiv v_0(x)$. Then $E$ is also an interior transmission eigenvalue with multiplicity at least $m-n$ in the sense of (11)–(13) for problem (1), where $V(x)=v_0(x)+v(x)$ and $v(x)$ is the potential in (2). The proof of Theorem 2 is similar to the proof of Theorem 1. There are many papers containing results of the type: “if the potential $V(x)$ is a sufficiently regular function in $D$, then the interior transmission eigenvalues are discrete and have finite multiplicities” (for instance, see [31]–[33]). On the other hand Theorem 2 for $v_0(x)\equiv 0$ yields our recent result in [22] that in dimension $d=2,3$, when the potential $V(x)$ reduces to a multipoint scatterer $v(x)$ of the form (2) all energies $E\in\mathbb{C}$ are transmission eigenvalues with infinite multiplicity. Finally, in the context of the scattering problem for equation (1) in $\mathbb{R}^d$, an energy $E$ is called a transmission eigenvalue if 1 is an eigenvalue of the scattering operator $\widehat S_E$ for (1). This can also be formulated as follows: there exists a function $u(\theta)\in L^2({\mathbb S}^{d-1})$, where ${\mathbb S}^{d-1}$ is the unit sphere in $\mathbb{R}^d$, such that (1) has a solution $\Psi(x)$ with asymptotic behaviour
$$
\begin{equation}
\Psi(x)=\int_{{\mathbb S}^{d-1}} e^{i|k|\theta x} u(\theta)\,d\theta+ o\biggl(\frac{1}{|x|^{(d-1)/2}}\biggr) \quad \text{as} \ |x|\to\infty.
\end{equation}
\tag{14}
$$
The dimension of the linear space of such functions $u(\theta)$ is called the multiplicity of the transmission eigenvalue. Note that for $V(x)\equiv 0$ and $d>1$ all real positive energies are transmission eigenvalues with infinite multiplicity. Similarly to transmission eigenvalues in a bounded domain $D$, many authors have published results of the form “if the potential $V(x)$ is a sufficiently regular function with compact support on $\mathbb{R}^d$, then the transmission eigenvalues are discrete and have finite multiplicities” (for instance, see [31]–[33]). On the other hand, in [5] we constructed real spherically symmetric potentials $v_0(x)$ in the Schwartz class on $\mathbb{R}^2$ such that for any fixed energy $E=E_0>0$ the corresponding scattering operator $\widehat S_E$ is the identity operator. As a consequence, for these potentials $E_0$ is a transmission eigenvalue with infinite multiplicity. In addition, the following analogue of Theorem 2 holds. Theorem 3. Let $E$ be a transmission eigenvalue with multiplicity $m>n$ for equation (1) in $\mathbb{R}^d$, where $V(x)\equiv v_0(x)$. Then $E$ is also a transmission eigenvalue for equation (1) for $V(x)=v_0(x)+v(x)$ in $\mathbb{R}^d$ with multiplicity at least $m-n$, where $v(x)$ is the potential in (2). The proof of Theorem 3 is similar to the proofs of Theorems 1 and 2. As a consequence, after adding any finite set of point scatterers to the potentials $v_0(x)$ from [5] mentioned above, the energy $E_0$ remains a transmission eigenvalue with infinite multiplicity. In conclusion we mention that the fact that the boundary conditions are linear and homogeneous, rather than the particular form of these conditions, is significant in Theorems 1–3. 
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\Bibitem{GriNov22} |
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