arXiv subjects
Publications and source records attributed to Michael Solomyak.
The problem of finding eigenvalue estimates for the Schrödinger operator turns out to be most complicated for the dimension 2. Some important results for this case have been obtained recently. We discuss these results and establish their counterparts for the operators on the combinatorial and metric graphs corresponding to the lattice Z^2.
A survey of estimates on the number $N_-(\BM_{\a G})$ of negative eigenvalues (bound states) of the Sturm-Liouville operator $\BM_{\a G}u=-u"-\a G$ on the half-line, as depending on the properties of the function $G$ and the value of the coupling parameter $\a>0$. The central result is \thmref{S1/2a} giving a sharp sufficient condition for the semi-classical behavior $N_-(\BM_{\a G})=O(\a^{1/2})$, and the necessary and sufficient conditions for a "super-classical" growth rate $N_-(\BM_{\a G})=O(\a^q)$ with any given $q>1/2$. Similar results for the problem on the whole $\R$ are also presented. Applications to the multi-dimensional spectral problems are discussed.
For a two-dimensional Schr\"odinger operator $H_{\alpha V}=-\Delta-\alpha V$ with the radial potential $V(x)=F(|x|), F(r)\ge 0$, we study the behavior of the number $N_-(H_{\alpha V})$ of its negative eigenvalues, as the coupling parameter $\alpha$ tends to infinity. We obtain the necessary and sufficient conditions for the semi-classical growth $N_-(H_{\alpha V})=O(\alpha)$ and for the validity of the Weyl asymptotic law.
The construction of "sparse potentials", suggested in \cite{RS09} for the lattice $\Z^d,\ d>2$, is extended to a wide class of combinatorial and metric graphs whose global dimension is a number $D>2$. For the Schrödinger operator $-\D-\a V$ on such graphs, with a sparse potential $V$, we study the behavior (as $\a\to\infty$) of the number $N_-(-\D-\a V)$ of negative eigenvalues of $-\D-\a V$. We show that by means of sparse potentials one can realize any prescribed asymptotic behavior of $N_-(-\D-\a V)$ under very mild regularity assumptions. A similar construction works also for the lattice $\Z^2$, where D=2.
The behavior of the discrete spectrum of the Schrödinger operator $-\D - V$, in quite a general setting, up to a large extent is determined by the behavior of the corresponding heat kernel $P(t;x,y)$ as $t\to 0$ and $t\to\infty$. If this behavior is powerlike, i.e., \[\|P(t;\cdot,\cdot)\|_{L^\infty}=O(t^{-δ/2}),\ t\to 0;\qquad \|P(t;\cdot,\cdot)\|_{L^\infty}=O(t^{-D/2}),\ t\to\infty,\] then it is natural to call the exponents $δ,D$ "{\it the local dimension}" and "{\it the dimension at infinity}" respectively. The character of spectral estimates depends on the relation between these dimensions. In the paper we analyze the case where $δ<D$ that was insufficiently studied before. Our applications concern the combinatorial and the metric graphs.
For the discrete Schrödinger operator we obtain sharp estimates for the number of negative eigenvalues.
We discuss estimates on the number $N_-(α)$ of negative eigenvalues of the Schrödinger operator $-Δ-αV$ on regular metric trees, as depending on the properties of the potential $V\ge 0$ and on the value of the large parameter $α$. We obtain conditions on $V$ guaranteeing the behavior $N_-(α)=O(α^p)$ for any given $p\ge 1/2$. For a special class of trees we show that these conditions are not only sufficient but also necessary. For $p>1/2$ the order-sharp estimates involve a (quasi-)norm of $V$ in some `weak' $L_p$- or $\ell_p(L_1)$-space. We show that the results can be easily derived from the ones of an earlier paper by Naimark and the author, Proc. London Math. Soc. (3) 80 (2000), 690-724. The results considerably improve the estimates found in the recent paper by Ekholm, Frank, and Kovařík, arXive:0710.5500.
We consider the Dirichlet Laplacian in a family of narrow unbounded domains. As the width of these domains goes to 0, we study the asymptotic behavior of the eigenvalues that lie below the essential spectrum and the asymptotic behavior of the corresponding eigenfunctions.
We derive a two-terms asymptotics for eigenvalues of the Dirichlet Laplacian in a narrow strip of variable width. The asymptotics is taken with respect to a small paprameter that characterizes the width of the strip.
A family $A_α$ of differential operators depending on a real parameter $α\ge 0$ is considered. This family was suggested by Smilansky as a model of an irreversible quantum system. We find the absolutely continuous spectrum $σ_{a.c.}$ of the operator $A_α$ and its multiplicity for all values of the parameter. The spectrum of $A_0$ is purely a.c. and admits an explicit description. It turns out that for $α<\sqrt 2$ one has $σ_{a.c.}(A_α)= σ_{a.c.}(A_0)$, including the multiplicity. For $α\ge\sqrt2$ an additional branch of absolutely continuous spectrum arises, its source is an auxiliary Jacobi matrix which is related to the operator $A_α$. This birth of an extra-branch of a.c. spectrum is the exact mathematical expression of the effect which was interpreted by Smilansky as irreversibility.
A family $\BA_\a$ of differential operators depending on a real parameter $\a$ is considered. The problem can be formulated in the language of perturbation theory of quadratic forms. The perturbation is only relatively bounded but not relatively compact with respect to the unperturbed form. The spectral properties of the operator $\BA_\a$ strongly depend on $\a$. In particular, for $\a<\sqrt2$ the spectrum of $\BA_\a$ below 1/2 is finite, while for $\a>\sqrt2$ the operator has no eigenvalues at all. We study the asymptotic behaviour of the number of eigenvalues as $\a\nearrow\sqrt2$. We reduce this problem to the one on the spectral asymptotics for a certain Jacobi matrix.
Some results on the approximation of functions from the Sobolev spaces on metric graphs by step functions are obtained. The estimates are uniform with respect to all graphs of a given finite length, and the constant factors in the inequalities are sharp.
Eigenvalue behavior for the equation -λy"=Vu on the edges of a graph G of final total length, with a non-negative weight function V and under the Kirchhoff matching conditions at the vertices and zero boundary condition at at least one point of G, is studied. It is shown that the eigenvalues satisfy an inequality which involves the length |G| and the total mass corresponding to V but otherwise does not depend on the graph. Applications and generalizations of this result are also discussed.
The Schrödinger operator on a metric tree is a family of ordinary differential operators on its edges complemented by certain matching conditions at the vertices. The regular trees are highly symmetric. This allows one to construct an orthogonal decomposition of the space L_2 on the tree which reduces the Schrödinger operator with any symmetric weight. Using this decomposition, we analyse the spectrum of such operators, including the free Laplacian, under various assumptions about the tree and the potential.