We consider the limit measures induced by the rescaled eigenfunctions of Schrödinger operators with even confining potentials. We show that the limit measure is supported on $$[-1,1]$$ and with the density proportional to $$(1-|x|^\beta )^{-1/2}$$ when the non-perturbed potential resembles $$|x|^\beta $$ , $$\beta >0$$ , for large x, and with the uniform density for super-polynomially growing potentials. We compare these results to analogous results in orthogonal polynomials and semiclassical defect measures.

We study the semigroup generated by the hypoelliptic Laplacian on the circle and the maximal bounded holomorphic extension of this semigroup. Using an orthogonal decomposition into harmonic oscillators with complex shifts, we describe the domain of this extension and we show that boundedness in a half plane corresponds to absolute convergence of the expansion of the semigroup in eigenfunctions. This relies on a novel integral formula for the spectral projections which also gives asymptotics for Laguerre polynomials in a large parameter regime.

We consider the perturbations of the harmonic oscillator operator by an odd pair of point interactions, z(δx−b−δx+b). The spectrum of such operators is analyzed as a set of roots of the properly constructed entire (meromorphic) function. If z = ir, r real, as r → ∞, the number of non-real eigenvalues tends to infinity.

A brief proof of the statement that the zero-set of a nontrivial real-analytic function in $d$-dimensional space has zero measure is provided.

This is a survey of results from the last 10 to 12 years about the structure of the spectra of Hill-Schrodinger and Dirac operators. Let L be a Hill operator or a one-dimensional Dirac operator on the interval [0, pi]. If L is considered with Dirichlet, periodic, or antiperiodic boundary conditions, then the corresponding spectra are discrete and, for sufficiently large vertical bar n vertical bar, close to n(2) in the Hill case or close to n in the Dirac case (n is an element of Z). There is one Dirichlet eigenvalue mu(n) and two periodic (if n is even) or antiperiodic (if n is odd) eigenvalues lambda(-)(n) and lambda(+)(n) (counted with multiplicity). Asymptotic estimates are given for the spectral gaps gamma(n) = lambda(+)(n) - lambda(-)(n) and the deviations delta(n) = mu(n) - lambda(+)(n) in terms of the Fourier coefficients of the potentials. Moreover, precise asymptotic expressions for gamma(n) and delta(n) are found for special potentials that are trigonometric polynomials.

The remarkable Russian mathematician Evgenii Alekseevich Gorin passed away on 4 October 2018. He was born in Moscow, on 28 January 1936. His father Aleksei Fedorovich Gorin (1903–1953), the eldest son from a poor peasant family with 11 children, was from the village of Verkhnee Khoroshovo in Kolomna Uyezd (a Moscow governorate). At the age of 13 he started working in a factory, developed into a highly skilled locksmith, and then enrolled in and graduated from the Bauman Higher Technical School in Moscow. In his peak years he was a well-known Moscow engineer and the director of several large factories; during World War II he was the head of the important Brake-System Plant. Gorin’s mother, Lidiya Semenovna Gorina (1912–1991), was born in the city of Nikolaev and married his father when she worked at the Brake-System Plant. Before World War II, already a mother of two children, she graduated from the Moscow State Pedagogical Institute (now Pedagogical University) with a diploma in the Russian language and literature, and for a long time she was a director of studies in extended education courses for school teachers. Evgenii Gorin met his future wife Irina Aleksandrovna Krsovskaya in the Faculty of Mechanics and Mathematics at Moscow State University, where she was one year behind him. Subsequently, she worked for many years in the Department of Applied Mathematics at the Moscow Institute (now University) of Civil Engineering as an associate professor. For almost sixty years she was Gorin’s devoted friend and life companion. Their son Andrei graduated from the Moscow Institute of Electronic Engineering (now Moscow Institute of Electronics and Mathematics) with a diploma in computer progamming. He is now a successful businessman. Gorin and his wife had three grandsons, and a great grandson was born shortly before his death. Gorin went to Moscow School no. 167 on Degtyarnyi Lane (now School no. 2054), where his wonderful school teacher El’frida Moiseevna Abezgauz planted in him an interest in and love of mathematics. As a high-school student, Gorin was successful

We exploit the so-called form-local subordination in the analysis of non-symmetric perturbations of unbounded self-adjoint operators with isolated simple positive eigenvalues. If the appropriate condition relating the size of gaps between the unperturbed eigenvalues and the strength of perturbation, measured by the form-local subordination, is satisfied, the root system of the perturbed operator contains a Riesz basis and usual asymptotic formulas for perturbed eigenvalues and eigenvectors hold. The power of the abstract perturbation results is demonstrated particularly on Schrödinger operators with possibly unbounded or singular complex potential perturbations.

We consider the operator L = −(d/dx) + xy + w(x)y in L(R), where w(x) = s [δ(x− b)− δ(x+ b)] , b 6= 0 real, s ∈ C. This operator has a discrete spectrum: eventually the eigenvalues are simple and (0.1) λn = (2n+ 1) + s 2 κ(n) n + ρ(n) where (0.2) κ(n) = 1 2π [

For fixed positive alpha and beta, and a fixed integer n, n >= 2, we consider the family of matrices diag(a(1), a(2), . . . , a(n)) - diag(b(1), b(2), . . ., b(n)) Sigma(*), where all the a(k)'s and b(k)'s are positive, the geometric mean of the a(k)'s and b(k)'s must be a and beta, respectively, and Sigma(*) denotes the permutation matrix corresponding to the cycle (1, 2, . . . , n). C. Johnson, Z. Price, and I. Spitkovsky conjectured that in this family, the number of eigenvalues in the left half plane is maximized by alpha I - beta Sigma(*); we prove this conjecture. Moreover, the complete range of possibilities for the number of eigenvalues in the left half-plane is demonstrated: if alpha < beta, then any odd number between 1 and the maximum, inclusive, is attainable. (C) 2017 Elsevier Inc. All rights reserved.

The completeness, minimality, and basis property in L 2[0, π] and L p[0, π], p ≠ 2, are considered for systems of dilated functions u n (x) = S(nx), n ∈ N, where S is the trigonometric polynomial S(x) = Σ k=0 m a k sin(kx), a 0 a m ≠ 0. A series of results are presented and several unanswered questions are mentioned.

It is shown that two conditions f(a+⋅)−f(⋅)∈Lp(R), and (sinb⋅)f(⋅)∈Lp(R) guarantee f∈Lp(R), 1≤p<∞, if and only if ab is not in (πZ).

Aleksander Pelczynski was a leader in functional analysis for more than half of a century.Olek (as he was known to most who knew him) worked in Banach space theory much of his later life but previously made serious contributions to infinite dimensional topology and the theory of nuclear Frechet spaces.He kept in touch with workers in all these vineyards and was frequently an inspiration to young workers with keen insights and suggestions.He was famous for his signature question, "What did you prove last night?"Olek wrote many papers now considered to be classics.In the seventies he concentrated on how Banach space theory interfaced with harmonic analysis, complex variables, and probability.He frequently expressed the opinion that Banach space theory was an area that needed to "test its wares in other areas of mathematical endeavor" and, as was usually the case, he was a leader in such efforts.For many years Olek was the main line of communications between functional analysts from the East and West.Many will remember a one-page statement and proof of Victor Lomonosov's startling theorem on invariant subspaces.This page was the result of Olek's dictating the result to Czeslaw Bessaga, who was in the United States at the time, and requesting that Czeslaw make copies and send them to "our friends in America."Stan Kwapien tells of Olek's early academic life:In 1950 Olek, along with Czeslaw Bessaga and Stefan Rolewicz, participated in the Mathematical Olympiad for high school students, which was organized in Poland for the first time.Later at the University of Warsaw they met an exceptional team of teachers, including Banach's closest collaborator, Stanislaw Mazur, whose seminar had a decisive impact on their future mathematics.As a PhD student (1957-58) Olek published fourteen papers, six jointly with Bessaga and one jointly with Bessaga and Rolewicz.One of these papers with Bessaga, "On bases and unconditional convergence in Banach spaces," is one of the most cited papers in functional analysis and is considered by many to be a classic in the area.Olek defended his dissertation in December 1958 after a trip with Orlicz to China, a reward for his achievements in mathematics.For Olek this trip was unforgettable, a trip "to the end of the earth."It is quite likely that the visit contributed to the decision of the Chinese government to implement functional analysis in the basic Chinese curriculum.Boris Mityagin remembers Olek's time in the Soviet Union:In November of 1959, Pelczynski came to Moscow State University for a half year as a visiting researcher.At the time he was interested in problems on nuclear spaces initiated by Kolmogorov and Gelfand.In 1955 Kolmogorov introduced invariants for Frechet spaces based on the growth of compact sets (entropy) in a space.Pelczynski suggested closely related invariants, later called approximative and diametral dimension.These helped explain why various spaces of differentiable functions were mutually isomorphic or not.I recall fondly when our families spent the summer of 1975 together in Peredelkino near Moscow.Olek's daughter, Kasis, 5, spoke Russian with her mother Svetlana and that summer perfected her skill with the Russian language.

We consider families of non-self-adjoint perturbations of self-adjoint Schrödinger operators with single-well potentials. The norms of spectral projections of these operators are found to grow at intermediate rates from arbitrarily slowly to exponentially rapidly.

октябрь т. 72, вып. 5 (437) УСПЕХИ МАТЕМАТИЧЕСКИХ НАУК Михаил Захарович Соломяк 31 июля 2016 г. ушел из жизни Михаил Захарович Соломяк, выдающийся математик, специалист в области функционального анализа и спектральной теории, один из создателей Ленинградской (Санкт-Петербургской) школы спектральной теории операторов.Михаил Захарович родился 16 мая 1931 г. в Ленинграде.Его отец был крупным инженером-строителем, мать преподавала русский язык и литературу.Во время войны первые блокадные месяцы семья оставалась в Ленинграде.Осенью 1941 г. отец руководил строительством укреплений на участке Ленинградского фронта.30 января 1942 г.Михаил Захарович с матерью эвакуировались по льду Ладожского озера.Отец выехал в феврале 1942 г.; он руководил эвакуацией строительного отдела Ждановского завода в город Зеленодольск Татарской АССР.Там отец возглавил строительство кораблестроительного завода.В 1945 г. семья вернулась в Ленинград.С восьмого класса Михаил Захарович был участником математического кружка, организованного Виктором Абрамовичем Залгаллером.Традиции этого кружка стали основой известной системы подготовки молодых талантов.В старших классах Михаил Захарович был победителем городской олимпиады по математике.Впоследствии он сам организовывал эти олимпиады и руководил кружком для школьников.Михаил Захарович учился на математико-механическом факультете Ленинградского государственного университета с 1948 по 1953 г.Его будущая жена Тамара училась на том же курсе.В университете Михаил Захарович увлекся функциональным анализом, предметом тогда

We analyze perturbations of the harmonic oscillator type operators in a Hilbert space \({\mathcal{H}}\), i.e. of the self-adjoint operator with simple positive eigenvalues μ k satisfying μ k+1 − μ k ≥ Δ > 0. Perturbations are considered in the sense of quadratic forms. Under a local subordination assumption, the eigenvalues of the perturbed operator become eventually simple and the root system contains a Riesz basis.

If A is a compact operator in a Banach space and some power A(q) is nuclear, we give a criterion of Z(d)-symmetry of its spectrum sigma A in terms of vanishing of the traces TrA(n) for all n, n >= 0, n not equal 0 mod d, sufficiently large.

We consider the operator \(Ly = - (d/dx)^{2}y + x^{2} y + w(x) y, \quad y \text { in} L^{2}(\mathbb {R}),\) where \(w(x) = s \delta (x - b) + t \delta (x + b) , \quad b \neq 0 \, \, \text {real}, \quad s, t \in \mathbb {C}\). This operator has a discrete spectrum: eventually the eigenvalues are simple. Their asymptotic is given. In particular, if s=−t, \(\lambda _{n} = (2n + 1) + s^{2}\, \frac {\kappa (n)}{n} + \rho (n) \label {eq:abstractlam}\) where \(\kappa (n) = \frac {1}{2\pi } \left [(-1)^{n + 1} \sin \left (2 b \sqrt {2n} \right ) - \frac {1}{2} \sin \left (4 b \sqrt {2n} \right ) \right ]\) and \(\vert \rho (n) \vert \leq C \frac {\log n}{n^{3/2}}. \label {eq:abstracterr}\) If \(\overline {s} = -t\), the number T(s) of non-real eigenvalues is finite, and \(T(s) \leq \left (C (1 + \vert s \vert ) \log (e + \vert s \vert ) \right )^{2}\). The analogue of the above asymptotic is given in the case of any two-point interaction perturbation.

The aim of this note is to characterize all pairs of sufficiently smooth functions for which the mean value in the Cauchy mean value theorem is taken at a point which has a well-determined position in the interval. As an application of this result, a partial answer is given to a question posed by Sahoo and Riedel.

Consider the Hill operator $L(v) = - d^2/dx^2 + v(x) $ on $[0,\pi]$ with Dirichlet, periodic or antiperiodic boundary conditions; then for large enough $n$ close to $n^2 $ there are one Dirichlet eigenvalue $\mu_n$ and two periodic (if $n$ is even) or antiperiodic (if $n$ is odd) eigenvalues $\lambda_n^-, \, \lambda_n^+ $ (counted with multiplicity). We describe classes of complex potentials $v(x)= \sum_{2\mathbb{Z}} V(k) e^{ikx}$ in weighted spaces (defined in terms of the Fourier coefficients of $v$) such that the periodic (or antiperiodic) root function system of $L(v) $ contains a Riesz basis if and only if $$V(-2n) \asymp V(2n) \quad \text{as} \;\; n \in 2\mathbb{N}\;\; (\text{or} \; n \in 1+ 2\mathbb{N}), \;\; n \to \infty.$$ For such potentials we prove that $\lambda_n^+ - \lambda_n^- \sim \pm 2\sqrt{V(-2n)V(2n)} $ and $$\mu_n - \frac{1}{2}(\lambda_n^+ + \lambda_n^-) \sim -\frac{1}{2} (V(-2n) + V(2n)).$$

We consider the Hill operator $$ Ly = - y^{\prime \prime} + v(x)y, \quad 0 \leq x \leq \pi, $$ subject to periodic or antiperiodic boundary conditions ($bc$) with potentials of the form $$ v(x) = a e^{-2irx} + b e^{2isx}, \quad a, b \neq 0, r,s \in \mathbb{N}, r\neq s. $$ It is shown that the system of root functions does not contain a basis in $L^2 ([0,\pi], \mathbb{C})$ if $bc$ are periodic or if $bc$ are antiperiodic and $r, s$ are odd or $r=1$ and $s \geq 3. $