Analysis as a Tool in Mathematical Physics

(Kronshtadt, Russia, 27 July 1936 – Auckland, New Zealand, 30 January 2016)1959, Leningrad University, Faculty of Physics, grad. 1958.

Pavlov’s contribution to science is not limited to his publications, he used to say that papers should be written for political reasons. Nevertheless, most of Pavlov’s ideas are reflected in his publications showing us different facets of his scientific personality.

The investigation of electron properties of polyatomic systems reduces, as a rule, to the spectral and scattering problems for the Schrödinger operator in $$ \mathbb L_2 \mathrm(R_3) $$ with an effective self-consistent potential $$ \mathit {V}(r)$$ that incorporates in some way the effect of electron–ion and electron–electron multi-particle interactions on a single valence electron.

Although I was not a direct student of Boris Sergeevich, he played a decisive role in my mathematical career. I got acquainted with him for the first time in our seminar on Complex Analysis in Leningrad.

Boris Pavlov, when I met him the first time, was a recent PhD student of Mikhail Birman at the Department of Physics of the LSU (Leningrad State University, now Saint-Petersburg State University). I suppose it happened around 1964–1965, when we both assisted on one of Birman’s beautiful advanced courses in spectral theory of selfadjoint operators (delivered, due to a bureaucratic caprice, at the old building of the History Department of LSU, near the Twelve Colleges Building on Vasilievsky Island).

Boris Pavlov’s mathematical quest was centered on the interplay between operator theory, complex analysis, and mathematical physics. More specifically, he had an amazing vision relating the spectral analysis of an operator to nontrivial phenomena in function theory and to relevant properties of associated physical systems. The bilateral shift, its restriction the unilateral shift, compressions thereof to coinvariant subspaces, and vector-valued versions of all these, where of course central players in the story.

Let A and A1 be unbounded selfadjoint operators in a Hilbert space $$ \mathcal H $$ . Following [3], we call A1 a singular perturbation of A if A and A1 have different domains $$ \mathcal D(A), D(A_1) $$ but $$ \mathcal D(A) \cap D(A_1) $$ is dense in $$ \mathcal H $$ and A = A1 on $$ \mathcal D(A) \cap D(A_1) $$ . In this note we specify without recourse to the theory of selfadjoint extensions of symmetric operators the conditions under which a given bounded holomorphic operator function in the open upper and lower half-planes is the resolvent of a singular perturbation A1 of a given selfadjoint operator A.For the special case when A is the standardly defined selfadjoint Laplace operator in L2( $$ \mathbb R^3 $$ ) we describe using the M.G. Krein resolvent formula a class of singular perturbations A1, which are defined by special selfadjoint boundary conditions on a finite or spaced apart by bounded from below distances infinite set of points in $$ \mathbb R^3 $$ and also on a bounded segment of straight line embedded into $$ \mathbb R^3 $$ by connecting parameters in the boundary conditions for A1 and the independent on A matrix or operator parameter in the Krein formula for the pair A, A1.

We study the leading coefficient in the asymptotic formula $$ \mathit N(R)=\frac{W}{\pi}R+O(1), R \rightarrow \infty $$ , for the resonance counting function $$ \mathit N(R) $$ of Schrödinger Hamiltonians with point interactions. For such Hamiltonians, the Weyl-type and non-Weyl-type asymptotics of $$ \mathit N(R) $$ was introduced recently in a paper by J. Lipovský and V. Lotoreichik (2017). In the present paper, we prove that the Weyl-type asymptotics is generic.

We consider a linear system composed of N+1 Schrödinger equations connected by point-mass-like interface conditions. We show that the system is exactly controllable with a Dirichlet boundary control at one end, and various homogeneous boundary conditions on the other end. The reachable set is characterized by spectral data. We then study the regularity of the reachable functions using a family of Riesz bases of asymmetric spaces.

For a scalar elliptic self-adjoint operator on a compact manifold without boundary we have two-term asymptotics for the number of eigenvalues between 0 and λ when λ → ∞, under an additional dynamical condition. (See [3, Theorem 3.5] for an early result in this direction.)In the case of an elliptic system of first order, the existence of two-term asymptotics was also established quite early and as in the scalar case Fourier integral operators have been the crucial tool. The complete computation of the coefficient of the second term was obtained only in the 2013 paper [2]. In the present paper we simplify that calculation. The main observation is that with the existence of two-term asymptotics already established, it suffices to study the resolvent as a pseudodifferential operator in order to identify and compute the second coefficient.

The aim of the paper is to derive spectral estimates on the eigenvalue moments of the magnetic Schrödinger operators defined on the two-dimensional disk with a radially symmetric magnetic field and radially symmetric electric potential.

In this note a representation formula for the scattering matrix of a pair of self-adjoint extensions of a non-densely defined symmetric operator with infinite deficiency indices is proved with the help of quasi boundary triples and their Weyl functions. This result is a generalization of a classical formula by V.A. Adamyan and B.S. Pavlov.

We investigate the behavior of large eigenvalues for the quantum Rabi Hamiltonian, i.e., for the Jaynes–Cummings model without the rotating wave approximation. The three-term asymptotics we obtain involves all the parameters of the model so that we can recover them from the behavior of its large eigenvalues.

This work deals with the functional model for a class of extensions of symmetric operators and its applications to the theory of wave scattering. In terms of Boris Pavlov’s spectral form of this model, we find explicit formulae for the action of the unitary group of exponentials corresponding to almost solvable extensions of a given closed symmetric operator with equal deficiency indices. On the basis of these formulae, we are able to construct wave operators and derive a new representation for the scattering matrix for pairs of such extensions in both self-adjoint and non-self-adjoint situations.

We determine which sets saturate the Szegő and Schiefermayr lower bounds on the norms of Chebyshev Polynomials. We also discuss sets that saturate our optimal Totik–Widom upper bound.

We shall study the solvability of pseudodifferential operators which are not of principal type. The operator will have complex principal symbol satisfying condition (Ψ) and we shall consider the limits of semibicharacteristics at the set where the principal symbol vanishes of at least second order. The convergence shall be as smooth curves, and we shall assume that the normalized complex Hamilton vector field of the principal symbol over the semicharacteristics converges to a real vector field. Also, we shall assume that the linearization of the real part of the normalized Hamilton vector field at the semibicharacteristic is tangent to and bounded on the tangent space of a Lagrangean submanifold at the semibicharacteristics, which we call a grazing Lagrangean space. Under these conditions one can invariantly define the imaginary part of the subprincipal symbol. If the quotient of the imaginary part of the subprincipal symbol with the norm of the Hamilton vector field switches sign from − to + on the bicharacteristics and becomes unbounded as they converge to the limit, then the operator is not solvable at the limit bicharacteristic.

Our paper is devoted to the oscillator semigroup, which can be defined as the set of operators whose kernels are centered Gaussian. Equivalently, they can be defined as the Weyl quantization of centered Gaussians. We use the Weyl symbol as the main parametrization of this semigroup. We derive formulas for the tracial and operator norm of the Weyl quantization of Gaussians. We identify the subset of Gaussians, which we call quantum degenerate, where these norms have a singularity.

This paper discusses resonance effects to advance a classical earthquake model, namely the celebrated M8 global test algorithm. This algorithm gives high confidence levels for prediction of Time Intervals of Increased Probability (TIP) of an earthquake. It is based on observation that almost 80% of earthquakes occur due to the stress accumulated from previous earthquakes at the location and stored in form of displacements against gravity and static elastic deformations of the plates. Nevertheless the M8 global test algorithm fails to predict some powerful earthquakes. In this paper we suggest the additional possibility of considering the dynamical storage of the elastic energy on the tectonic plates due to resonance beats of seismogravitational oscillations (SGO) modes of the plates. We make sure that the tangential compression in the middle plane of an “active zone” of a tectonic plate may tune its SGO modes to the resonance condition of coincidence the frequencies of the corresponding localized modes with the delocalized SGO modes of the complement. We also consider the beats arising between the modes under a small perturbations of the plates, and, assuming that the discord between the perturbed and unperturbed resonance modes is strongly dominated by the discord between the non-resonance modes estimate the energy transfer coefficient.

We characterize positivity preserving, translation invariant, linear operators in $$ \mathit L^p (\mathbb R^n)^m , \mathit p \epsilon [1,\infty), m,n \epsilon \mathbb R $$ .

We consider directed weighted graphs and define various families of path counting functions. Our main results are explicit formulas for the main term of the asymptotic growth rate of these counting functions, under some irrationality assumptions on the lengths of all closed orbits on the graph. In addition we assign transition probabilities to such graphs and compute statistics of the corresponding random walks. Some examples and applications are reviewed.

We obtain sufficient conditions that ensure block diagonalization (by a direct rotation) of sign-indefinite symmetric sesquilinear forms as well as the associated operators that are semi-bounded neither from below nor from above. In the semi-bounded case, we refine the obtained results and, as an example, revisit the block Stokes operator from fluid dynamics.

Quasi-periodic solutions of a nonlinear polyharmonic equation for the case 4l > n + 1 in $$ \mathbb R^n $$ , n > 1, are studied. This includes Gross–Pitaevskii equation in dimension two (l = 1, n = 2). It is proven that there is an extensive “non-resonant” set $$ \mathcal G \subset \mathbb R^n $$ such that for every $$ \overrightarrow{k} \epsilon \mathcal G $$ there is a solution asymptotically close to a plane wave $$ \mathcal Ae^ \mathit i \langle \overrightarrow{k},\overrightarrow{x} \rangle as |\overrightarrow{k}| \rightarrow \infty $$ , given A is sufficiently small.

We study the integral transform which appeared in a different form in Akhiezer’s textbook “Lectures on Integral Transforms”.

We study operators obtained by coupling an n×n random matrix from one of the Gaussian ensembles to the discrete Laplacian. We find the joint distribution of the eigenvalues and resonances of such operators. This is one of the possible mathematical models for quantum scattering in a complex physical system with one semi-infinite lead attached.

The review of the results obtained in the spectral analysis for a class of integral-difference operators and their application to physical processes, obtained since late 1990s until now, is provided. Some fresh not yet published results in the field are also presented. We demonstrate the physical background and the logical structure of the corresponding studies. The discussion includes both the 1D case and the recent generalization to higher dimensions. We trace the links with different fields of mathematics. In particular, a new class of special functions that naturally appear as the kernels of the mentioned operators, is discussed. The open problems are highlighted and further possible encouraging investigations are proposed.

We consider the inverse problem for integral-difference operators underlying the dynamics of the matter relaxation in external attractive fields for quasi-1-dim structures (graphs). Under specific simplifying conditions the solution of this inverse problem is obtained. The further possible generalization for a broader set of graphs and other domains is discussed and the corresponding encouraging investigations are proposed.

Spin-depending scattering is studied on a two-dimensional quantum network constructed of a quantum well with three semi-infinite quantum wires (an input wire and two terminals ) attached to it. The spin-orbital interaction causing selective scattering in the network is described by the Rashba Hamiltonian. The transmission of electrons across the well from the input quantum wire to terminals is caused by the excitation of a resonance oscillatory spin-mode in the well. For thin quantum networks we suggest also an approximate formula which defines the resonance transmission of electrons across the quantum well, depending on the shape of the standing spin-mode in the quantum well.

Quantum graph with the Landau operator (Schrödinger operator with a magnetic field) at the edges is considered. The Kirchhoff condition is assumed at the internal vertices. We derive conditions for the graph structure ensuring the completeness of the resonance states on finite subgraphs obtained by cutting all infinite leads of the initial graph. Due to the use of a functional model, the problem reduces to factorization of the characteristic matrix-function. The result is compared with the corresponding completeness theorem for the Schrödinger quantum graph.

It was shown by M.I. Zelikin (2007) that the spectrum of a nuclear operator in a Hilbert space is central-symmetric iff the traces of all odd powers of the operator equal zero. B. Mityagin (2016) generalized Zelikin’s criterium to the case of compact operators (in Banach spaces) some of which powers are nuclear, considering even a notion of so-called $$ \mathbb Z_d $$ -symmetry of spectra introduced by him. We study α-nuclear operators generated by the tensor elements of so-called α-projective tensor products of Banach spaces, introduced in the paper (α is a quasi-norm). We give exact generalizations of Zelikin’s theorem to the cases of $$ \mathbb Z_d $$ -symmetry of spectra of α-nuclear operators (in particular, for s-nuclear and for (r, p)-nuclear operators). We show that the results are optimal.

Given the Laplacian on a planar, convex domain with piecewise linear boundary subject to mixed Dirichlet–Neumann boundary conditions, we provide a sufficient condition for its lowest eigenvalue to dominate the lowest eigenvalue of the Laplacian with the complementary boundary conditions (i.e., with Dirichlet replaced by Neumann and vice versa). The application of this result to triangles gives an affirmative partial answer to a recent conjecture. Moreover, we prove a further observation of similar flavor for right triangles.

The paper develops a theory of spectral boundary value problems from the perspective of general theory of linear operators in Hilbert spaces. An abstract form of spectral boundary value problem with generalized boundary conditions is suggested and results on its solvability complemented by representations of weak and strong solutions are obtained. Existence of a closed linear operator defined by a given boundary condition and description of its domain are studied in detail. These questions are addressed on the basis of Krein’s resolvent formula derived from the explicit representations of solutions also obtained here. Usual resolvent identities for two operators associated with two different boundary conditions are written in terms of the so called M-function. Abstract considerations are complemented by illustrative examples taken from the theory of partial differential operators. Other applications to boundary value problems of analysis and mathematical physics are outlined.