One-dimensional Schrodinger operators with singular periodic potentials

We study the one-dimensional Schrodinger operators $$ S(q)u:=-u+q(x)u,quad uin mathrm{Dom}left(S(q)right), $$ with $1$-periodic real-valued singular potentials $q(x)in H_{operatorname{per}}^{-1}(mathbb{R},mathbb{R})$ on the Hilbert space $L_{2}left(mathbb{R}right)$. We show equivalence of five basic definitions of the operators $S(q)$ and prove that they are self-adjoint. A new proof of continuity of the spectrum of the operators $S(q)$ is found. Endpoints of spectrum gaps are precisely described.

Semiclassical second microlocal propagation of regularity and integrable systems

We develop a second-microlocal calculus of pseudodifferential operators in the semiclassical setting. These operators test for Lagrangian regularity of semiclassical families of distributions on a manifold $X$ with respect to a Lagrangian submanifold of $T^*X.$ The construction of the calculus, closely analogous to one performed by Bony in the setting of homogeneous Lagrangians, proceeds via the consideration of a model case, that of the zero section of $T^*mathbb{R}^n,$ and conjugation by appropriate Fourier integral operators. We prove a propagation theorem for the associated wavefront set analogous to Hormanders theorem for operators of real principal type. As an application, we consider the propagation of Lagrangian regularity on invariant tori for quasimodes (e.g. eigenfunctions) of an operator with completely integrable classical hamiltonian. We prove a secondary propagation result for second wavefront set which implies that even in the (extreme) case of Lagrangian tori with all frequencies rational, provided a nondegeneracy assumption holds, Lagrangian regularity either spreads to fill out a whole torus or holds nowhere locally on it.

On a class of non-self-adjoint periodic eigenproblems with boundary and interior singularities

We prove that the spectrum of a certain PT-symmetric periodic problem is purely real. Our results extend to a larger class of potentials those recently found by Brian Davies [math.SP/0702122] and John Weir [arXiv:0711.1371].

Bethe-Sommerfeld conjecture for pseudodifferential perturbation

We consider a periodic pseudodifferential operator $H=(-Delta)^l+A$ ($l>0$) in $R^d$ which satisfies the following conditions: (i) the symbol of $H$ is smooth in $x$, and (ii) the perturbation $A$ has order smaller than $2l-1$. Under these assumptions, we prove that the spectrum of $H$ contains a half-line.

Correspondence of the eigenvalues of a non-self-adjoint operator to those of a self-adjoint operator

We prove that the eigenvalues of a certain highly non-self-adjoint operator that arises in fluid mechanics correspond, up to scaling by a positive constant, to those of a self-adjoint operator with compact resolvent; hence there are infinitely many real eigenvalues which accumulate only at $pm infty$. We use this result to determine the asymptotic distribution of the eigenvalues and to compute some of the eigenvalues numerically. We compare these to earlier calculations by other authors.

Absolutely continuous spectrum of a Schrodinger operator on a tree

We give sufficient conditions for the presence of the absolutely continuous spectrum of a Schrodinger operator on a regular rooted tree without loops (also called regular Bethe lattice or Cayley tree).

On the nature of ill-posedness of the forward-backward heat equation

We study the Cauchy problem with periodic initial data for the forward-backward heat equation defined by the J-self-adjoint linear operator L depending on a small parameter. The problem has been originated from the lubrication approximation of a viscous fluid film on the inner surface of the rotating cylinder. For a certain range of the parameter we rigorously prove the conjecture, based on the numerical evidence, that the set of eigenvectors of the operator $L$ does not form a Riesz basis in $L^2 (-pi,pi)$. Our method can be applied to a wide range of the evolutional problems given by $PT-$symmetric operators.

On Symmetries in the Theory of Finite Rank Singular Perturbations

For a nonnegative self-adjoint operator $A_0$ acting on a Hilbert space $mathfrak{H}$ singular perturbations of the form $A_0+V, V=sum_{1}^{n}{b}_{ij}<psi_j,cdot>psi_i$ are studied under some additional requirements of symmetry imposed on the initial operator $A_0$ and the singular elements $psi_j$. A concept of symmetry is defined by means of a one-parameter family of unitary operators $sU$ that is motivated by results due to R. S. Phillips. The abstract framework to study singular perturbations with symmetries developed in the paper allows one to incorporate physically meaningful connections between singular potentials $V$ and the corresponding self-adjoint realizations of $A_0+V$. The results are applied for the investigation of singular perturbations of the Schr{o}dinger operator in $L_2(dR^3)$ and for the study of a (fractional) textsf{p}-adic Schr{o}dinger type operator with point interactions.

Two dimensional Berezin-Li-Yau inequalities with a correction term

We improve the Berezin-Li-Yau inequality in dimension two by adding a positive correction term to its right-hand side. It is also shown that the asymptotical behaviour of the correction term is almost optimal. This improves a previous result by Melas.

Maximization of the second positive Neumann eigenvalue for planar domains

We prove that the second positive Neumann eigenvalue of a bounded simply-connected planar domain of a given area does not exceed the first positive Neumann eigenvalue on a disk of a twice smaller area. This estimate is sharp and attained by a sequence of domains degenerating to a union of two identical disks. In particular, this result implies the Polya conjecture for the second Neumann eigenvalue. The proof is based on a combination of analytic and topological arguments. As a by-product of our method we obtain an upper bound on the second eigenvalue for conformally round metrics on odd-dimensional spheres.