We perturb one-dimensional Dirac operators on a bounded interval subject to Dirichlet boundary conditions by potentials with Fourier coefficients exhibiting power-decay. As a consequence of Paley-Zygmund theorem, this broad family of potentials comprises distributions that are neither integrable functions nor measures. We localize the spectrum of the perturbed operator and, as the main result, show that its eigensystem generates a Riesz basis.

We investigate massive one-dimensional Dirac operators perturbed by diagonal matrix potentials of the form $i V$ where the function $V$ is real-valued and unbounded at infinity. For such operators we find an $L^2$-realization with non-empty resolvent set using generalized coercivity and Schur complement dominance techniques. In the prototypical Airy-Dirac case $V(x)=x$, $x \in \mathbb{R}$, we derive the precise asymptotic behavior of the resolvent as the spectral parameter tends to infinity and also as the mass $m$ tends to $0$. Finally, we find the asymptotics of the resolvent norm for general potentials $V$ in terms of the Airy-Dirac resolvent, which in particular yields an asymptotic shape of $\varepsilon$-pseudospectral curves and establishes the optimality of the pseudospectral region found in [32].

We study the damped wave equation with a damping coefficient which is possibly singular and unbounded at infinity. In general, zero belongs to the spectrum of the corresponding generator, which prevents a uniform (exponential) decay for the energy. However, for initial conditions in a suitable subspace, a detailed analysis of the resolvent norm for low frequencies leads to sharp polynomial time-decay rates for the solution and its energy.

We study the existence of negative eigenvalues for two-dimensional Schrödinger operators with real-valued potentials in the weak coupling regime. In his pioneering paper [Simon 1976] from half a century ago, Simon was the first to describe the unique negative eigenvalue emerging from the threshold of the essential spectrum of one- and two-dimensional Schrödinger operators. The aim of this paper is to extend Simon's results in two dimensions to a broader class of potentials, allowing for both stronger singularities and slower decay at infinity, at the cost of losing uniqueness of weakly coupled eigenvalues.

For a broad class of polynomial potentials V, with an important and instructive representative being V(x) = x^2a + i x^b, x ∈ℝ, a, b ∈ℕ, we show that the system of spectral projections {P_n}_n of an anharmonic operator L = - (d/ dx)^2 + V(x) does not generate a (Riesz) basis in L^2(ℝ) if a - 1 < b < 2a. Moreover, for σ= [b - (a - 1)]/(1 + a) and γ> 0 small enough, lim sup_n P_n/ exp(γn^σ) = ∞. Proofs are based on two groups of results which are of great interest on their own: (a) relationship between behavior (growth) of the norms of projections P_n and of the resolvent (z - L)^-1 outside of the spectrum σ(L); (b) partial fraction decompositions of special meromorphic functions 1/F where F(w) = ∏_k=1^∞( 1 + w/a_k), a_k+1≥ a_k>0, k ∈ℕ, and the generalization of the first resolvent identity.

We extend the notion of generalized boundary triples and their Weyl functions from extension theory of symmetric operators to adjoint pairs of operators, and we provide criteria on the boundary parameters to induce closed operators with a nonempty resolvent set. The abstract results are applied to Schrödinger operators with complex L^p -potentials on bounded and unbounded Lipschitz domains with compact boundaries.

We study the behaviour of the norm of the resolvent for non-self-adjoint operators of the form $A := -\partial_x + W(x)$ , with $W(x) \ge 0$ , defined in ${L^2}({\mathbb{R}})$ . We provide a sharp estimate for the norm of its resolvent operator, $\| (A - \lambda)^{-1} \|$ , as the spectral parameter diverges $(\lambda \to +\infty)$ . Furthermore, we describe the C0-semigroup generated by −A and determine its norm. Finally, we discuss the applications of the results to the asymptotic description of pseudospectra of Schrödinger and damped wave operators, and also the optimality of abstract resolvent bounds based on Carleman-type estimates.

We study the discrete eigenvalues emerging from the threshold of the essential spectrum of one or two-dimensional Schrödinger operators with complex-valued L^p-potentials in a weak coupling regime. We derive necessary and sufficient conditions on the potential for the existence or absence of discrete eigenvalues in this regime and also analyze their uniqueness and algebraic multiplicity. Our results can be viewed as natural non-self-adjoint extensions of the well-known classical weak coupling phenomenon for self-adjoint Schrödinger operators with real-valued potentials going back half a century to Simon's famous paper [Simon 1976].

Pseudomodes of non-self-adjoint Schrödinger operators corresponding to large pseudoeigenvalues are constructed. The approach is non-semiclassical and extendable to other types of models including the damped wave equation and Dirac operators.

We study one-dimensional Schrödinger operators H=−∂x2+V with unbounded complex potentials V and derive asymptotic estimates for the norm of the resolvent, Ψ(λ):=‖(H−λ)−1‖, as |λ|→+∞, separately considering λ∈RanV and λ∈R+. In each case, our analysis yields an exact leading order term and an explicit remainder for Ψ(λ) and we show these estimates to be optimal. We also discuss several extensions of the main results, their interrelation with some aspects of semigroup theory and illustrate them with examples.

Diverging eigenvalues in domain truncations of Schrödinger operators with complex potentials are analyzed and their asymptotic formulas are obtained. Our approach also yields asymptotic formulas for diverging eigenvalues in the strong coupling regime for the imaginary part of the potential.

New eigenvalue enclosures for the block operator problem arising in the study of stability of the Ekman boundary layer are proved. This solves an open problem in [ 19 ] on the existence of open sets of eigenvalues in domains of Fredholmness of the analyzed operator family.

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 analyze pseudospectra of the generator of the damped wave equation with unbounded damping. We show that the resolvent norm diverges as $\Re z \to - \infty$. The highly non-normal character of the operator is a robust effect preserved even when a strong potential is added. Consequently, spectral instabilities and other related pseudospectral effects are present.

We analyze the spectral properties and peculiar behavior of solutions of a damped wave equation on a finite interval with a singular damping of the form $\alpha/x$, $\alpha>0$. We establish the exponential stability of the semigroup for all positive $\alpha$, and determine conditions for the spectrum to consist of a finite number of eigenvalues. As a consequence, we fully characterize the set of initial conditions for which there is extinction of solutions in finite time. Finally, we propose two open problems related to extremal decay rates of solutions.

For one-dimensional Schroedinger operators with complex-valued potentials, we construct pseudomodes corresponding to large pseudoeigenvalues. Our (non-semi-classical) approach results in substantial progress in achieving optimal conditions and conclusions as well as in covering a wide class of previously inaccessible potentials, including discontinuous ones.

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 analyze eigenvalues emerging from thresholds of the essential spectrum of one-dimensional Dirac operators perturbed by complex and non-symmetric potentials. In the general non-self-adjoint setting, we establish the existence and asymptotics of weakly coupled eigenvalues and Lieb-Thirring inequalities. As physical applications, we investigate the damped wave equation and armchair graphene nanoribbons.

We analyze new phenomena arising in linear damped wave equations on unbounded domains when the damping is allowed to become unbounded at infinity. We prove the generation of a contraction semigroup, study the relation between the spectra of the semigroup generator and the associated quadratic operator function, the convergence of non-real eigenvalues in the asymptotic regime of diverging damping on a subdomain, and we investigate the appearance of essential spectrum on the negative real axis. We further show that the presence of the latter prevents exponential estimates for the semigroup and turns out to be a robust effect that cannot be easily canceled by adding a positive potential. These analytic results are illustrated by examples.