Mathematics

In this paper the discrete eigenvalues of elliptic second order differential operators in L 2 ( R n ) , n∈N , with singular δ - and δ ′ -interactions are studied. We show the self-adjointness of these operators and derive equivalent formulations for the eigenvalue problems involving boundary integral operators. These formulations are suitable for the numerical computations of the discrete eigenvalues and the corresponding eigenfunctions by boundary element methods. We provide convergence results and show numerical examples.

We prove that no eigenvalue of the clamped disk can have multiplicity greater than six. Our method of proof is based on a new recursion formula, linear algebra arguments and a transcendency theorem due to Siegel and Shidlovskii.

We consider magnetic Schrödinger operators on a bounded region Ω with the smooth boundary ∂Ω in Euclidean space R d . In reference to the result from Weyl's asymptotic law and Pólya's conjecture, P. Li and S. -T. Yau(1983) (resp. P. Kröger(1992)) found the lower (resp. upper) bound d d+2 (2π ) 2 (Vol( S d−1 )Vol(Ω) ) −2/d k 1+2/d for the k -th (resp. ( k+1 )-th) eigenvalue of the Dirichlet (resp. Neumann) Laplacian. We show in this paper that this bound relates to the upper bound for k -th excited state energy eigenvalues of magnetic Schrödinger operators with the compact resolvent. Moreover, we also investigate and mention the gap between two energies of particles on the magnetic field. For that purpose, we extend the results by Li, Yau and Kröger to the magnetic cases with no or Robin boundary conditions on the basis of their ideas and proofs.

We consider Schrödinger operators of the form H R =− d 2 /d x 2 +q+iγ χ [0,R] for large R>0 , where q∈ L 1 (0,∞) and γ>0 . Bounds for the maximum magnitude of an eigenvalue and for the number of eigenvalues are proved. These bounds complement existing general bounds applied to this system, for sufficiently large R .

In this paper, we study the bounds for discrete Steklov eigenvalues on trees via geometric quantities. For a finite tree, we prove sharp upper bounds for the first nonzero Steklov eigenvalue by the reciprocal of the size of the boundary and the diameter respectively. We also prove similar estimates for higher order Steklov eigenvalues.

Several recent papers have focused their attention in proving the correct analogue to the Lieb-Thirring inequalities for non self-adjoint operators and in finding bounds on the distribution of their eigenvalues in the complex plane. This paper provides some improvement in the state of the art in this topic. Precisely, we address the question of finding quantitative bounds on the discrete spectrum of the perturbed Lamé operator of elasticity − Δ ∗ +V in terms of L p -norms of the potential. Original results within the self-adjoint framework are provided too.

The Burchnall-Chaundy theory concerns the classification of all pairs of commuting ordinary differential operators. We phrase this theory in the language of spectral data for integrable systems. In particular, we define spectral data for rank 1 commutative algebras A of ordinary differential operators. We solve the inverse problem for such data, i.e. we prove that the algebra A is (essentially) uniquely determined by its spectral data. The isomorphy type of A is uniquely determined by the underlying spectral curve.

We study spectral properties of two-dimensional canonical systems y ′ (t)=zJH(t)y(t) , t∈[a,b) , where the Hamiltonian H is locally integrable on [a,b) , positive semidefinite, and Weyl's limit point case takes place at b . We answer the following questions explicitly in terms of H : Is the spectrum of the associated selfadjoint operator discrete ? If it is discrete, what is its asymptotic distribution ? Here asymptotic distribution means summability and limit superior conditions relative to comparison functions growing sufficiently fast. Making an analogy with complex analysis, this corresponds to convergence class and type w.r.t.\ proximate orders having order larger than 1 . It is a surprising fact that these properties depend only on the diagonal entries of H . In 1968 this http URL~Branges posed the following question as a fundamental problem: Which Hamiltonians are the structure Hamiltonian of some\\ de~Branges space ? We give a complete and explicit answer.

Stimulated by the category theorems of Eisner and Serény in the setting of unitary and isometric C 0 -semigroups on separable Hilbert spaces, we prove category theorems for Schrödinger semigroups. Specifically, we show that, to a given class of Schrödinger semigroups, Baire generically the semigroups are strongly stable but not exponentially stable. We also present a typical spectral property of the corresponding Schrödinger operators.

In recent years there has been a growing interest in companion matrices. There is a deep knowledge of sparse companion matrices, in particular it is known that every sparse companion matrix can be transformed into a unit lower Hessenberg matrix of a particularly simple type by any combination of transposition, permutation similarity and diagonal similarity. The latter is not true for the companion matrices that are non-sparse, although it is known that every non-sparse companion matrix is nonderogatory. In this work the non-sparse companion matrices that are unit lower Hessenberg will be described. A natural generalization is also considered.