Eigenvalue asymptotics for strong δ-interactions supported on curves with corners

Asymptotics of large eigenvalues for some discrete unbounded Jacobi matrices

Our topological setting is a smooth compact manifold of dimension two or higher with smooth boundary. Although this underlying topological structure is smooth, the Riemannian metric tensor is only assumed to be bounded and measurable. This is known as a rough Riemannian manifold. For a large class of boundary conditions we demonstrate a Weyl law for the asymptotics of the eigenvalues of the Laplacian associated to a rough metric. Moreover, we obtain eigenvalue asymptotics for weighted Laplace equations associated to a rough metric. Of particular novelty is that the weight function is not assumed to be of fixed sign, and thus the eigenvalues may be both positive and negative. Key ingredients in the proofs were demonstrated by Birman and Solomjak nearly fifty years ago in their seminal work on eigenvalue asymptotics. In addition to determining the eigenvalue asymptotics in the rough Riemannian manifold setting for weighted Laplace equations, we also wish to promote their achievements which may have further applications to modern problems.

We consider an eigenvalue problem for a quasilinear nonautonomous second-order differential equation with a cubic nonlinearity. The problem is posed on an interval with boundary conditions of the first kind and with an auxiliary (local) condition at one of the endpoints of the interval. We prove that the problem in question has infinitely many negative and infinitely many positive eigenvalues. The corresponding linear problem has infinitely many negative and finitely many (or none) positive eigenvalues. Moreover, the first terms of the asymptotics of the negative eigenvalues of the nonlinear and linear problems coincide, while the asymptotics of the positive eigenvalues of the nonlinear problem is expressed in terms of a transcendental function of the eigenvalue number. The results are derived with the use of a nonclassical approach.

In this paper, we obtain a remainder term for the asymptotics of the counting function associated with a Schrodinger operator, involving an indefinite weight and defined on an unbounded domain n in JR.n, with Dirichlet or Neumann boundary conditions. More precisely, we consider, for example, the following eigenvalue problem: (-.t.. + q)u = )..ruin n, with u = 0 on an, where the potential q is positive and the weight function r changes sign in n. Such indefinite problems-which occur naturally by linearization of many nonlinear elliptic equations-are of broad interest in mathematical physics. We establish asymptotic estimates with a remainder term for N()..), the number of positive (resp., negative) eigenvalues < ).. (resp., > )..), as).. -+ oo, when q(x) -+ +oo, r(x)jq(x) -+ 0 at infinity, and under certain technical assumptions. We also obtain analogous results for more general SchrOdinger operators. This extends previous joint works of the authors dealing only with the leading term (Trans. Amer. Math. Soc., 295 (1986), 305-324) and with the remainder term for elliptic operators on bounded domains (Arch. Rational Mech. Anal., 98 (1987), 329-356).

In this paper, three different uniqueness data are investigated to reconstruct the potential function in the Sturm-Liouville boundary value problem in the normal form. Taking account of R\"{o}hrl's objective function, the steepest descent method is used in the computation of potential functions. To decrease the volume of computation, we propose a theorem to precalculate the minimization parameter that is required in the optimization. Further, we propose a novel time-saving algorithm in which the obligation of using the asymptotics of eigenvalues and eigenfunctions and the appropriateness of selected boundary conditions are also eliminated. As partial data, we take two spectra, the set of the $j$th elements of the infinite numbers of spectra obtained by changing boundary conditions in the problem, and one spectrum with the set of terminal velocities. In order to show the efficiency of the proposed method, numerical results are given for three test potentials which are smooth, nonsmooth continuous, and noncontinuous, respectively.

We consider one-dimensional Dirac operatorLP,U with Birkhoff regular boundary conditions and summable potential P(x) on[0, ]. We introduce strongly and weakly regular operators. In both cases, asymptotic formulas for eigenvalues are found. In these formulas, we obtain main asymptotic terms and estimates for the second term. We specify these estimates depending on the functional class of the potential: Lp[0,] with p [1,2] and the Besov space Bp,p'[0,] with p [1,2] and (0,1/p). Additionally, we prove that our estimates are uniform on balls Pp,R Then we get asymptotic formulas for normalized eigenfunctions in the strongly regular case with the same residue estimates in uniform metric on x [0,]. In the weakly regular case, the eigenvalues 2n and 2n+1 are asymptotically close and we obtain similar estimates for two-dimensional Riesz projectors. Next, we prove the Riesz basis property in the space (L2[0,])2 for a system of eigenfunctions and associated functions of an arbitrary strongly regular operatorLP,U. In case of weak regularity, the Riesz basicity of two-dimensional subspaces is proved.&#x0D; In parallel with theLP,U operator, we consider the SturmLiouville operator Lq,U generated by the differential -y'' + q(x)y expressionwith distribution potential q of first-order singularity (i.e., we assume that the primitive u = q(1) belongs to L2[0, ]) and Birkhoff-regular boundary conditions. We reduce to this case -(1y')'+i(y)'+iy'+0y, operators of more general form where '1,,0(-1)L2and 10. For operator Lq,U, we get the same results on the asymptotics of eigenvalues, eigenfunctions, and basicity as for operator LP,U .&#x0D; Then, for the Dirac operator LP,U, we prove that the Riesz basis constant is uniform over the ballsPp,R for p1 or 0. The problem of conditional basicity is naturally generalized to the problem&#x0D; of equiconvergence of spectral decompositions in various metrics. We prove the result on equiconvergence by varying three indices: fL[0,] (decomposable function), PL[0,] (potential), and Sm-Sm00,m in L[0,] (equiconvergence of spectral decompositions in the corresponding norm). In conclusion, we prove theorems on conditional and unconditional basicity of the system of eigenfunctions and associated functions of operator LP,U in the spaces L[0,],2, and in various Besov spaces Bp,q[0,].

For a particular family of long-range potentials $V$, we prove that the eigenvalues of the indefinite Sturm--Liouville operator $A = \mathrm{sign}(x)(-\Delta + V(x))$ accumulate to zero asymptotically along specific curves in the complex plane. Additionally, we relate the asymptotics of complex eigenvalues to the two-term asymptotics of the eigenvalues of associated self-adjoint operators.

In the present paper we study the stability problem for a stretched tube conveying fluid with boundary control. The abstract spectral problem concerns operator pencils of the forms taking values in different Hilbert product spaces. Thorough analysis is made of the existence, location, multiplicities, and asymptotics of eigenvalues in the complex plane and Riesz basisness of the corresponding eigenfunctions and eigenvectors. Well‐posedness of the closed‐loop system represented by the initial‐value problem for the abstract equation is established in the framework of ‐semigroups as well as expansions of the solutions in terms of eigenvectors and stability of the closed‐loop system. For the parameters of the problem we give new regions, larger than those in the literature, in which a stretched tube with flow, simply supported at one end, with a boundary controller applied at the other end, can be exponentially stabilised.

Semiclassical approximation addresses the important relationship between quantum and classical mechanics. There has been a very strong development in the mathematical theory, mainly thanks to methods of microlocal analysis. This book develops the basic methods, including the WKB-method, stationary phase and h-pseudodifferential operators. The applications include results on the tunnel effect, the asymptotics of eigenvalues in relation to classical trajectories and normal forms, plus slow perturbations of periodic Schrödinger operators appearing in solid state physics. No previous specialized knowledge in quantum mechanics or microlocal analysis is assumed, and only general facts about spectral theory in Hilbert space, distributions, Fourier transforms and some differential geometry belong to the prerequisites. This book is addressed to researchers and graduate students in mathematical analysis, as well as physicists who are interested in rigorous results. A fairly large fraction can be (and has been) covered in a one semester course.

We consider eigenvalues of elliptic boundary value problems, written in variational form, when the leading coefficients are perturbed by terms which are small in some integral sense. We obtain asymptotic formulae. The main specific of these formulae is that the leading term is different from that in the corresponding formulae when the perturbation is small inL∞-norm.

The paper studies spectral theory of Schrödinger operators H=ℏ2Δ+V on the sphere from the standpoint of integrability and separation. Our goal is to uncover the fine structure of spec H, i.e., asymptotics of eigenvalues and spectral clusters, determine their relation to the underlying geometry and classical dynamics and apply this data to the inverse spectral problem on the sphere. The prototype model is the celebrated Neumann Hamiltonian p2+V with quadratic potential V on Sn. We show that the quantum Neumann Hamiltonian (Schrödinger operator H) remains an integrable and find an explicit set of commuting integrals. We also exhibit large classes of separable potentials {V} based on ellipsoidal coordinates on Sn. Several approaches to spectral theory of such Hamiltonians are outlined. The semiclassical problem (small ℏ) involves the EKB(M)-quantization of the classical Neumann flow along with its invariant tori, Maslov indices, etc., all made explicit via separation of variables. Another approach exploits Stäckel–Robertson separation of the quantum Hamiltonian and reduction to certain ODE problems: the Hill’s and the generalized Lamè equations. The detailed analysis is carried out for S2, where the ODE becomes the perturbed classical Lamè equation and the Schrödinger eigenvalues are expressed through the Lamè eigendata.

We determine asymptotics of eigenvalues and eigenfunctions of a discontinuous boundary value problem with a spectral parameter in the transmission condition.

We study a fourth-order differential operator with a sign-alternating weight function with separated boundary conditions. For large values of the spectral parameter the asymptotics of the solutions to the corresponding differential equations is derived. The study of boundary conditions makes it possible to obtain an equation for eigenvalues of the considered differential operator. The indicator diagram of this equation is studied. The asymptotics of eigenvalues in various sectors of the indicator diagram is obtained.

We consider the Schrödinger operator Hy=−y″+(p+q)y with a periodic potential p plus a compactly supported potential q on the real line. The spectrum of H consists of an absolutely continuous part plus a finite number of simple eigenvalues below the spectrum and in each spectral gap γn≠∅,n≥1. We prove the following results: (1) the distribution of resonances in the disk with large radius is determined, (2) the asymptotics of eigenvalues and antibound states are determined at high energy gaps, (3) if H has infinitely many open gaps in the continuous spectrum, then for any sequence (κ)1∞,κn∈{0,2}, there exists a compactly supported potential q with ∫Rq dx=0 such that H has κn eigenvalues and 2−κn antibound states (resonances) in each gap γn for n large enough.

On the asymptotics of eigenvalues and singular numbers of linear smoothing operators