Eigenvalue Asymptotics and a Trace Formula for a Fourth-Order Differential Operator

were constructed in [1] for a real integrable function q(x) on [0, `] and for arbitrary n = 1, 2, 3, . . . and m = 0, 1, 2, . . . Here λn is the nth eigenvalue, sn ≡ √ λn, yn(x) is the nth normalized eigenfunction, and λn,m(q), sn,m(q), and yn,m(q, x) are quantities that can be constructed explicitly via the function q(x). In particular, the λn,0 ≡ (nπ/`) are the eigenvalues of the degenerate problem, the yn,0(x) ≡ √ 2/` sin(nπx/`)

We define and study the properties of Darboux-type transformations between Sturm--Liouville problems with boundary conditions containing rational Herglotz--Nevanlinna functions of the eigenvalue parameter (including the Dirichlet boundary conditions). Using these transformations, we obtain various direct and inverse spectral results for these problems in a unified manner, such as asymptotics of eigenvalues and norming constants, oscillation of eigenfunctions, regularized trace formulas, and inverse uniqueness and existence theorems.

A fourth-order regular ordinary differential operator with eigenvalue dependent boundary conditions is considered. This problem is realized by a quadratic operator pencil with self-adjoint operators. The location of the eigenvalues is discussed and the first four terms of the eigenvalue asymptotics are evaluated explicitly.

The purpose of writing this article is to show some spectral properties of the Bessel operator equation, with spectral parameter-dependent boundary condition. This problem arises upon separation of variables in heat or wave equations, when one of the boundary conditions contains partial derivative with respect to time. To illustrate the problem and the proof in detail, as a first step, the corresponding operator's discreteness of the spectrum is proved. Then, the nature of the eigenvalue distribution is established. Finally, based on these results, a regularized trace formula for the eigenvalues is obtained.

Nonlinear eigenvalue problems are transformed to fixed point problems with operators expanding a cone in a Banach space. Using techniques developed by J. Sprekels and the author for this type of equations we get bounds for the first eigenvalue and the corresponding eigenelement. The method is applied to second and fourth order differential operators. Its high degree of accuracy is demonstrated by means of numerical examples.

We prove the existence of solutions for some semilinear elliptic equations in the appropriate H^4 spaces using the fixed point technique where the elliptic equation contains fourth order differential operators with and without Fredholm property, generalizing the results of the preceding work [22].

This paper deals with a problem of nondestructive testing for a composite system formed by the connection of a steel beam and a reinforced concrete beam. The small vibrations of the composite beam are described by a differential system where a coupling takes place between longitudinal and bending motions. The motion is governed in space by two second order and two fourth order differential operators, which are coupled in the lower order terms by the shearing,k, and axial, μ, stiffness coefficients of the connection. The coefficientskand μ define the mechanical model of the connection between the steel beam and the concrete beam and contain direct information on the integrity of the system. In this paper we study the inverse problem of determiningkand μ by mixed data. The inverse problem is transformed to a variational problem for a cost function which includes boundary measurements of Neumann data and also some interior measurements. By computing the Gateaux derivatives of the functional, an algorithm based on the projected gradient method is proposed for identifying the unknown coefficients. The results of some numerical simulations on real steel-concrete beams are presented and discussed.

The polynomials which are orthogonal with respect to the indefinite inner product \[ \langle {f,g} \rangle = \int_{ - 1}^1 {f(x)} \overline {g(x)} \frac{\alpha } {2}dx + \frac{1} {2}f(1)\overline {g(1)} + \frac{1} {2}f( - 1)\overline {g( - 1)} \] with $\alpha < 0$ are shown to span the Pontryagin (Krein) space generated by the inner product. The polynomials are eigenfunctions associated with a selfadjoint, fourth order differential operator.

In [Spectral asymmetry and Riemannian geometry. III, Math. Proc. Cambridge Philos. Soc. 79 (1976) 71–99] Atiyah, Patodi and Singer introduced spectral flow for elliptic operators on odd-dimensional compact manifolds. They argued that it could be computed from the Fredholm index of an elliptic operator on a manifold of one higher dimension. A general proof of this fact was produced by Robbin–Salamon [The spectral flow and the Maslov index, Bull. London Math. Soc. 27(1) (1995) 1–33, MR 1331677]. In [F. Gesztesy, Y. Latushkin, K. Makarov, F. Sukochev and Y. Tomilov, The index formula and the spectral shift function for relatively trace class perturbations, Adv. Math. 227(1) (2011) 319–420, MR 2782197], a start was made on extending these ideas to operators with some essential spectrum as occurs on non-compact manifolds. The new ingredient introduced there was to exploit scattering theory following the fundamental paper [A. Pushnitski, The spectral flow, the Fredholm index, and the spectral shift function, in Spectral Theory of Differential Operators, American Mathematical Society Translations: Series 2, Vol. 225 (American Mathematical Society, Providence, RI, 2008), pp. 141–155, MR 2509781]. These results do not apply to differential operators directly, only to pseudo-differential operators on manifolds, due to the restrictive assumption that spectral flow is considered between an operator and its perturbation by a relatively trace-class operator. In this paper, we extend the main results of these earlier papers to spectral flow between an operator and a perturbation satisfying a higher pth Schatten class condition for [Formula: see text], thus allowing differential operators on manifolds of any dimension [Formula: see text]. In fact our main result does not assume any ellipticity or Fredholm properties at all and proves an operator theoretic trace formula motivated by [M.-T. Benameur, A. Carey, J. Phillips, A. Rennie, F. Sukochev and K. Wojciechowski, An analytic approach to spectral flow in von Neumann algebras, in Analysis, Geometry and Topology of Elliptic Operators (World Scientific Publisher, Hackensack, NJ, 2006), pp. 297–352, MR 2246773; A. Carey, H. Grosse and J. Kaad, On a spectral flow formula for the homological index, Adv. Math. 289 (2016) 1106–1156, MR 3439708]. We illustrate our results using Dirac type operators on [Formula: see text] for arbitrary [Formula: see text] (see Sec. 8). In this setting Theorem 6.4 substantially extends Theorem 3.5 of [A. Carey, F. Gesztesy, H. Grosse, G. Levitina, D. Potapov, F. Sukochev and D. Zanin, Trace formulas for a class of non-Fredholm operators: a review, Rev. Math. Phys. 28(10) (2016) 1630002, MR 3572626], where the case d = 1 was treated.

The time evolution of the scattering and spectral data is obtained for the differential operator , where u(x, t) and v(x, t) are real-valued potentials decaying exponentially as x → ±∞ at each fixed t. The result is relevant in a crucial step of the inverse scattering transform method that is used in solving the initial-value problem for a pair of coupled nonlinear partial differential equations satisfied by u(x, t) and v(x, t).

A pulse or a sharp front in concentration of a tracer or a substrate in the blood within a vessel becomes dispersed while being transported along a vessel. Cross-stream mixing and pulsations in flow with the heartbeat cause the dispersion to be less than would occur with a parabolic velocity profile (Newtonian flow). These characteristics allow intravascular mass transport to be described well by a simple two-parameter differential operator, which is a one-dimensional representation of the rather complex real situation. The operator consists of two components in series, a pure delay and a fourth-order linear differential operator. The latter is merely two underdamped second-order operators in series, with fixed relationships between the natural frequencies and damping coefficients. The operator is useful because it provides a transport function with skewness and kurtosis suitable to intravascular transport where the mean velocity profile is blunter than in Newtonian parabolic flow. The parameters of the operator are its mean transit time, t, and its relative dispersion, RD, which is the standard deviation of the response impulse divided by t. The operator transport function describes blood transport through the human leg arterial system, where the RD values are approximately 15-20%.

This paper studies the existence and uniqueness of the classical solution of inverse problems for a fourth-order hyperbolic equation with a complex-valued coefficient with Dirichlet and Neumann boundary conditions. Using the method of separation of variables, formal solutions are obtained in the form of a Fourier series in terms of the eigenfunctions of a non-self-adjoint fourth-order ordinary differential operator. The proofs of the uniform convergence of the Fourier series are based on estimates of the norms of the derivatives of the eigenfunctions of a fourth-order ordinary differential operator and the uniform boundedness of the Riesz bases of the eigenfunctions.

The short-time heat kernel expansion of elliptic operators provides a link between local and global features of classical geometries. For many geometric structures related to (non-)involutive distributions, the natural differential operators tend to be Rockland, hence hypoelliptic. In this paper, we establish a universal heat kernel expansion for formally self-adjoint non-negative Rockland differential operators on general closed filtered manifolds. The main ingredient is the analysis of parametrices in a recently constructed calculus adapted to these geometric structures. The heat expansion implies that the new calculus, a more general version of the Heisenberg calculus, also has a non-commutative residue. Many of the well-known implications of the heat expansion such as, the structure of the complex powers, the heat trace asymptotics, the continuation of the zeta function, as well as Weyl’s law for the eigenvalue asymptotics, can be adapted to this calculus. Other consequences include a McKean–Singer type formula for the index of Rockland differential operators. We illustrate some of these results by providing a more explicit description of Weyl’s law for Rumin–Seshadri operators associated with curved BGG sequences over 5-manifolds equipped with a rank-two distribution of Cartan type.

In the present section we discuss simple problems concerning the asymptotics of eigenvalues of self-adjoint operators which arise in analysis, geometry and mathematical physics.

We study the convergence of spectral expansions of functions of the class W 1 (G), p ≥ 1, G = (0, 1), in the eigenfunctions of an ordinary differential operator of even order with integrable coefficients. Sufficient conditions for absolute and uniform convergence are obtained and the rate of uniform convergence of these expansions on the interval G is found.