New trace formula for the matrix Sturm-Liouville equation with eigenparameter dependent boundary conditions

Sturm–Liouville problems with discontinuities at two points

Transfer operators and Hankel transforms between relative trace formulas, II: Rankin–Selberg theory

Transfer operators and Hankel transforms between relative trace formulas, I: Character theory

Inverse Sturm–Liouville problems with eigenparameter-dependent boundary conditions are considered. Theorems analogous to those of both Hochstadt and Gelfand and Levitan are proved. In particular, let ly = (1/r)(−(py′)′+qy), l ˜ y = ( 1 / r ˜ ) ( - ( p ˜ y ′ ) ′ + q ˜ y ) , Δ = [ a b c d ] and ∑ = [ r s t u ] where det Δ = δ > 0, c ≠ 0, det ∑ > 0, t ≠ 0 and (cs + dr − au − tb)2 < 4(cr − ta)(ds − ub). Denote by (l; α; Δ) the eigenvalue problem ly = λy with boundary conditions y(0)cosα+y′(0)sinα = 0 and (aλ+b)y(1) = (cλ+d)(py′)(1). Define ( l ˜ ; α; Δ) as above but with l replaced by l ˜ . Let wn denote the eigenfunction of (l; α; Δ) having eigenvalue λn and initial conditions wn(0) = sin α and pw′n(0) = −cos α and let γn = −awn(1)+cpw′n(1). Define w ˜ n and γ ˜ n similarly. As sample results, it is proved that if (l; α; Δ) and ( l ˜ ; α; Δ) have the same spectrum, and (l; α; Σ) and ( l ˜ ; α; Σ) have the same spectrum or ∫ | w n | 0 1 r d t + ( | γ n | 2 / δ ) = ∫ | w ˜ n | 0 1 r ˜ d t + ( | γ ˜ n | 2 / δ ) for all n, then q/r = q ˜ / r ˜ .

In this study, we present a problem which consists quantum difference equation with matrix value impulsive conditions and eigenparameter dependent boundary condition. Our examination is based on the concept of main spectral and scattering properties of this problem. We introduce Jost solution, Jost function and scattering function of the problem. Then, we find resolvent operator, Green function, eigenvalues and continuous spectrum of the operator generated by same problem.

The regularized trace formula of first order for the ”weighted” Sturm-Liouville equation with point δ-interaction is obtained.

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.

Linear Partial Differential Equations with Multiple Involutive Characteristics O. Liess, L. Rodino. Gevrey and Analytic Hypoellipticity D.S. Tartakoff. Higher Microlocalization and Propagation of Singularities O. Liess. Conormality and Lagrangian Properties in Diffractive Boundary Value Problems P. Laubin. Parametrized Pseudodifferential Operators and Geometric Invariants G. Grubb. Boundary Value Problems and Edge Pseudodifferential Operators B.-W. Schulze. Wodzicki's Noncommutative Residue and Traces for Operator Algebras on Manifolds with Conical Singularities E. Schrohe. Lower Bounds for Pseudodifferential Operators C. Parenti, A. Parmeggiani. Weyl Formula for Globally Hypoelliptic Operator in Rn E. Buzano. Splitting in Large Dimension and Infrared Estimates B. Helffer. Microlocal Exponential Estimates and Applications to Tunneling A. Martinez. A Trace Formula and Review of Some Estimates for Resonances J. Sjostrand.

We shall establish an identity between distributions on different real reductive groups. The distributions arise from the trace formula. They represent the main archimedean terms in both the invariant and stable forms of the trace formula. The identity will be an essential part of the comparison of these formulas. As such, it is expected to lead to reciprocity laws among automorphic representations on different groups. Our techniques are analytic. We shall show that the difference of the two sides of the proposed identity is the solution of a homogeous boundary value problem. More precisely, we shall show that it satisfies a system of linear differential equations, that it obeys certain boundary conditions around the singular set, and that it is asymptotic to zero. We shall then show that any such solution vanishes.

The partial differential equation for vertical, one‐phase, unsaturated moisture flow in soils is employed as a mathematical model for infiltration rate. Solution of this equation for the rainfall‐ponding upper boundary condition is proposed as a sensitive means to describe infiltration rate as a dependent upper boundary condition. A nonlinear Crank‐Nicholson implicit finite difference scheme is used to develop a solution to this equation that predicts infiltration under realistic upper boundary and soil matrix conditions. The kinematic wave approximation to the equations of unsteady overland flow on cascaded planes is solved by a second order explicit difference scheme. The difference equations of infiltration and overland flow are then combined into a model for a simple watershed that employs computational logic so that boundary conditions match at the soil surface. The mathematical model is tested by comparison with data from a 40‐foot laboratory soil flume fitted with a rainfall simulator and with data from the USDA Agricultural Research Service experimental watershed at Hastings, Nebraska. Good agreement is obtained between measured and predicted hydrographs, although there are some differences in recession lengths. The results indicate that a theoretically based model can be used to describe simple watershed response when appropriate physical parameters can be obtained.

An analysis of the number and form of the required inflow-outflow boundary conditions for the full two-dimensional time-dependent nonlinear acoustic system in subsonic mean flow is performed. The explicit predictor-corrector method of MacCormack (1969) is used. The methodology is tested on both uniform and sheared mean flows with plane and nonplanar sources. Results show that the acoustic system requires three physical boundary conditions on the inflow and one on the outflow boundary. The most natural choice for the inflow boundary conditions is judged to be a specification of the vorticity, the normal acoustic impedance, and a pressure gradient-density gradient relationship normal to the boundary. Specification of the acoustic pressure at the outflow boundary along with these inflow boundary conditions is found to give consistent reliable results. A set of boundary conditions developed earlier, which were intended to be nonreflecting is tested using the current method and is shown to yield unstable results for nonplanar acoustic waves.

In this study, first, a formula for regularized sums of eigenvalues of a Sturm‐Liouville problem with retarded argument at 2 points of discontinuity which contains a spectral parameter in the boundary conditions is obtained. After that, oscillation properties of the related problem is investigated. Finally, under the condition that a subset of nodal points is dense in definition set, the potential function is determined.

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.

Eigenvalues of a general class of boundary value problem with derivative-dependent nonlinearity

In this paper a non-polynomial quintic spline function is applied to the numerical solution of a certain fourth-order, two-point boundary-value problem occurring in plate deflection theory. Direct methods of orders two, four, and six have been obtained which lead to five-diagonal linear systems. Boundary formulae of various orders have been developed to retain the bandwidth of the coefficient matrix as five. Convergence analysis of the sixth-order method is given. Numerical results are provided to demonstrate the superiority of our methods.