D. R. Yafaev
We discuss applications of the M. G. Kre\uın theory of the spectral shift function to the multi-dimensional Schrödinger operator as well as specific properties of this function, for example, its high-energy asymptotics. Trace identities for the Schrödinger operator are derived.
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We introduce a new canonical trace on odd class logarithmic pseudo-differential operators on an odd dimensional manifold, which vanishes on commutators. When restricted to the algebra of odd class classical pseudo-differential operators our trace coincides with the canonical trace of Kontsevich and Vishik. Using the new trace we construct a new determinant of odd class classical elliptic pseudo-differential operators. This determinant is multiplicative up to sign whenever the multiplicative anomaly formula for usual determinants of Kontsevich-Vishik and Okikiolu holds. When restricted to opera…
Using simple commutator relations, we obtain several trace identities involving eigenvalues and eigenfunctions of an abstract self-adjoint operator acting in a Hilbert space. Applications involve abstract universal estimates for the eigenvalue gaps. As particular examples, we present simple proofs of the classical universal estimates for eigenvalues of the Dirichlet Laplacian (Payne-Polya-Weinberger, Hile-Protter, etc.), as well as of some known and new results for other differential operators and systems. We also suggest an extension of the methods to the case of non-self-adjoint operators.
The goal of this paper is to prove that on surfaces with asymptotically cusp ends the relative determinant of pairs of Laplace operators is well defined. We consider a surface with cusps (M,g) and a metric h on the surface that is a conformal transformation of the initial metric g. We prove the existence of the relative determinant of the pair $(Δ_{h},Δ_{g})$ under suitable conditions on the conformal factor. The core of the paper is the proof of the existence of an asymptotic expansion of the relative heat trace for small times. We find the decay of the conformal factor at infinity for which …
After recalling a fundamental identity relating traces and modified Fredholm determinants, we apply it to a class of half-line Schrödinger operators $(- d^2/dx^2) + q$ on $(0,\infty)$ with purely discrete spectra. Roughly speaking, the class considered is generated by potentials $q$ that, for some fixed $C_0 > 0$, $\varepsilon > 0$, $x_0 \in (0, \infty)$, diverge at infinity in the manner that $q(x) \geq C_0 x^{(2/3) + \varepsilon_0}$ for all $x \geq x_0$. We treat all self-adjoint boundary conditions at the left endpoint $0$.
We study the spectral properties of the Laplace type operator on the circle. We discuss various approximations for the heat trace, the zeta function and the zeta-regularized determinant. We obtain a differential equation for the heat kernel diagonal and a recursive system for the diagonal heat kernel coefficients, which enables us to find closed approximate formulas for the heat trace and the functional determinant which become exact in the limit of infinite radius. The relation to the generalized KdV hierarchy is discussed as well.
Sub-Riemannian Selberg Trace Formulae for Compact Quotients of SL(2,R) and Determinants of Sub-Laplacians
本文证明了SL(2,R)紧商群的亚黎曼Selberg迹公式，通过SO(2)纤维的傅里叶分解将热迹计算约化为双曲平面上Maass拉普拉斯算子的Selberg迹公式，并利用该公式计算亚拉普拉斯算子的zeta正则化行列式，得到简洁的行列式表达式。
The principal aim in this paper is to develop an effective and unified approach to the computation of traces of resolvents (and resolvent differences), Fredholm determinants, $ζ$-functions, and $ζ$-function regularized determinants associated with linear operators in a Hilbert space. In particular, we detail the connection between Fredholm and $ζ$-function regularized determinants. Concrete applications of our formalism to general (i.e., three-coefficient) regular Sturm-Liouville operators on compact intervals with various (separated and coupled) boundary conditions, and Schrödinger operators …
We prove trace identities for commutators of operators, which are used to derive sum rules and sharp universal bounds for the eigenvalues of periodic Schroedinger operators and Schroedinger operators on immersed manifolds. In particular, we prove bounds on the eigenvalue lambda_{N+1} in terms of the lower spectrum, bounds on ratios of means of eigenvalues, and universal monotonicity properties of eigenvalue moments, which imply sharp versions of Lieb-Thirring inequalities. In the geometric context we derive a version of Reilly's inequality, bounding the eigenvalue lambda_{N+1} of the Laplace-B…
We derive explicit Krein resolvent identities for generally singular Sturm-Liouville operators in terms of boundary condition bases and the Lagrange bracket. As an application of the resolvent identities obtained, we compute the trace of the resolvent difference of a pair of self-adjoint realizations of the Bessel expression $-d^2/dx^2+(ν^2-(1/4))x^{-2}$ on $(0,\infty)$ for values of the parameter $ν\in[0,1)$ and use the resulting trace formula to explicitly determine the spectral shift function for the pair.
We use the boundary triplet approach to extend the classical concept of perturbation determinants to a more general setup. In particular, we examine the concept of perturbation determinants to pairs of proper extensions of closed symmetric operators. For an ordered pair of extensions we express the perturbation determinant in terms of the abstract Weyl function and the corresponding boundary operators. A crucial role in our approach plays so-called almost solvable extensions. We obtain trace formulas for pairs of self-adjoint, dissipative and other pairs of extensions and express the spectral …
We present a concise explicit expression for the heat trace coefficients of spheres. Our formulas yield certain combinatorial identities which are proved following ideas of D. Zeilberger. In particular, these identities allow to recover in a surprising way some known formulas for the heat trace asymptotics. Our approach is based on a method for computation of heat invariants developed in math.DG/9905073.
Discrete Space-Time Wave Kernels and Trace Identities on Regular Graphs
针对(q+1)-正则图上的仿射Laplace型算子，推导了离散时空波动方程的基本解（波核）的显式公式，并利用波核的唯一性证明了新的迹型恒等式，推导出三角函数和与切比雪夫多项式求和的封闭形式。
The aim of this paper is twofold: On one hand we discuss an abstract approach to symmetrized Fredholm perturbation determinants and an associated trace formula for a pair of operators of positive-type, extending a classical trace formula. On the other hand, we continue a recent systematic study of boundary data maps, that is, 2 \times 2 matrix-valued Dirichlet-to-Neumann and more generally, Robin-to-Robin maps, associated with one-dimensional Schrödinger operators on a compact interval [0,R] with separated boundary conditions at 0 and R. One of the principal new results in this paper reduces a…
For a purely imaginary sign-definite perturbation of a self-adjoint operator, we obtain exponential representations for the perturbation determinant in both upper and lower half-planes and derive respective trace formulas.
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