### A family of exactly solvable radial quantum systems on space of non-constant curvature with accidental degeneracy in the spectrum.

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Let $R$ be a Riemannian manifold without a biharmonic Green function defined on it and $\Omega $ a domain in $R$. A necessary and sufficient condition is given for the existence of a biharmonic Green function on $\Omega $.

We study the method of layer potentials for manifolds with boundary and cylindrical ends. The fact that the boundary is non-compact prevents us from using the standard characterization of Fredholm or compact pseudo-differential operators between Sobolev spaces, as, for example, in the works of Fabes-Jodeit-Lewis and Kral-Wedland . We first study the layer potentials depending on a parameter on compact manifolds. This then yields the invertibility of the relevant boundary integral operators in the...

Let $R\left(\lambda \right)$ be the error term in Weyl’s law for a 3-dimensional Riemannian Heisenberg manifold. We prove that ${\int}_{1}^{T}{\left|R\left(t\right)\right|}^{2}dt=c{T}^{\frac{5}{2}}+{O}_{\delta}\left({T}^{\frac{9}{4}+\delta}\right)$, where $c$ is a specific nonzero constant and $\delta $ is an arbitrary small positive number. This is consistent with the conjecture of Petridis and Toth stating that $R\left(t\right)={O}_{\delta}\left({t}^{\frac{3}{4}+\delta}\right)$.The idea of the proof is to use the Poisson summation formula to write the error term in a form which can be estimated by the method of the stationary phase. The similar result will be also proven in the $2n+1$-dimensional case.

We apply the Feynman-Kac formula to compute the λ-Poisson kernels and λ-Green functions for half-spaces or balls in hyperbolic spaces. We present known results in a unified way and also provide new formulas for the λ-Poisson kernels and λ-Green functions of half-spaces in ℍⁿ and for balls in real and complex hyperbolic spaces.

We show that $n$-dimensional $(n\u2a7e2)$ complete and noncompact metric measure spaces with nonnegative weighted Ricci curvature in which some Caffarelli-Kohn-Nirenberg type inequality holds are isometric to the model metric measure $n$-space (i.e. the Euclidean metric $n$-space). We also show that the Euclidean metric spaces are the only complete and noncompact metric measure spaces of nonnegative weighted Ricci curvature satisfying some prescribed Sobolev type inequality.

We prove two-sided estimates of heat kernels on non-parabolic Riemannian manifolds with ends, assuming that the heat kernel on each end separately satisfies the Li-Yau estimate.