Well-posed problems for the fractional Laplace equation with integral boundary conditions

In this remark we study the boundary-value problems for a fractional analogue of the Laplace equation with integral boundary conditions in rectangular and half-strip domains. We prove the existence and uniqueness of solutions by using the spectral decomposition method

The Fourier transform and convolutions generated by a differential operator with boundary condition on a segment

We introduce the concepts of the Fourier transform and convolution generated by an arbitrary restriction of the differentiation operator in the space $L_{2}(0,b).$ In contrast to the classical convolution, the introduced convolution explicitly depends on the boundary condition that defines the domain of the operator $L.$ The convolution is closely connected to the inverse operator or to the resolvent. So, we first find a representation for the resolvent, and then introduce the required convolution.Comment: 15 page

Nonharmonic analysis of boundary value problems

In this paper we develop the global symbolic calculus of pseudo-differential operators generated by a boundary value problem for a given (not necessarily self-adjoint or elliptic) differential operator. For this, we also establish elements of a non-self-adjoint distribution theory and the corresponding biorthogonal Fourier analysis. We give applications of the developed analysis to obtain a-priori estimates for solutions of operators that are elliptic within the constructed calculus.Comment: 54 pages, updated version, to appear in IMR

On transparent boundary conditions for the high--order heat equation

In this paper we develop an artificial initial boundary value problem for the high-order heat equation in a bounded domain $\Omega$. It is found an unique classical solution of this problem in an explicit form and shown that the solution of the artificial initial boundary value problem is equal to the solution of the infinite problem (Cauchy problem) in $\Omega$.Comment: 9 page

A Simple Approach to Ascertain the Magnitudes of the Coefficients of 2 p-Atomic Orbitals in Each 1t-Electron Molecular Orbital of the Linear Polyenes or Polyenyl Systems

A simple approach is suggesred to ascertain the magnitudes o[the coefficients o[all interacting 2p-atomic orbitals (2pAO) in each Tt-eledron moleeuw.r arbital o[the linearpolyenes ar polyenyl systems, C.H,.2' with n ranging from 2 to 2

On nonlinear damped wave equations for positive operators, I : discrete spectrum

In this paper, we study a Cauchy problem for the nonlinear damped wave equations for a general positive operator with discrete spectrum. We derive the exponential in time decay of solutions to the linear problem with decay rate depending on the interplay between the bottom of the operator's spectrum and the mass term. Consequently, we prove global in time well-posedness results for semilinear and for more general nonlinear equations with small data. Examples are given for nonlinear damped wave equations for the harmonic oscillator, for the twisted Laplacian (Landau Hamiltonian), and for the Laplacians on compact manifolds

A Simple Technique to Ascertain the Phase Relationships between the Various Atomic Orbitals in Each Pi-Molecular Orbital for the Linear Polyenes

A simple technique is suggested for the construction of phase relationships of interacting 2p-atomic orbitals (2P-AO) in each molecular orbital for pi-electrons (Pi-MO) in the linear polyenes, CnHn+2 with n ranging from 2 to 25

Laser and microwave spectroscopy of even-parity Rydberg states of neutral ytterbium and Multichannel Quantum Defect Theory analysis

New measurements of high-lying even parity $6sns\, {}^1 \! S_0$ and $6snd\,{}^{3,1}\!D_2$ levels of neutral $^{174}$Yb are presented in this paper. Spectroscopy is performed by a two-step laser excitation from the ground state $4f^{14}6s^2 \, {}^1 \! S_0$, and the Rydberg levels are detected by using the field ionization method. Additional two-photon microwave spectroscopy is used to improve the relative energy accuracy where possible. The spectroscopic measurements are complemented by a multichannel quantum defect theory (MQDT) analysis for the J=0 and the two-coupled J=2 even parity series. We compare our results with the previous analysis of Aymar {\it{et al}} \cite{Aymar_1980} and analyze the observed differences. From the new MQDT models, a revised value for the first ionization limit $I_{6s}=50443.07041(25)$ cm$^{-1}$ is proposed.Comment: 15 pages, 3 figure