# Mathematical Methods in the Applied Sciences

Mathematical Methods in the Applied Sciences > 42 > 15 > 4981 - 4998

*H*= −Δ +

*W*+

*W*

_{s}on ${R}^{n}$ with regular potentials $W\in {L}^{\infty}\left({R}^{n}\right)$ and singular potentials ${W}_{s}\in D\prime \left({R}^{n}\right)$ with supports on unbounded enough smooth hypersurfaces Γ. In particular, we consider singular potentials that are linear combinations of

*δ*−functions on Γ and its normal derivatives. We consider extensions of

*H*as symmetric operators in ${L}^{2}\left({R}^{n}\right)$ with domain ${C}_{0}^{\infty}({R}^{n}\u20e5\Gamma )$ to self‐adjoint...

Mathematical Methods in the Applied Sciences > 42 > 15 > 4939 - 4956

Mathematical Methods in the Applied Sciences > 42 > 15 > 5008 - 5028

Mathematical Methods in the Applied Sciences > 42 > 15 > 4899 - 4901

Mathematical Methods in the Applied Sciences > 42 > 15 > 5052 - 5059

Mathematical Methods in the Applied Sciences > 42 > 15 > 5106 - 5117

Mathematical Methods in the Applied Sciences > 42 > 15 > 5072 - 5093

*S*

_{q}on $R$ with a bounded potential

*q*supported on the segment $\left({h}_{0},{h}_{1}\right)$ and a singular potential supported at the ends

*h*

_{0},

*h*

_{1}. We consider an extension of the operator

*S*

_{q}in ${L}^{2}\left(R\right)$ defined by the Schrödinger operator ${H}_{q}=-\frac{{d}^{2}}{d{x}^{2}}+q$ and matrix point conditions at the ends

*h*

_{0},

*h*

_{1}. By using the spectral parameter power series method, we derive...

Mathematical Methods in the Applied Sciences > 42 > 15 > 4971 - 4980

Mathematical Methods in the Applied Sciences > 42 > 15 > 5029 - 5039

Mathematical Methods in the Applied Sciences > 42 > 15 > 4957 - 4970

Mathematical Methods in the Applied Sciences > 42 > 15 > 4925 - 4938

Mathematical Methods in the Applied Sciences > 42 > 15 > 4999 - 5007

Mathematical Methods in the Applied Sciences > 42 > 15 > 4902 - 4908

*L*be the

*n*‐th order linear differential operator

*L*

*y*=

*ϕ*

_{0}

*y*

^{(n)}+

*ϕ*

_{1}

*y*

^{(n−1)}+⋯+

*ϕ*

_{n}

*y*with variable coefficients. A representation is given for

*n*linearly independent solutions of

*L*

*y*=

*λ*

*r*

*y*as power series in

*λ*, generalizing the SPPS (spectral parameter power series) solution that has been previously developed for

*n*=2. The coefficient functions in these series are obtained by recursively iterating a simple integration...

Mathematical Methods in the Applied Sciences > 42 > 15 > 4909 - 4924

Mathematical Methods in the Applied Sciences > 42 > 15 > 5060 - 5071

Mathematical Methods in the Applied Sciences > 42 > 15 > 5094 - 5105

Mathematical Methods in the Applied Sciences > 42 > 15 > 5040 - 5051

Mathematical Methods in the Applied Sciences > 42 > 16 > 5626 - 5634

Mathematical Methods in the Applied Sciences > 42 > 16 > 5595 - 5606

Mathematical Methods in the Applied Sciences > 42 > 16 > 5535 - 5550

*x*

^{2}=

*k*

*x*+ 1 for

*k*is any positive integer which is the characteristic equation of the recurrence relation of

*k*‐Fibonacci (

*k*‐Lucas) numbers. This paper is about the metallic ratio in ${Z}_{p}$. We define

*k*‐Fibonacci and

*k*‐Lucas numbers in ${Z}_{p}$, and we show that metallic ratio can be calculated in ${Z}_{p}$ if and only if

*p*≡ ± 1 mod (

*k*

^{2}+ 4), which is...