Efficient Spectral Methods for Some Singular Eigenvalue Problems

Abstract

We propose and analyze some efficient spectral/spectral element methods to solve singular eigenvalue problems related to the Schrödinger operator with an inverse-power potential. For the Schrödinger eigenvalue problem $-\Delta u +V(x)u=\lambda u$ with a regular potential $V(x)=c_1|x|^{-1}$, we first design an efficient spectral method on a ball of any dimension by adopting the Sobloev-orthogonal basis functions with respect to the Laplacian operator to overwhelm the homogeneous inverse potential and to eliminate the singularity of the eigenfunctions. Then we extend this spectral method to arbitrary polygonal domains by the mortar element method with each corner covered by a circular sector and origin covered by a circular disc. Furthermore, for the Schrödinger eigenvalue problem with a singular potential $V(x)=c_3|x|^{-3}$, we devise a novel spectral method by modifying the former Sobloev-orthogonal bases to fit the stronger singularity. As in the case of $|x|^{-1}$ potential, this approach can be extended to arbitrary polygonal domains by the mortar element method as well. Finally, for the singular elliptic eigenvalue problem $-\frac{\partial ^2}{\partial x^2}u-\frac{1}{x^2}\frac{\partial ^2}{\partial y^2}u =\lambda u$ on rectangles, we propose a novel spectral method by using tensorial bases composed of the $L^2$- and $H^1$-simultaneously orthogonal functions in the y-direction and the Sobolev-orthogonal functions with respect to the Schrödinger operator with an inverse-square potential in the x-direction. Numerical experiments indicate that all our methods possess exponential orders of convergence, and are superior to the existing polynomial based spectral/spectral element methods and hp-adaptive methods.

Appendix A. Entries of Stiffness and Mass Matrices Corresponding to (5.2)

Lemma A.1

Denote $\Phi _{k,\ell }^{n}(x)= \frac{2k+2\beta _n}{k+2\beta _n} \Phi _{k,\ell }^{-1,n}(x)$. For $m,n,k,j\in \mathbb {N}_0, 1\le \ell \le a^d_n, 1\le \iota \le a^d_m$, it holds that

$$\begin{aligned}&(\nabla \Phi _{k,\ell }^{n}, \nabla \Phi _{j,\iota }^{m})_{\mathbb {B}^d} + c_2^2 (\Phi _{k,\ell }^{n}, \Phi _{j,\iota }^{m})_{r^{-2},\mathbb {B}^d} =\omega _d \delta _{m,n}\delta _{\ell ,\iota }\nonumber \\&\quad \times {\left\{ \begin{array}{ll} \frac{2\beta _n^2-d\beta _n+4\beta _n-d+3}{2\beta _n+2}, &{} k=j=0,\\ \frac{(k+\beta _n)(8\beta ^4_n+16k\beta ^3_n+(20k^2-6)\beta ^2_n+(12k^3-8k)\beta _n+3k^4-4k^2+1)}{(k-1+\beta _n)(2k+1+2\beta _n)(2k-1+2\beta _n)(k+1+\beta _n)}, &{}k=j\ge 1,\\ \frac{(16k+8)\beta ^3_n+(24k^2+24k+4)\beta ^2_n+(16k^3+24k^2+4k-2)\beta _n+4k^4+8k^3+2k^2-2k-1}{(2k+3+2\beta _n)(2k+1+2\beta _n)(2k-1+2\beta _n)}, &{} j=k+1,\\ \frac{(k+1)^2(2\beta _n+k+1)^2}{2(2k+3+2\beta _n)(2k+1+2\beta _n)(k+1+\beta _n)}, &{} j=k+2,\\ \frac{(16j+8)\beta ^3_n+(24j^2+24j+4)\beta ^2_n+(16j^3+24j^2+4j-2)\beta _n+4j^4+8j^3+2j^2-2j-1}{(2j+3+2\beta _n)(2j+1+2\beta _n)(2j-1+2\beta _n)}, &{} k=j+1,\\ \frac{(j+1)^2(2\beta _n+j+1)^2}{2(2j+3+2\beta _n)(2j+1+2\beta _n)(j+1+\beta _n)}, &{} k=j+2,\\ 0, &{} \text {otherwise}, \end{array}\right. } \end{aligned}$$

(A.1)

$$\begin{aligned}&(\Phi _{k,\ell }^{n}, \Phi _{j,\iota }^{m})_{r^{-1},\mathbb {B}^d}=\omega _d \delta _{m,n}\delta _{\ell ,\iota }\nonumber \\&\quad \times {\left\{ \begin{array}{ll} \frac{1}{2k+2\beta _n+3 }, &{} k=j=0,\\ \frac{8(k+\beta _n)(2\beta _n(4\beta _n^3+4\beta _n^2k+4\beta _nk^2+2k^3-3k-5\beta _n)+k^4-3k^2+2)}{(2k+2\beta _n+1)_3(2k+2\beta _n-3)_3}, &{} k=j\ge 1, \\ \frac{-16\beta _n^4+(4k^2+4k+20)\beta _n^2+(4k^3+6k^2+2k)\beta _n+k^4+2k^3+k^2-4}{(2k+2\beta _n+1)(2k+2\beta _n+3)_2(2k+2\beta _n-2)_2}, &{} j=k+1,\\ -\frac{2(k+1)(k+2\beta _n+1)(4\beta _n^2+2\beta _nk+2\beta _n+k^2+2k-2)}{(2k+2\beta _n+1)_3(2k+2\beta _n+5)(2k+2\beta _n-1)}, &{} j = k+2, \\ -\frac{(k+2\beta _n+1)_2(k+1)_2}{(2k+2\beta _n+1)_5}, &{} j = k+3,\\ \frac{-16\beta _n^4+(4j^2+4j+20)\beta _n^2+(4j^3+6j^2+2j)\beta _n+j^4+2j^3+j^2-4}{(2j+2\beta _n+1)(2j+2\beta _n+3)_2(2j+2\beta _n-2)_2}, &{} k=j+1,\\ -\frac{2(j+1)(j+2\beta _n+1)(4\beta _n^2+2\beta _nj+2\beta _n+j^2+2j-2)}{(2j+2\beta _n+1)_3(2j+2\beta _n+5)(2j+2\beta _n-1)}, &{} k = j+2, \\ -\frac{(j+2\beta _n+1)_2(j+1)_2}{(2j+2\beta _n+1)_5}, &{} k = j+3,\\ 0, &{} \text {otherwise}, \end{array}\right. }\end{aligned}$$

(A.2)

$$\begin{aligned}&(\Phi _{k,\ell }^{n}, \Phi _{j,\iota }^{m})_{\mathbb {B}^d}=(\Phi _{j,\iota }^{m}, \Phi _{k,\ell }^{n})_{\mathbb {B}^d}\nonumber \\&\quad = \omega _d \delta _{m,n}\delta _{\ell ,\iota }\times {\left\{ \begin{array}{ll} \frac{1}{2k+2\beta _n+4}, &{} k=j=0,\\ \frac{(\mathcal {C}^0_{n,k})^2}{2k+2\beta _n+4} + \frac{(\mathcal {C}^{-1}_{n,k})^2}{2k+2\beta _n+2} + \frac{(\mathcal {C}^{-2}_{n,k})^2}{2k+2\beta _n} + \frac{(\mathcal {C}^{-3}_{n,k})^2}{2k+2\beta _n-2} + \frac{(\mathcal {C}^{-4}_{n,k})^2}{2k+2\beta _n-4}, &{} k=j\ge 1,\\ \frac{ \mathcal {C}^0_{n,k}\mathcal {C}^{-1}_{n,k+1}}{2k+2\beta _n+4} + \frac{\mathcal {C}^{-1}_{n,k+1}\mathcal {C}^{-2}_{n,k+1}}{2k+2\beta _n+2} + \frac{\mathcal {C}^{-2}_{n,k+1}\mathcal {C}^{-3}_{n,k+1}}{2k+2\beta _n} + \frac{\mathcal {C}^{-3}_{n,k+1}\mathcal {C}^{-4}_{n,k+1}}{2k+2\beta _n-2}, &{} j=k+1,\\ \frac{ \mathcal {C}^0_{n,k}\mathcal {C}^{-2}_{n,k+2}}{2k+2\beta _n+4} + \frac{\mathcal {C}^{-1}_{n,k}\mathcal {C}^{-3}_{n,k+2}}{2k+2\beta _n+2} + \frac{\mathcal {C}^{-2}_{n,k}\mathcal {C}^{-4}_{n,k+2}}{2k+2\beta _n}, &{} j=k+2,\\ \frac{ \mathcal {C}^0_{n,k}\mathcal {C}^{-3}_{n,k+3}}{2k+2\beta _n+4} + \frac{\mathcal {C}^{-1}_{n,k}\mathcal {C}^{-4}_{n,k+3}}{2k+2\beta _n+2}, &{} j=k+3,\\ \frac{ \mathcal {C}^0_{n,k}\mathcal {C}^{-4}_{n,k+4}}{2k+2\beta _n+4}, &{} j=k+4,\\ 0, &{} |j-k|>4, \end{array}\right. } \end{aligned}$$

(A.3)

and

$$\begin{aligned} (\Phi _{k,\ell }^{n}, \Phi _{j,\iota }^{m})_{r^{-3},\mathbb {B}^d}= \omega _d \delta _{m,n}\delta _{\ell ,\iota }\times {\left\{ \begin{array}{ll} \frac{1}{2k+2\beta _n+1}+\frac{1}{2k+2\beta _n-1}, &{} k=j\ge 1,\\ \frac{1}{2\beta _n+1}, &{} k=j=0,\\ -\frac{1}{2k+2\beta _n+1}, &{} j=k+1,\\ -\frac{1}{2j+2\beta _n+1}, &{} k=j+1,\\ 0, &{} \text {otherwise}, \end{array}\right. } \end{aligned}$$

(A.4)

where

$$\begin{aligned}&\mathcal {C}^0_{n,k} = \frac{(k+2\beta _n+1)_3}{(2k+2\beta _n+1)_3}, \quad \mathcal {C}^{-1}_{n,k} = \frac{(k+2\beta _n+1)_2(2k-3-2\beta _n)}{(2k+2\beta _n-1)(2k+2\beta _n+1)(2k+2\beta _n+3)},\\&\mathcal {C}^{-2}_{n,k} = \frac{-12(k-1)(k+2\beta _n+1)(\beta _n+1)(k+\beta _n)}{(2k+2\beta _n-2)_2(2k+2\beta _n+1)_2}, \qquad \mathcal {C}^{-4}_{n,k} = \frac{-(k-3)_3}{(2k+2\beta _n-3)_3}, \\&\mathcal {C}^{-3}_{n,k} = \frac{-(k-2)_2(2k+6\beta _n+3)}{(2k+2\beta _n-1)(2k+2\beta _n+1)(2k+2\beta _n-3)}. \end{aligned}$$

Proof

To prove this lemma, we temporarily set $\phi _{k,n}(r)= \frac{2k+2\beta _n}{k+2\beta _n}J^{-1,2\beta _n}_k(2r-1)r^{\beta _n+2-d/2}$.

It is readily obtained from “Appendix A” in [19] that

$$\begin{aligned} \begin{aligned}&(\nabla \Phi _{k,\ell }^{n}, \nabla \Phi _{j,\iota }^{m})_{\mathbb {B}^d} + c^2 (\Phi _{k,\ell }^{n}, \Phi _{j,\iota }^{m})_{r^{-2},\mathbb {B}^d}\\&\quad =\omega _d \delta _{m,n}\delta _{\ell ,\iota } \int _0^1 \partial _r \big [ r^{d/2-1-\beta _n}\phi _{k,n}(r)\big ] \big [ r^{d/2-1-\beta _n}\phi _{j,n}(r)\big ]r^{2\beta _n+1} \mathrm {d}r \\&\qquad + \omega _d \delta _{m,n}\delta _{\ell ,\iota }(\beta _n+1-d/2)\delta _{k,0}\delta _{k,j}. \end{aligned} \end{aligned}$$

(A.5)

Before proceeding the proof of (A.1), we resort to the following identity on generalized Jacobi polynomials,

$$\begin{aligned} \begin{aligned} J_k^{\alpha ,\beta }(2r-1)r=&\frac{(k+1)(k+\alpha +\beta +1)}{(2k+\alpha +\beta +1)_2}J_{k+1}^{\alpha ,\beta }(2r-1)+ \frac{(k+\alpha )(k+\beta )}{(2k+\alpha +\beta )_2}J_{k}^{\alpha ,\beta }(2r-1) \\&+ \frac{2k^2+2(\alpha +\beta +1)k+(\alpha +\beta )(\beta +1)}{(2k+\alpha +\beta )(2k+\alpha +\beta +2)}J_{k-1}^{\alpha ,\beta }(2r-1), \end{aligned} \end{aligned}$$

(A.6)

which is derived from (2.7) by a simple calculation. Then (A.1) is an immediate consequence of (A.5), (2.8), (A.6) and (2.9).

Next, it is easy to show that

$$\begin{aligned} \Big (\Phi _{k,\ell }^{n}, \Phi _{j,\iota }^{m}\Big )_{r^{-1},\mathbb {B}^d}= & {} \frac{(2k+2\beta _n)(2j+2\beta _m)}{(k+\beta _n)(j+2\beta _m)}\int _{\mathbb {S}^{d-1}}Y^n_{\ell }(\xi )Y^m_{\iota }(\xi )\mathrm {d}\sigma (\xi )\nonumber \\&\times \int _0^1J_k^{-1,2\beta _n}(2r-1)J_j^{-1,2\beta _m}(2r-1)r^{\beta _n+\beta _m+2}\mathrm {d}r\nonumber \\= & {} \omega _d\delta _{m,n}\delta _{\ell ,\iota }\frac{(2k+2\beta _n)(2j+2\beta _m)}{(k+\beta _n)(j+2\beta _m)}\nonumber \\&\times \int _0^1J_k^{-1,2\beta _n}(2r-1)J_j^{-1,2\beta _m}(2r-1)r^{\beta _n+\beta _m+2}\mathrm {d}r\nonumber \\= & {} \frac{1}{2^{2\beta _n+3}}\omega _d\delta _{m,n}\delta _{\ell ,\iota }\frac{(2k+2\beta _n)(2j+2\beta _m)}{(k+\beta _n) (j+2\beta _m)}\nonumber \\&\times \int _{-1}^1J_k^{-1,2\beta _n}(\rho )J_j^{-1,2\beta _m}(\rho )(\rho +1)^{2\beta _n+2}\mathrm {d}\rho , \end{aligned}$$

(A.7)

where the last equality sign is derived from variable substitution $\rho =2r-1$. Using (3.6) repeatedly together with (2.6) yields

$$\begin{aligned} J_k^{\alpha ,\beta }(x)= & {} \frac{(k+\alpha +\beta +1)_2}{(2k+\alpha +\beta +1)_2}J_k^{\alpha +1,\beta +1}(x)\\&+\frac{(\alpha -\beta )(k+\alpha +\beta +1)}{(2k+\alpha +\beta )(2k+\alpha +\beta +2)}J_{k-1}^{\alpha +1,\beta +1}(x) - \frac{(k+\alpha )(k+\beta )}{(2k+\alpha +\beta )_2}J_{k-2}^{\alpha +1,\beta +1}(x)\\= & {} \frac{(k+\alpha +\beta +1)_3}{(2k+\alpha +\beta +1)_3}J_k^{\alpha +1,\beta +2}(x)+\frac{(k+\alpha +\beta +1)_2(k+2\alpha -\beta )}{(2k+\alpha +\beta )_2(2k+\alpha +\beta +3)}J_{k-1}^{\alpha +1,\beta +2}(x)\\&- \frac{(k+\alpha )(k+\alpha +\beta +1)(k-\alpha +2\beta +1)}{(2k+\alpha +\beta -1)(2k+\alpha +\beta +1)_2}J_{k-2}^{\alpha +1,\beta +2}(x)\\&-\frac{(k+\alpha -1)_2(k+\beta )}{(2k+\alpha +\beta -1)_3}J_{k-3}^{\alpha +1,\beta +2}(x)\\= & {} \frac{(k+\alpha +\beta +1)_4}{(2k+\alpha +\beta +1)_4}J_k^{\alpha +1,\beta +3}(x)+\frac{(k+\alpha +\beta +1)_3(2k+3\alpha -\beta )}{(2k+\alpha +\beta )_3(2k+\alpha +\beta +4)}J_{k-1}^{\alpha +1,\beta +3}(x)\\&+ \frac{3(k+\alpha )(k+\alpha +\beta +1)_2(\alpha -\beta -1)}{(2k+\alpha +\beta -1)_2(2k+\alpha +\beta +2)_2}J_{k-2}^{\alpha +1,\beta +3}(x)\\&-\frac{(k+\alpha -1)_2(k+\alpha +\beta +1)(2k-\alpha +3\beta +2)}{(2k+\alpha +\beta -2)(2k+\alpha +\beta )_3}J_{k-3}^{\alpha +1,\beta +3}\\&-\frac{(k+\alpha -2)_3(k+\beta )}{(2k+\alpha +\beta -2)_4}J_{k-4}^{\alpha +1,\beta +3} \end{aligned}$$

which implies (A.2) together with (A.7) and (2.9).

At last, (A.3) and (A.4) can be readily obtained by using similar arguments as (A.2), we omit the details here. $\square$