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On the numerical solution of the eigenvalue problem in fractional quantum mechanics
Accepted Manuscript On the numerical solution of the eigenvalue problem in fractional quantum mechanics Alejandro Guerrero, Miguel Angel Moreles PII: DOI: Reference:
S1007-5704(14)00276-7 http://dx.doi.org/10.1016/j.cnsns.2014.06.013 CNSNS 3228
To appear in:
Communications in Nonlinear Science and Numerical Simulation
Please cite this article as: Guerrero, A., Moreles, M.A., On the numerical solution of the eigenvalue problem in fractional quantum mechanics, Communications in Nonlinear Science and Numerical Simulation (2014), doi: http:// dx.doi.org/10.1016/j.cnsns.2014.06.013
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On the numerical solution of the eigenvalue problem in fractional quantum mechanicsI Alejandro Guerreroa , Miguel Angel Morelesb,1,∗ a CIMAT,
Jalisco S/N, Valenciana, Guanajuato GTO 36240, Mexico of Chemical and Petroleum Engineering, University of Calgary, 2500 University Drive NW, Calgary, Alberta, Canada T2N 1N4
b Department
Abstract In this work we propose a Control Volume Function Approximation (CVFA) method to solve equations involving the fractional Laplacian. The function approximation part is carried out with Radial Basis Function (RBF) interpolation. The physical application of interest is the eigenvalue problem for the time independent fractional Schr¨ odinger equation. Fractional derivatives are considered in the Riesz potentials sense. Keywords: Fractional quantum mechanics, Fractional Laplacian, Control Volume Method, Radial Basis Functions
1. Introduction In recent years, fractional calculus has become a topic of great research activity. This is due to the fact that generalizations of integer to fractional derivatives in connection with physical systems, have found applications in most 5
fields. See for instance the historical survey presented in Machado, Kiryakova & Mainardi [1]. The application of interest in this work is to fractional quantum mechanics. Here, the Schr¨ odinger equation is generalized by considering fractional powers I Partially
supported by CONACYT author Email address: [email protected] (Miguel Angel Moreles) 1 Permanent address: CIMAT, Jalisco S/N, Valenciana, Guanajuato GTO 36240, Mexico, [email protected] ∗ Corresponding
Preprint submitted to Communications in Nonlinear Science and Numerical SimulationJune 13, 2014
of the Laplacian, the fractional Laplacian, in the Hamiltonian. A physically 10
sound generalization is that of Laskin [2, 3, 4]. Therein a derivation of the fractional Schr¨ odinger equation is developed based on Feynman path integrals over Levy trajectories. Following these works, there has been an explosion on research regarding fractional Schr¨odinger equations. For instance, in Wang & Xu [5] a fractional Schr¨ odinger equation with space-time fractional derivatives
15
is considered. A solution is proposed by the method of integral transforms. Application to the free particle and square well potential is presented. Of related interest is Tarasov [6], where a fractional generalization of Heisenberg equation is developed, yielding a generalization to quantum Hamiltonian systems. An alternative approach, the D-deformed calculus, to model fractional-dimension
20
systems is introduced in Matos-Abigue [7]. The free particle and the harmonic oscillator are considered as systems in the so called framework of D-deformed quantum mechanics, the process resembles to the classical case. Finally, on the theoretical side, Ionescu & Pusateri [8] carry out a well-posedness analysis for a nonlinear fractional Schr¨ odinger equation in one dimension.
25
This work is from a numerical perspective. Our purpose is to introduce a numerical method to solve the time independent Schr¨odinger equation, namely, h
Dα −~2 ∆
α/2
i + V (r) ψ(r) = Eψ(r),
1 < α ≤ 2.
Here Dα is a generalized diffusion coefficient, and the fractional Laplacian is defined in terms of the Fourier Transform, namely
2
−~ ∆
α/2
ψ(r) =
Z
1 3
(2π)
p · r α d3 p exp i |p| ψ(p). ~
Numerical methods to solve this eigenvalue problem are scarce at best. For 30
the one dimensional case, a collocation method is presented in Amore et al [9]. Within the method, sampling functions, sampling points and interval of definition of the wave functions, are to be determined according to the prescribed boundary conditions. There is no apparent extension to higher dimensions. In Zoia, Rosso & Kardar [10], a matrix representation of the fractional Laplacian
2
35
is introduced. The eigenvalues and eigenfunctions of such a matrix, converge to the eigenvalues and eigenfunctions of the fractional Laplacian when the size of the matrix tends to infinity. Apparently, the method requires large matrix sizes. For instance, in Kwa´snicki [11], the method is used to approximate the eigenvalues associated to the infinite well potential. Therein, a matrix of size
40
O(103 ) is required. A difficulty on approximating the Fractional Laplacian, or any fractional derivative for that matter, is its non local property. Consequently, one is led to full and large matrix systems. Moreover, on bounded domains boundary conditions need to be dealt carefully. To address these issues, we propose a method
45
based on the Control Volume Function Approximation Method (CVFA), see Li, Chen & Huan [12]. After integrating on control volumes, function approximation is carried out on the boundaries by local interpolation. Our proposal is to use Radial Basis Function (RBF) for global interpolation. It is well known that RBF approximations lead to ill-posed full matrix systems, but give satisfactory
50
results on coarse discretizations. As we shall see, the latter, and the global feature of RBF are well suited to approximate the fractional Laplacian in the eigenvalue problem. We provide numerical results for the one dimensional cases in the works above. We remark that in all examples it suffices to solve O(102 ) matrix problems. The matrix eigenvalue problem is solved simply by the inverse
55
power method and deflation. The outline is as follows. In Section 2 we introduce the classical definition of the fractional Laplacian in terms of Riesz potentials and the Fourier Transform. The restriction to bounded domains is as in Musina & Nazarov [13]. In Section 3 we introduce
60
a Control Volume Radial Basis Function (CVRBF) Method for the eigenvalue problem, in any dimension. We use the simplest radial function, φ(r) = r, for approximation. The matrix problem for the one dimensional case is constructed in Section 4. Numerical results are presented in Section 5. We solve the eigenvalue problem for the infinite well potential, the harmonic potential and the
65
quartic anharmonic potential. In Section 6 we draw conclusions and comment 3
of future work.
2. The fractional Laplacian 2.1. Riesz potentials and the fractional Laplacian The results in this section are well known, proofs are found for instance in 70
Helgason [14]. We shall use the definition of Fourier Transform as follows Z fb(k) ≡ F(f )(k) = exp (−ik · x) f (x)dx, Rd
with inverse
F
−1
(F )(x) =
Z
1 d
(2π)
exp (ik · x) F (k)dk Rd
For s ≥ 0 the Sobolev space H s (Rd ) ≡ H s is defined as customary by Z s s 2 H = {f ∈ L : 1 + |k|2 |fb(k|2 dk < +∞}. Rd 0
Let S and S the spaces of tempered functions and tempered distributions respectively. For the case s < 0, the Sobolev space is defined as follows 0
s
Z
H = {f ∈ S :
1 + |k|2
s
|fb(k|2 dk < +∞}.
Rd
By means of the Fourier Transform, the fractional Laplacian operator, (−∆)s 75
is defined for f ∈ H s by h i s 2s (−∆) f = F −1 |k| F(f ) . Let us recall the Riesz potential 1 I (f ) (x) = Hd (γ) γ
Z
γ−d
f (y) |x − y|
with Γ (γ/2) . Hd (γ) = 2γ π d/2 Γ d−γ 2 4
dy
If γ − d ∈ / 2Z+ we can write (I γ f ) (x) = f ∗
1 rγ−d Hd (γ)
(x),
f ∈ S.
We have the following result Proposition. If f ∈ S Rd then γ 7−→ (I γ f ) (x) extends to a holomorphic function in the set Cd = {γ ∈ C : γ − d ∈ / 2Z+ } . Also I 0 f = lim I α f = f α→0
and I γ ∆f = ∆I γ f = −I γ−2 f.
(1)
In terms of the Riesz potential the fractional Laplacian is expressed in the form s
(−∆) f = I −2s f, 80
−s − d ∈ / 2Z+ .
2.2. Fractional Hamiltonian on bounded domains Let Ω be a Lipschitz bounded domain in Rd . We shall consider (−∆)s f defined on the space e s (Ω) = {f ∈ H s : supp f ⊂ Ω} ¯ H s
For 0 < s < 1 the spectrum of (−∆) is discrete and contained in (0, ∞). See Musina & Nazarov [13]. 85
Hereafter we let s = α/2,
(−∆)
α/2
α > 0. In light of (1) we can write f = I −α f = (−∆) I 2−α f.
(2)
This expression is the point of departure for the Control Volume method to be introduced below.
5
¯ c and sufficiently smooth on Consider a potential function V (x) infinite on Ω Ω. The eigenvalue problem of interest is h i α/2 (−∆) + V (x) ψ = E ψ. 90
(3)
whee ψ is the wave function, and E the energy level. e α/2 (Ω). Because of the Sobolev EmNotice that ψ is sought in the space H bedding Theorem, if α > 1, ψ can be identified with a continuous function ¯ c . Consequently we append zero Dirichlet conditions to the that vanishes on Ω eigenvalue problem (3), namely
x ∈ ∂Ω.
ψ(x) = 0,
95
We shall see that this approximation provides highly satisfactory numerical results.
3. A Control volume method with RBF interpolation 3.1. The CVFA Method ¯ c . We consider We have that the wave function ψ(x) in (3) is zero for x ∈ Ω h
(−∆)
α/2
i + V (x) ψ(x) = E ψ(x),
x ∈ Ω, (4) x ∈ ∂Ω.
ψ(x) = 0,
A partition Th of Ω consists of a set of (open) control volumes Vi : Ω=
N [
V i,
Vi ∩ Vj = ∅,
i=1
Define the boundary of each Vi by ∂Vi =
Ni [
eik ,
k=1 100
where Ni is the number of edges eik on ∂Vi . 6
i 6= j.
On each control volume integrate the equation and using (2) we may apply the divergence theorem in the first term to obtain Z −
∇ I
2−α
Z
ψ · νi dσ +
∂Vi
−
Ni Z X k=1
Z V (x)ψ(x) dx = E
Vi
∇ I 2−α ψ · νik dσ +
eik
ψ(x) dx, Vi
Z
Z V (x)ψ(x) dx = E
ψ(x) dx,
Vi
(5)
Vi
On each eik ⊂ ∂Vi , an interpolant ψik is used to approximate ψ ψik (x) =
Rik X
i ψk,j ϕik,j (x),
x ∈ eik ,
j=1
where Rik is the number of interpolation nodes xik,j for eik and these nodes can be located on or surrounding Vi . 105
The coordinate functions ϕik,j are defined as follows:
ϕik,j (x) =
1
at node xik,j
0
at other nodes
We have for the integral term in the sum the approximation R eik
R ik P j=1
∇ I 2−α ψik (x) · νik dσ R eik
=
i ∇ I 2−α ϕik,j (x) · νik dσ ψk,j
i The other terms are also approximated in terms of the nodal values ψk,j ,
leading to a matrix eigenvalue problem. We show this in examples below. 3.2. Coordinate functions by RBF interpolation 110
Let us describe the choice of the coordinate functions ϕik,j (x). The purpose in to make the approximation of ∇ I 2−α ϕik,j (x) as simple as possible. Suppose X ⊂ Ω ⊂ Rd , Ω a bounded set. X = {x1 , x2 , . . . , xN } . Suppose further that we are given {y1 , y2 , . . . , yN } , data values of a function f ∈ C(Ω), 7
i.e., f (xj ) = yj , 1 ≤ j ≤ N. The problem is to find an interpolating function 115
sf,X (x) such that sf,X (xj ) = yj ,
1 ≤ j ≤ N.
(6)
The interpolating function to be used is of the form
sf,X (x) =
N X
αj φ (|x − xj |) + β0 .
(7)
j=1
Here φ(r) = r, r = |x|, is a conditionally positive definite radial function or order 1. Consequently, the system of equations (6) is complemented with N X
αj = 0.
j=1
This guarantees existence and uniqueness of the coefficients in (7), see Wendland 120
[15]. Our aim is to test the method with the simplest radial function. Extension to other choices of radial functions is straightforward. Noteworthy, the thin-plate spline, ϕ(r) = r2 log r is used in Li, Chen & Huan [12].
4. The One dimensional eigenvalue problem 125
Let us consider the interval D = (a, b) and a partition {a = x0 , x1 , . . . , xN , xN +1 = b}. The control segments are Vi = xi− 12 , xi+ 12 with boundaries n o ∂Vi = xi− 12 , xi+ 12 . n o n o In each boundary xi− 12 , xi+ 12 , we shall consider all points in the partition as the neighboring points. Hence, the interpolant is of the form
φh (x) =
N +1 X
αj |x − xj | + β0
j=0
8
Let ψi ≈ ψ(xi ). From (5) we obtain the eigenvalue problem 1 [Aα + Λ] ψ = E ψ |Vi | 130
(8)
Here |Vi | is the length of the control segment, and Λ is a diagonal matrix with diagonal entries Z Λi,i =
V (x)dx. Vi
The computation of Aα , the sub matrix associated to the Fractional Laplacian, is shown below. In the examples that follow, we consider a regular grid. 135
We remark that to approximate the first smallest eigenvalues of (8), we simply use the inverse power method and deflation. 4.1. The fractional Laplacian matrix Extending by zero outside the interval (a, b) we write the Riesz potential for a function in the form 2−α 2−α I 2−α f (x) = I− f (x) + I+ f (x)
where 2−α I− f (x) =
x
1 H1 (2 − α)
Z
1 H1 (2 − α)
Z
|x − y|1−α f (y) dy,
(9)
|x − y|1−α f (y) dy,
(10)
a
and 2−α I+ f (x) =
140
b
x
For k = 1, 2, . . . , N the coordinate functions are of the form
φk (x) =
N +1 X
αkj |x − xj | + αkN +1
(11)
j=0
Thus the matrix components are (Aα )i,k
x 1 i+ 2 d 2−α =− I φk (x) dx x 1 i−
9
2
(12)
We are led to consider d 2−α dx I− φk ,
d 2−α φk . dx I+
We have
N
X d 2−α I∓ φk (x) = dx j=1
d 2−α d 2−α I∓ |x − xj | αkj + I∓ 1 αkN +1 dx dx
(13)
The computation of each term in the sum is straightforward. We obtain
d 2−α dx I−
d 2−α dx I−
1 2−α (x
|x − xj | = (xj − a)(x − a)1−α −
|x − xj | = (xj − a)(x − a)1−α +
1 2−α
− a)2−α ,
x < xj ,
2(x − xj )2−α − (x − a)2−α ,
x > xj ,
and d 2−α I 1 = (x − a)1−α . dx − 145
Similarly
d 2−α dx I+
|x − xj | =
d 2−α dx I+
1 2−α
−2(xj − x)2−α + (b − x)2−α − (b − xj )(b − x)1−α ,
|x − xj | = −(b − xj )(b − x)1−α +
1 2−α (b
and d 2−α I 1 = −(b − x)1−α . dx − It remains to evaluate (13) at the edge points xi−1/2 ,
xi+1/2
and substitute in (12). 10
− x)2−α ,
x > xj ,
x < xj ,
5. One dimensional Fractional Quantum Mechanics 5.1. Infinite potential well let D = (−1, 1). In this case
150
V (x) =
0,
x ∈ D, (14)
∞,
otherwise.
The problem to consider is α/2 d2 − 2 ψ(x) = E ψ(x), dx
x∈D
(15)
We apply the method above to approximate the first 10 eigenvalues. In Table 1 we list the results and relative errors in comparison with the method of Zoia, Rosso & Kardar [10]. For the latter we use the values reported in Kwa´snicki 155
[11]. Therein a system 5000 × 5000 is solved. An order of magnitude larger than the 200 × 200 system used here. Table 1 (Infinite potential well) α
E1
E2
E3
E4
E5
E6
E7
E8
E9
E10
1.5
1.5998
5.0669
9.6075
15.039
21.217
28.071
35.532
43.559
52.111
61.159
0.0017
0.0016
0.0016
0.0016
0.0015
0.0015
0.0015
0.0014
0.0014
0.0013
1.8
2.0501
7.5079
15.809
26.739
40.135
55.891
73.918
94.147
116.515
140.974
(1e-3)
0.9695
0.9538
0.9216
0.8823
0.8341
0.7744
0.7033
0.6214
0.5266
0.4206
1.9
2.2448
8.5984
18.72
32.47
49.73
70.42
94.48
121.86
152.51
186.38
(1e-3)
0.7162
0.6736
0.6335
0.5816
0.5207
0.4451
0.3552
0.2501
0.1301
0.0048
1.99
2.4438
9.7332
21.828
38.707
60.355
86.760
117.91
153.79
194.40
239.71
(1e-3)
0.4311
0.3973
0.3521
0.2889
0.2100
0.1133
0.0011
0.3281
0.2834
0.4507
11
Figure 1: First wave functions, ψi (x), i = 1, 2, 3, 4, for the infinite well potential. α = 1.5 solid line. α = 2 dashed line
As remarked above, we use the inverse power method and deflation. Thus, we obtain also the eigenvectors which approximate the wave functions. For 160
illustration we plot the first four in Figure 1 for α = 1.5. In all examples that follow the classical case α = 2 is included. Remark. In the terminology of Musina & Nazarov [13], we are approximating the eigenvalue problem for the Dirichlet fractional Laplacian. Therein it is also considered the eigenvalue problem for the Navier fractional Laplacian. Let us
165
denote by EnD , and EnN the corresponding eigenvalues. The Navier case is solved in Laskin [3]. The normalized wave functions are n πx o even ψ2m+1 (x) = cos (2m + 1) , 2 12
m = 0, 1, . . . ,
(16)
and n πx o odd ψ2m (x) = sin (2m) , 2
m = 1, 2, . . . .
(17)
These are the functions plotted in Figure 1 corresponding to α = 2. The Navier eigenvalues are EnN = 170
π α 2
nα ,
n = 1, 2, . . . .
(18)
Consequently, the Navier fractional Laplacian is defined as a spectral power of the Laplacian. As pointed out in Musina & Nazarov [13], this is not the case for the Dirichlet fractional Laplacian. For the latter, to our knowledge, there are no analytic expressions for wave functions and eigenvalues. See the in depth study of the eigenvalue problem in Kwa´snicki [11].
175
For eigenvalues it is known that EnN > EnD .
(19)
See Corollary 4 in Musina & Nazarov [13]. It is readily seen that our estimates in Table 1 satisfy this inequality. 5.2. Fractional Harmonic Oscillator Let V (x) the quadratic potential V (x) = x2 , 180
x ∈ R.
We consider the fractional quantum harmonic oscillator "
d2 − 2 dx
#
α/2 +x
2
ψ(x) = E ψ(x),
x ∈ R.
b α/2 (DL ). We solve a Let DL = (−L, L). For approximation we seek ψ ∈ H 200 × 200 system for α = 3/2 and L = 5.5. Wave functions are plotted in Figure 2. In Table 2 we show numerical results in comparison to the method of Amore et al [9], and the WKB estimates in Laskin [4], namely EnW KB
π = B(1/2, 1/α + 1)
1 n− 2
13
2α 2+α
,
n = 1, 2, . . . .
(20)
Figure 2: First wave functions, ψi (x), i = 1, 2, 3, 4, for the harmonic potential. α = 1.5 solid line. α = 2 dashed line
185
Where B(·, ·) is the beta function. Table 2 (harmonic oscillator) α
E1
E2
E3
1.5
1.0011
2.7076
4.1762
rel. error
0.0000756
0.0004866
0.0007545
rel. error WKB
0.0496
0.0024
0.0021
Remark.
Nontrivial issues of the collocation method of Amore et al [9], are
the choice of L and the number of sampling points, N . It is noticed that the number of sampling points increases significantly with L. First they start with 190
a number N of collocation functions, then a number of sampling points and an
14
optimal choice of L are determined. Refinement may be required. In contrast, our method only requires a good estimate for the decay of the wave function. 5.3. Fractional anharmonic Oscillator Our last example is the anharmonic oscillator associated to the potential 195
V (x) = x4 . For α = 4/3 it is known that a good approximation for the eigenvalues is provided by the asymptotic W KB estimates. The predicted spectrum is evenly spaced and is given by EnW KB =
2π Γ(1/4)Γ(7/4)
1 n+ . 2
(21)
We apply the Control Volume method with L = 3 and solve a 300 × 300 200
system. Results are listed in Table 3, α = 4/3 En , n = 1, 2, 3, ...10 and relative errors with respect to WKB. The first wave functions are plotted in Figure 3. Table 3 (anharmonic oscillator) α
E1
E2
E3
E4
E5
E6
E7
E8
E9
E10
4/3
0.9947
2.8204
4.7065
6.6022
8.4794
10.369
12.249
14.136
16.016
17.902
0.0550
0.0028
0.0016
0.0004
0.0007
0.0001
0.0006
0.0004
0.0007
0.0006
Results in Table 3 show that the eigenvalues of the anharmonic oscillator follow the W KB line (21), see Figure 4. 205
Remark.
Equations (20) and (21), are special cases of the WKB general
equation derived in the pioneering work of Laskin [4]. Accuracy of the WKB estimates, even in small eigenvalues, is remarkable.
6. Conclusions and future work We have introduced a Control Volume Radial Basis Function method to solve 210
the eigenvalue problem associated to the time independent Scr¨odinger equation.
15
Figure 3: First wave functions, ψi (x), i = 1, 2, 3, 4, for the anharmonic potential. α = 4/3 solid line. α = 2 dashed line
16
Figure 4: CVRBF vs. WKB for α = 4/3
17
The method has been tested on one dimensional problems with satisfactory results. In order to test the applicability of the method, we have used for interpolation the simplest radial function. In line with RBF approximation theory, it is 215
straightforward to test other radial functions, as well as node location strategies. Currently, we are working on applications of the method to higher dimensions and irregular domains. The latter is possible thanks to the control volume approach. Hopefully, RBF interpolation will make the method practical in higher dimensions.
220
It is apparent that the proposed method is practical, and its implementation is straightforward. The approximation theory supporting the method is left for future investigations.
Acknowledgements Part of this work was carried out while M. A. Moreles was visiting the 225
Deparment of Chemical and Petroleum Engineering, University of Calgary. M. A. Moreles is thankful for their hospitality. [1] J. T. Machado, V. Kiryakova, F. Mainardi: Recent history of fractional calculus; Commun Nonlinear Sci Numer Simulat; 16 (2011) 1140-1153 [2] N. Laskin: Fractional quantum mechanics and Levy path integrals; Phys.
230
Rev. A Vol 268, 298-305. (2000) [3] N. Laskin:Fractals and quantum mechanics; Chaos: An Interdisciplinary Journal of Nonlinear Science 10.4 (2000): 780-790. [4] N. Laskin: Fractional Schr¨ odinger equation; Phys. Rev. E Vol 66 056108. (2002)
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[5] A. Wang, M. Xu: Generalized fractional Schr¨ odinger equation with spacetime fractional derivatives; J. Math. Phys 48, 043502 (2007)
18
[6] V. E. Tarasov: Fractional Heisenberg equation; Phys. Letters A 372 (2008) 2984-2988 [7] A. Matos-Abiague: 240
Deformation of quantum mechanics in fractional-
dimensional space; J. Phys. A: Math. Gen. 34 (2001) 11059-11068 [8] A. D. Ionescu, F. Pusateri: Nonlinear fractional Schr¨ odinger equations in one dimension; J. Functional Analysis 266 (2014) 139-176 [9] P. Amore, F. M. Fern´ andez, C. P. Hofmann, R. A. S´aenz, R. A. (2010). Collocation method for fractional quantum mechanics. Journal of Mathematical
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Physics, 51(12), 122101. [10] A. Zoia, A. Rosso, M. Kardar: Fractional Laplacian in bounded domains; Phys. Rev. E 76(2) (2007) 061121 [11] M. Kwa´snicki, M. (2012). Eigenvalues of the fractional Laplace operator in the interval. Journal of Functional Analysis, 262(5), 2379-2402.
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[12] B. Li, Z. Chen, G. Huan: Control volume function approximation methods and their applications to modeling porous media flow ; Advances in Water Resources 26, 435 - 444. (2003) [13] Musina, R., Nazarov, A. I. (2013). On fractional Laplacians. Communications in Partial Differential Equations, (just-accepted).
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[14] S. Helgason: The Radon Transform, 2nd ed; Birkh¨auser; Boston. (1999) [15] H. Wendland: Scattered Data Approximation; Cambridge University Press, Cambridge. (2005)
19
Highlights of: “On the numerical solution of the eigenvalue problem in fractional quantum mechanics” • • • • •
A novel method to solve the eigenvalue problem in fractional quantum mechanics is proposed Homogeneous Dirichlet boundary conditions are introduced for the fractional Laplacian defined on appropriate Sobolev Spaces The method is control volume based, thus it applies to irregular domains Eigenfunction approximations is in terms of Radial Basis Functions (RBF) interpolation The method is applied to the potentials: infinite well, quadratic (harmonic oscillator), quartic (anharmonic oscillator)
S1007-5704(14)00276-7 http://dx.doi.org/10.1016/j.cnsns.2014.06.013 CNSNS 3228
To appear in:
Communications in Nonlinear Science and Numerical Simulation
Please cite this article as: Guerrero, A., Moreles, M.A., On the numerical solution of the eigenvalue problem in fractional quantum mechanics, Communications in Nonlinear Science and Numerical Simulation (2014), doi: http:// dx.doi.org/10.1016/j.cnsns.2014.06.013
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
On the numerical solution of the eigenvalue problem in fractional quantum mechanicsI Alejandro Guerreroa , Miguel Angel Morelesb,1,∗ a CIMAT,
Jalisco S/N, Valenciana, Guanajuato GTO 36240, Mexico of Chemical and Petroleum Engineering, University of Calgary, 2500 University Drive NW, Calgary, Alberta, Canada T2N 1N4
b Department
Abstract In this work we propose a Control Volume Function Approximation (CVFA) method to solve equations involving the fractional Laplacian. The function approximation part is carried out with Radial Basis Function (RBF) interpolation. The physical application of interest is the eigenvalue problem for the time independent fractional Schr¨ odinger equation. Fractional derivatives are considered in the Riesz potentials sense. Keywords: Fractional quantum mechanics, Fractional Laplacian, Control Volume Method, Radial Basis Functions
1. Introduction In recent years, fractional calculus has become a topic of great research activity. This is due to the fact that generalizations of integer to fractional derivatives in connection with physical systems, have found applications in most 5
fields. See for instance the historical survey presented in Machado, Kiryakova & Mainardi [1]. The application of interest in this work is to fractional quantum mechanics. Here, the Schr¨ odinger equation is generalized by considering fractional powers I Partially
supported by CONACYT author Email address: [email protected] (Miguel Angel Moreles) 1 Permanent address: CIMAT, Jalisco S/N, Valenciana, Guanajuato GTO 36240, Mexico, [email protected] ∗ Corresponding
Preprint submitted to Communications in Nonlinear Science and Numerical SimulationJune 13, 2014
of the Laplacian, the fractional Laplacian, in the Hamiltonian. A physically 10
sound generalization is that of Laskin [2, 3, 4]. Therein a derivation of the fractional Schr¨ odinger equation is developed based on Feynman path integrals over Levy trajectories. Following these works, there has been an explosion on research regarding fractional Schr¨odinger equations. For instance, in Wang & Xu [5] a fractional Schr¨ odinger equation with space-time fractional derivatives
15
is considered. A solution is proposed by the method of integral transforms. Application to the free particle and square well potential is presented. Of related interest is Tarasov [6], where a fractional generalization of Heisenberg equation is developed, yielding a generalization to quantum Hamiltonian systems. An alternative approach, the D-deformed calculus, to model fractional-dimension
20
systems is introduced in Matos-Abigue [7]. The free particle and the harmonic oscillator are considered as systems in the so called framework of D-deformed quantum mechanics, the process resembles to the classical case. Finally, on the theoretical side, Ionescu & Pusateri [8] carry out a well-posedness analysis for a nonlinear fractional Schr¨ odinger equation in one dimension.
25
This work is from a numerical perspective. Our purpose is to introduce a numerical method to solve the time independent Schr¨odinger equation, namely, h
Dα −~2 ∆
α/2
i + V (r) ψ(r) = Eψ(r),
1 < α ≤ 2.
Here Dα is a generalized diffusion coefficient, and the fractional Laplacian is defined in terms of the Fourier Transform, namely
2
−~ ∆
α/2
ψ(r) =
Z
1 3
(2π)
p · r α d3 p exp i |p| ψ(p). ~
Numerical methods to solve this eigenvalue problem are scarce at best. For 30
the one dimensional case, a collocation method is presented in Amore et al [9]. Within the method, sampling functions, sampling points and interval of definition of the wave functions, are to be determined according to the prescribed boundary conditions. There is no apparent extension to higher dimensions. In Zoia, Rosso & Kardar [10], a matrix representation of the fractional Laplacian
2
35
is introduced. The eigenvalues and eigenfunctions of such a matrix, converge to the eigenvalues and eigenfunctions of the fractional Laplacian when the size of the matrix tends to infinity. Apparently, the method requires large matrix sizes. For instance, in Kwa´snicki [11], the method is used to approximate the eigenvalues associated to the infinite well potential. Therein, a matrix of size
40
O(103 ) is required. A difficulty on approximating the Fractional Laplacian, or any fractional derivative for that matter, is its non local property. Consequently, one is led to full and large matrix systems. Moreover, on bounded domains boundary conditions need to be dealt carefully. To address these issues, we propose a method
45
based on the Control Volume Function Approximation Method (CVFA), see Li, Chen & Huan [12]. After integrating on control volumes, function approximation is carried out on the boundaries by local interpolation. Our proposal is to use Radial Basis Function (RBF) for global interpolation. It is well known that RBF approximations lead to ill-posed full matrix systems, but give satisfactory
50
results on coarse discretizations. As we shall see, the latter, and the global feature of RBF are well suited to approximate the fractional Laplacian in the eigenvalue problem. We provide numerical results for the one dimensional cases in the works above. We remark that in all examples it suffices to solve O(102 ) matrix problems. The matrix eigenvalue problem is solved simply by the inverse
55
power method and deflation. The outline is as follows. In Section 2 we introduce the classical definition of the fractional Laplacian in terms of Riesz potentials and the Fourier Transform. The restriction to bounded domains is as in Musina & Nazarov [13]. In Section 3 we introduce
60
a Control Volume Radial Basis Function (CVRBF) Method for the eigenvalue problem, in any dimension. We use the simplest radial function, φ(r) = r, for approximation. The matrix problem for the one dimensional case is constructed in Section 4. Numerical results are presented in Section 5. We solve the eigenvalue problem for the infinite well potential, the harmonic potential and the
65
quartic anharmonic potential. In Section 6 we draw conclusions and comment 3
of future work.
2. The fractional Laplacian 2.1. Riesz potentials and the fractional Laplacian The results in this section are well known, proofs are found for instance in 70
Helgason [14]. We shall use the definition of Fourier Transform as follows Z fb(k) ≡ F(f )(k) = exp (−ik · x) f (x)dx, Rd
with inverse
F
−1
(F )(x) =
Z
1 d
(2π)
exp (ik · x) F (k)dk Rd
For s ≥ 0 the Sobolev space H s (Rd ) ≡ H s is defined as customary by Z s s 2 H = {f ∈ L : 1 + |k|2 |fb(k|2 dk < +∞}. Rd 0
Let S and S the spaces of tempered functions and tempered distributions respectively. For the case s < 0, the Sobolev space is defined as follows 0
s
Z
H = {f ∈ S :
1 + |k|2
s
|fb(k|2 dk < +∞}.
Rd
By means of the Fourier Transform, the fractional Laplacian operator, (−∆)s 75
is defined for f ∈ H s by h i s 2s (−∆) f = F −1 |k| F(f ) . Let us recall the Riesz potential 1 I (f ) (x) = Hd (γ) γ
Z
γ−d
f (y) |x − y|
with Γ (γ/2) . Hd (γ) = 2γ π d/2 Γ d−γ 2 4
dy
If γ − d ∈ / 2Z+ we can write (I γ f ) (x) = f ∗
1 rγ−d Hd (γ)
(x),
f ∈ S.
We have the following result Proposition. If f ∈ S Rd then γ 7−→ (I γ f ) (x) extends to a holomorphic function in the set Cd = {γ ∈ C : γ − d ∈ / 2Z+ } . Also I 0 f = lim I α f = f α→0
and I γ ∆f = ∆I γ f = −I γ−2 f.
(1)
In terms of the Riesz potential the fractional Laplacian is expressed in the form s
(−∆) f = I −2s f, 80
−s − d ∈ / 2Z+ .
2.2. Fractional Hamiltonian on bounded domains Let Ω be a Lipschitz bounded domain in Rd . We shall consider (−∆)s f defined on the space e s (Ω) = {f ∈ H s : supp f ⊂ Ω} ¯ H s
For 0 < s < 1 the spectrum of (−∆) is discrete and contained in (0, ∞). See Musina & Nazarov [13]. 85
Hereafter we let s = α/2,
(−∆)
α/2
α > 0. In light of (1) we can write f = I −α f = (−∆) I 2−α f.
(2)
This expression is the point of departure for the Control Volume method to be introduced below.
5
¯ c and sufficiently smooth on Consider a potential function V (x) infinite on Ω Ω. The eigenvalue problem of interest is h i α/2 (−∆) + V (x) ψ = E ψ. 90
(3)
whee ψ is the wave function, and E the energy level. e α/2 (Ω). Because of the Sobolev EmNotice that ψ is sought in the space H bedding Theorem, if α > 1, ψ can be identified with a continuous function ¯ c . Consequently we append zero Dirichlet conditions to the that vanishes on Ω eigenvalue problem (3), namely
x ∈ ∂Ω.
ψ(x) = 0,
95
We shall see that this approximation provides highly satisfactory numerical results.
3. A Control volume method with RBF interpolation 3.1. The CVFA Method ¯ c . We consider We have that the wave function ψ(x) in (3) is zero for x ∈ Ω h
(−∆)
α/2
i + V (x) ψ(x) = E ψ(x),
x ∈ Ω, (4) x ∈ ∂Ω.
ψ(x) = 0,
A partition Th of Ω consists of a set of (open) control volumes Vi : Ω=
N [
V i,
Vi ∩ Vj = ∅,
i=1
Define the boundary of each Vi by ∂Vi =
Ni [
eik ,
k=1 100
where Ni is the number of edges eik on ∂Vi . 6
i 6= j.
On each control volume integrate the equation and using (2) we may apply the divergence theorem in the first term to obtain Z −
∇ I
2−α
Z
ψ · νi dσ +
∂Vi
−
Ni Z X k=1
Z V (x)ψ(x) dx = E
Vi
∇ I 2−α ψ · νik dσ +
eik
ψ(x) dx, Vi
Z
Z V (x)ψ(x) dx = E
ψ(x) dx,
Vi
(5)
Vi
On each eik ⊂ ∂Vi , an interpolant ψik is used to approximate ψ ψik (x) =
Rik X
i ψk,j ϕik,j (x),
x ∈ eik ,
j=1
where Rik is the number of interpolation nodes xik,j for eik and these nodes can be located on or surrounding Vi . 105
The coordinate functions ϕik,j are defined as follows:
ϕik,j (x) =
1
at node xik,j
0
at other nodes
We have for the integral term in the sum the approximation R eik
R ik P j=1
∇ I 2−α ψik (x) · νik dσ R eik
=
i ∇ I 2−α ϕik,j (x) · νik dσ ψk,j
i The other terms are also approximated in terms of the nodal values ψk,j ,
leading to a matrix eigenvalue problem. We show this in examples below. 3.2. Coordinate functions by RBF interpolation 110
Let us describe the choice of the coordinate functions ϕik,j (x). The purpose in to make the approximation of ∇ I 2−α ϕik,j (x) as simple as possible. Suppose X ⊂ Ω ⊂ Rd , Ω a bounded set. X = {x1 , x2 , . . . , xN } . Suppose further that we are given {y1 , y2 , . . . , yN } , data values of a function f ∈ C(Ω), 7
i.e., f (xj ) = yj , 1 ≤ j ≤ N. The problem is to find an interpolating function 115
sf,X (x) such that sf,X (xj ) = yj ,
1 ≤ j ≤ N.
(6)
The interpolating function to be used is of the form
sf,X (x) =
N X
αj φ (|x − xj |) + β0 .
(7)
j=1
Here φ(r) = r, r = |x|, is a conditionally positive definite radial function or order 1. Consequently, the system of equations (6) is complemented with N X
αj = 0.
j=1
This guarantees existence and uniqueness of the coefficients in (7), see Wendland 120
[15]. Our aim is to test the method with the simplest radial function. Extension to other choices of radial functions is straightforward. Noteworthy, the thin-plate spline, ϕ(r) = r2 log r is used in Li, Chen & Huan [12].
4. The One dimensional eigenvalue problem 125
Let us consider the interval D = (a, b) and a partition {a = x0 , x1 , . . . , xN , xN +1 = b}. The control segments are Vi = xi− 12 , xi+ 12 with boundaries n o ∂Vi = xi− 12 , xi+ 12 . n o n o In each boundary xi− 12 , xi+ 12 , we shall consider all points in the partition as the neighboring points. Hence, the interpolant is of the form
φh (x) =
N +1 X
αj |x − xj | + β0
j=0
8
Let ψi ≈ ψ(xi ). From (5) we obtain the eigenvalue problem 1 [Aα + Λ] ψ = E ψ |Vi | 130
(8)
Here |Vi | is the length of the control segment, and Λ is a diagonal matrix with diagonal entries Z Λi,i =
V (x)dx. Vi
The computation of Aα , the sub matrix associated to the Fractional Laplacian, is shown below. In the examples that follow, we consider a regular grid. 135
We remark that to approximate the first smallest eigenvalues of (8), we simply use the inverse power method and deflation. 4.1. The fractional Laplacian matrix Extending by zero outside the interval (a, b) we write the Riesz potential for a function in the form 2−α 2−α I 2−α f (x) = I− f (x) + I+ f (x)
where 2−α I− f (x) =
x
1 H1 (2 − α)
Z
1 H1 (2 − α)
Z
|x − y|1−α f (y) dy,
(9)
|x − y|1−α f (y) dy,
(10)
a
and 2−α I+ f (x) =
140
b
x
For k = 1, 2, . . . , N the coordinate functions are of the form
φk (x) =
N +1 X
αkj |x − xj | + αkN +1
(11)
j=0
Thus the matrix components are (Aα )i,k
x 1 i+ 2 d 2−α =− I φk (x) dx x 1 i−
9
2
(12)
We are led to consider d 2−α dx I− φk ,
d 2−α φk . dx I+
We have
N
X d 2−α I∓ φk (x) = dx j=1
d 2−α d 2−α I∓ |x − xj | αkj + I∓ 1 αkN +1 dx dx
(13)
The computation of each term in the sum is straightforward. We obtain
d 2−α dx I−
d 2−α dx I−
1 2−α (x
|x − xj | = (xj − a)(x − a)1−α −
|x − xj | = (xj − a)(x − a)1−α +
1 2−α
− a)2−α ,
x < xj ,
2(x − xj )2−α − (x − a)2−α ,
x > xj ,
and d 2−α I 1 = (x − a)1−α . dx − 145
Similarly
d 2−α dx I+
|x − xj | =
d 2−α dx I+
1 2−α
−2(xj − x)2−α + (b − x)2−α − (b − xj )(b − x)1−α ,
|x − xj | = −(b − xj )(b − x)1−α +
1 2−α (b
and d 2−α I 1 = −(b − x)1−α . dx − It remains to evaluate (13) at the edge points xi−1/2 ,
xi+1/2
and substitute in (12). 10
− x)2−α ,
x > xj ,
x < xj ,
5. One dimensional Fractional Quantum Mechanics 5.1. Infinite potential well let D = (−1, 1). In this case
150
V (x) =
0,
x ∈ D, (14)
∞,
otherwise.
The problem to consider is α/2 d2 − 2 ψ(x) = E ψ(x), dx
x∈D
(15)
We apply the method above to approximate the first 10 eigenvalues. In Table 1 we list the results and relative errors in comparison with the method of Zoia, Rosso & Kardar [10]. For the latter we use the values reported in Kwa´snicki 155
[11]. Therein a system 5000 × 5000 is solved. An order of magnitude larger than the 200 × 200 system used here. Table 1 (Infinite potential well) α
E1
E2
E3
E4
E5
E6
E7
E8
E9
E10
1.5
1.5998
5.0669
9.6075
15.039
21.217
28.071
35.532
43.559
52.111
61.159
0.0017
0.0016
0.0016
0.0016
0.0015
0.0015
0.0015
0.0014
0.0014
0.0013
1.8
2.0501
7.5079
15.809
26.739
40.135
55.891
73.918
94.147
116.515
140.974
(1e-3)
0.9695
0.9538
0.9216
0.8823
0.8341
0.7744
0.7033
0.6214
0.5266
0.4206
1.9
2.2448
8.5984
18.72
32.47
49.73
70.42
94.48
121.86
152.51
186.38
(1e-3)
0.7162
0.6736
0.6335
0.5816
0.5207
0.4451
0.3552
0.2501
0.1301
0.0048
1.99
2.4438
9.7332
21.828
38.707
60.355
86.760
117.91
153.79
194.40
239.71
(1e-3)
0.4311
0.3973
0.3521
0.2889
0.2100
0.1133
0.0011
0.3281
0.2834
0.4507
11
Figure 1: First wave functions, ψi (x), i = 1, 2, 3, 4, for the infinite well potential. α = 1.5 solid line. α = 2 dashed line
As remarked above, we use the inverse power method and deflation. Thus, we obtain also the eigenvectors which approximate the wave functions. For 160
illustration we plot the first four in Figure 1 for α = 1.5. In all examples that follow the classical case α = 2 is included. Remark. In the terminology of Musina & Nazarov [13], we are approximating the eigenvalue problem for the Dirichlet fractional Laplacian. Therein it is also considered the eigenvalue problem for the Navier fractional Laplacian. Let us
165
denote by EnD , and EnN the corresponding eigenvalues. The Navier case is solved in Laskin [3]. The normalized wave functions are n πx o even ψ2m+1 (x) = cos (2m + 1) , 2 12
m = 0, 1, . . . ,
(16)
and n πx o odd ψ2m (x) = sin (2m) , 2
m = 1, 2, . . . .
(17)
These are the functions plotted in Figure 1 corresponding to α = 2. The Navier eigenvalues are EnN = 170
π α 2
nα ,
n = 1, 2, . . . .
(18)
Consequently, the Navier fractional Laplacian is defined as a spectral power of the Laplacian. As pointed out in Musina & Nazarov [13], this is not the case for the Dirichlet fractional Laplacian. For the latter, to our knowledge, there are no analytic expressions for wave functions and eigenvalues. See the in depth study of the eigenvalue problem in Kwa´snicki [11].
175
For eigenvalues it is known that EnN > EnD .
(19)
See Corollary 4 in Musina & Nazarov [13]. It is readily seen that our estimates in Table 1 satisfy this inequality. 5.2. Fractional Harmonic Oscillator Let V (x) the quadratic potential V (x) = x2 , 180
x ∈ R.
We consider the fractional quantum harmonic oscillator "
d2 − 2 dx
#
α/2 +x
2
ψ(x) = E ψ(x),
x ∈ R.
b α/2 (DL ). We solve a Let DL = (−L, L). For approximation we seek ψ ∈ H 200 × 200 system for α = 3/2 and L = 5.5. Wave functions are plotted in Figure 2. In Table 2 we show numerical results in comparison to the method of Amore et al [9], and the WKB estimates in Laskin [4], namely EnW KB
π = B(1/2, 1/α + 1)
1 n− 2
13
2α 2+α
,
n = 1, 2, . . . .
(20)
Figure 2: First wave functions, ψi (x), i = 1, 2, 3, 4, for the harmonic potential. α = 1.5 solid line. α = 2 dashed line
185
Where B(·, ·) is the beta function. Table 2 (harmonic oscillator) α
E1
E2
E3
1.5
1.0011
2.7076
4.1762
rel. error
0.0000756
0.0004866
0.0007545
rel. error WKB
0.0496
0.0024
0.0021
Remark.
Nontrivial issues of the collocation method of Amore et al [9], are
the choice of L and the number of sampling points, N . It is noticed that the number of sampling points increases significantly with L. First they start with 190
a number N of collocation functions, then a number of sampling points and an
14
optimal choice of L are determined. Refinement may be required. In contrast, our method only requires a good estimate for the decay of the wave function. 5.3. Fractional anharmonic Oscillator Our last example is the anharmonic oscillator associated to the potential 195
V (x) = x4 . For α = 4/3 it is known that a good approximation for the eigenvalues is provided by the asymptotic W KB estimates. The predicted spectrum is evenly spaced and is given by EnW KB =
2π Γ(1/4)Γ(7/4)
1 n+ . 2
(21)
We apply the Control Volume method with L = 3 and solve a 300 × 300 200
system. Results are listed in Table 3, α = 4/3 En , n = 1, 2, 3, ...10 and relative errors with respect to WKB. The first wave functions are plotted in Figure 3. Table 3 (anharmonic oscillator) α
E1
E2
E3
E4
E5
E6
E7
E8
E9
E10
4/3
0.9947
2.8204
4.7065
6.6022
8.4794
10.369
12.249
14.136
16.016
17.902
0.0550
0.0028
0.0016
0.0004
0.0007
0.0001
0.0006
0.0004
0.0007
0.0006
Results in Table 3 show that the eigenvalues of the anharmonic oscillator follow the W KB line (21), see Figure 4. 205
Remark.
Equations (20) and (21), are special cases of the WKB general
equation derived in the pioneering work of Laskin [4]. Accuracy of the WKB estimates, even in small eigenvalues, is remarkable.
6. Conclusions and future work We have introduced a Control Volume Radial Basis Function method to solve 210
the eigenvalue problem associated to the time independent Scr¨odinger equation.
15
Figure 3: First wave functions, ψi (x), i = 1, 2, 3, 4, for the anharmonic potential. α = 4/3 solid line. α = 2 dashed line
16
Figure 4: CVRBF vs. WKB for α = 4/3
17
The method has been tested on one dimensional problems with satisfactory results. In order to test the applicability of the method, we have used for interpolation the simplest radial function. In line with RBF approximation theory, it is 215
straightforward to test other radial functions, as well as node location strategies. Currently, we are working on applications of the method to higher dimensions and irregular domains. The latter is possible thanks to the control volume approach. Hopefully, RBF interpolation will make the method practical in higher dimensions.
220
It is apparent that the proposed method is practical, and its implementation is straightforward. The approximation theory supporting the method is left for future investigations.
Acknowledgements Part of this work was carried out while M. A. Moreles was visiting the 225
Deparment of Chemical and Petroleum Engineering, University of Calgary. M. A. Moreles is thankful for their hospitality. [1] J. T. Machado, V. Kiryakova, F. Mainardi: Recent history of fractional calculus; Commun Nonlinear Sci Numer Simulat; 16 (2011) 1140-1153 [2] N. Laskin: Fractional quantum mechanics and Levy path integrals; Phys.
230
Rev. A Vol 268, 298-305. (2000) [3] N. Laskin:Fractals and quantum mechanics; Chaos: An Interdisciplinary Journal of Nonlinear Science 10.4 (2000): 780-790. [4] N. Laskin: Fractional Schr¨ odinger equation; Phys. Rev. E Vol 66 056108. (2002)
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[5] A. Wang, M. Xu: Generalized fractional Schr¨ odinger equation with spacetime fractional derivatives; J. Math. Phys 48, 043502 (2007)
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[6] V. E. Tarasov: Fractional Heisenberg equation; Phys. Letters A 372 (2008) 2984-2988 [7] A. Matos-Abiague: 240
Deformation of quantum mechanics in fractional-
dimensional space; J. Phys. A: Math. Gen. 34 (2001) 11059-11068 [8] A. D. Ionescu, F. Pusateri: Nonlinear fractional Schr¨ odinger equations in one dimension; J. Functional Analysis 266 (2014) 139-176 [9] P. Amore, F. M. Fern´ andez, C. P. Hofmann, R. A. S´aenz, R. A. (2010). Collocation method for fractional quantum mechanics. Journal of Mathematical
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Physics, 51(12), 122101. [10] A. Zoia, A. Rosso, M. Kardar: Fractional Laplacian in bounded domains; Phys. Rev. E 76(2) (2007) 061121 [11] M. Kwa´snicki, M. (2012). Eigenvalues of the fractional Laplace operator in the interval. Journal of Functional Analysis, 262(5), 2379-2402.
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[12] B. Li, Z. Chen, G. Huan: Control volume function approximation methods and their applications to modeling porous media flow ; Advances in Water Resources 26, 435 - 444. (2003) [13] Musina, R., Nazarov, A. I. (2013). On fractional Laplacians. Communications in Partial Differential Equations, (just-accepted).
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[14] S. Helgason: The Radon Transform, 2nd ed; Birkh¨auser; Boston. (1999) [15] H. Wendland: Scattered Data Approximation; Cambridge University Press, Cambridge. (2005)
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Highlights of: “On the numerical solution of the eigenvalue problem in fractional quantum mechanics” • • • • •
A novel method to solve the eigenvalue problem in fractional quantum mechanics is proposed Homogeneous Dirichlet boundary conditions are introduced for the fractional Laplacian defined on appropriate Sobolev Spaces The method is control volume based, thus it applies to irregular domains Eigenfunction approximations is in terms of Radial Basis Functions (RBF) interpolation The method is applied to the potentials: infinite well, quadratic (harmonic oscillator), quartic (anharmonic oscillator)