Post on 02-Feb-2018
Solution & visualization in The Sturm – Liouville problem
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JASS 2009Alexandra Zykova
superviser: PhD Vadim Monakhov
Department of Computational PhysicsSPSU
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Outline
1. Introduction. The regular Sturm - Liouville problem.
2. The Two – Center Problem in quantum mechanics.
3. SLEIGN23.1 Manual for program package SLEIGN23.2 SLEIGN2 with BARSIC
),()()()()()')'()((
baxxyxwxyxqxyxp
∈=+− λ
(a, b) – open interval
{p, q, w} - coefficients defined on the open interval ( a, b)
λ - spectral parameter
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1. Introduction. The regular Sturm - Liouville problem.
Boundary conditions:
• Regular (R)• Singular (S)• Separated• Coupled
Regular boundary conditions:
• separated boundary conditions
0)()('0)()('
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=+=+
byBbyBayAayA
• coupled boundary conditions
))('())('()()(
bpyapybyay=
=
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Eigen solution of the regular Sturm - Liouville problem -
},{ yλ
eigenvalue, for which differential equation hasnontrivial solution
−λ
−y eigenfunction, corresponds to eigenvalue, satisfies boundary conditions
ψ
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The regular Sturm - Liouville problem & Schrödinger problem
),(),()()()()')'()(( baxxyxwxyxqxyxp ∈=+− λ
(a, b) – is the integration interval and the boundaryconditions
),(),()()(')'( baxxyxyxqxy ∈=+− λ
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(a, b) – is the integration interval and the boundaryconditions
2. The Two – Center Problem in quantum mechanics.
0);()(2);(2
2
1
1 =−−+Δ RrrZ
rZERr ψψ
1r 2r 1Z2Z
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eme ==h
and - nuclear charges
and – distance between electron and ,
1Z
2Z
)(REE= - energy term
R - internuclear distance
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Prolate spheroidal coordinate system
)/arctan(,, 2121 xyR
rrR
rr=
−=
+= ϕηζ
- coordinates of charge
- coordinates of charge
1Z
1,1 −== ηζ 2Z1,1 +== ηζ
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ϕηζϕηζψψ immqmkkqmkqmj eRYRXRNR );();()();,,( ==
- set of quantum numbers
- principal quantum number
- orbital quantum number
- magnetic quantum number
},,{ mqkj =
k
q
m
∫ =V
mmqqkkmqkkqm dVRR '''''' );,,();,,(* δδδϕηξψϕηξψ
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0);(]1
)1([);()1(
0);(]1
)1([);()1(
2
222
~2
2
2222
=−
−+−−−+−
=−
−+−−−+−
RYmbpRYdd
dd
RXmapRXdd
dd
mqmq
mkmk
ηη
ηηληη
ηη
ζζ
ζζλζζ
ζζ
RZZbRZZa
pRE
p jj
)(,)(
)0(2
1221
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−=+=
>−= - energy parameter
- charge parameter
),()( apmkζλλ = ),()(
~bpmq
ηλλ =
),(),( )()( bpap mqmkηζ λλ =
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, - separation constants
Quasiradial equation
),1[,0);(,);1(
0);(]1
)1([);()1( 2
2222
∞∈∞<
=−
−+−−−+−
→∞→
ζζ
ζζ
ζζλζζ
ζζ
ζ
RXRX
RXmapRXdd
dd
mkmk
mkmk
Jaffé expansion
- transformation of variable
- transformation of equilibrium points
- transformation of interval
)1/()1( +−= ζζt
1,01,1, +→∞→+∞+→−→tζ
)1;0[);1[ ∈→∞∈ tζ
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ss
s
mpmk geRX )
11()1()1();(
1
2/2
+−
+−= Σ∞
=
−
ζζζζζ σζ
)1(2/ +−= mpaδ
- three - termed relation between coefficients sg
011 =+− −+ ssssss ggg γβα
)1)(1(2)1)(()2(2
)1)(1(
δδγλδδδβ
α
−−−−−=+−++−−+=
+++=
msspmmpss
mss
s
s
s
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Eigenvalues of the problem are found from the condition of nullifying of the continued fraction.
0...),,(
2
211
100 =
−−
−=
βγαβ
γαβλapF
)()41(41
2
2
1
1
spO
sp
sss
ss +−→∞→−
−
ββγα
Coefficients of three – termed relation converge for all It follows from the equation above.
0>p
The ratio of the series coefficients
)(211
spO
sp
s
s
s
gg
+−→∞→
+
provides the convergence of Jaffé expansion on the complete interval )1;0[∈t or );1[ ∞∈ζ 14
Quasiangular equation
0);(]1
)1([);()1( 2
222
~2 =
−−+−−−+− RYmbpRY
dd
dd
mqmq ζη
ηηληη
ηη
11;);1( +≤≤−∞<± ηRYmq
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Baber – Hasse expansion
∑∞
=+
−=0
)();(s
mmss
pmq PceRY ηη η
- relation between three – terms coefficients sc
011 =+− −+ ssssss ccc δχρ
1)(2)](2[
)1)((3)(2
))1(2)(12(
−+++
=
−+++=++
++−++=
msmspbs
msmsms
mspbms
s
s
s
δ
λχ
ρ
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The ratio of succeeding series coefficients at large indexes:
0~1 →∞→
+
ss
s
sp
cc
Solutions of can be found from the equation
0...),,(
2
211
100
)( =
−−
−=
kk
kbpFδρ
δρλη
),(),( )()( bpap qmkmηζ λλ =
Continuous fraction converges due to following limit:
2
1
1 2⎟⎠⎞
⎜⎝⎛→
∞→−
−
sp
sss
ss
χχδρ
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General-purpose programs for computing the eigenvalues and eigenfunctions of Sturm - Liouville problem
• Program SLEIGN has been developed by Bailey, Gordon and Shampine , programming language FORTRAN
• Code in the NAG Library has been developed by Pryce and Marletta , programming language FORTRAN
• Program SLEDGE has been developed by Fulton and Pruess , programming language FORTRAN
• Program SLEIGN2 has been developed by Bailey Everitt and Zettl , programming language FORTRAN
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3. SLEIGN2
To meet the needs of numerical computing techniques was made the following assumptions:
1. The interval ( a, b) of R may be bounded or unbounded2. p, q and w are real-valued functions on (a, b)3. p, q and w piecewise continuous on (a, b)4. p and w strictly positive on (a, b)
Conditions on the coefficients:Minimal conditions:
Smoothness conditions:
),(,, 11 baLwqp ∈−
),(,,', baCwqpp ∈
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0)(),( >xwxp
0)(),( >xwxp
Endpoint classification
To classify endpoints a and b, it is convenient to choose a point ),( bac ∈
• а is Regular (R), if
p, q, w - piecewise continuous on [ a, c] p ( a ) > 0,w ( a ) > 0
•А is Singular (S), ifa or
a R, but ∫ +∞=++−c
a
dxxwxqxp )}()())({( 1∈
±∞=
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∞+<<∞− a
• The singular endpoint a is Limit Point (LP) if for some real at least one solution of differential equation satisfies the condition
• The endpoint a is Weakly Regular (WR) if &
∫ +∞=c
a
dxyw 2
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∫ ∞+<++−c
a
dxwqp )( 1
λ
a<∞−
• The singular endpoint is Limit-Circle Non-Oscillatory(LCNO) if for some real value of spectral parameter ALL real-valued solutions satisfy the conditions
and has at most a finite number of zeros in (a, c ]
• The singular endpoint is Limit-Circle Oscillatory(LCO)if for some real value of spectral parameter ALL real-valued solutions satisfy the conditions
and has an infinite number of zeros in (a, c ]
),( λxy
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λ
∞+<∫2c
a
yw
∞+<∫2c
a
yw
),( λxy
λ),( λxy
),( λxy
SLP problems are classified into various classes based on the classification of the endpoints and on whether the boundary conditions are separated (S) or coupled (C). We have the following categories:
1. R/R, Separated2. R/R, Coupled3. R/LCNO LCNO/R, Separated4. R/LCNO LCNO/R, Coupled5. R/LCO LCO/R, Separated6. R/LCO LCO/R, Coupled7. LCNO /LCO LCO/ LCNO LCO/ LCO, Separated 8. LCNO /LCO LCO/ LCNO LCO/ LCO, Coupled9. LP/R LP/LCNO LP/LCO R/LP LCNO/LP LCO/LP10.LP/LP
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The algorithm in SLEIGN2
• Initial interval (a, b) is converted to interval (-1,+1) in the SLEIGN2 package
• The computation procedure is implemented by the use of Prüfer Transform.
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Prüfer Transform.
))(cos()()()'())(sin()()(
xxxpyxxxyθρ
θρ=
=
Differential equation for ρ & θ :
))(cos())(sin())()()(()(/)('))((sin))()(())((cos)()('
1
221
xxxqxwxpxxxxqxwxxpxθθλρρ
θλθθ
+−=
−+=−
−
)/()()/()(
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BBarctgnbAAarctga
−=−−=πθ
θ
Boundary conditions for θ :
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The following disadvantages were found in the program package SLEIGN2:
• The interface of the program is organized on the base of console dialog. This approach considerably increase the time for defining the problem paramters.
• The program is unstable towards the input : if numbers are taken in the incorrect format (say, with comma instead of dot) or letters are taking instead of number. In this case the program is terminated and it is required to start work from the very beginning.
• Additional program MAKEPQW is required in order to create own examples.
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Solution in BARSIC (Business And Research Scientific Interactive Calculator)
• Numerical algorithms from SLEIGN2 remain unchanged.Subroutines of SLEIGN2 package (programming language FORTRAN) are compiled into ‘dll’ file (‘so’ files in case of OSLinux ) and then they are called from BARSIC programs.
• Additional functions for calculation of first and second derivatives were created (It’s necessary to write them in FORTRAN when SLEIGN2 is used directly)
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The comparison between SLEIGN2 and well known mathematical packages (Mathematica, Maple, COMSOL Multiphysics) shows that SLEIGN2 is more efficient from the point of view time of computation and numerical errors. Moreover, is not possible to solve problem with coupled conditions in Maple and Mathematica.
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References and resources:
1. Werener O. Amerin, Andress M. Hinz, David B. Pearson “Sturm – Liouville Theory. Past an Present”
2. J.D. Pryce “Numerical solution of Sturm-Liouville Problems” (Oxford University Press; 1993)
3. http://www.docstoc.com/docs/2594692/SLEIGN2
4. http://www.math.niu.edu/SL2
5. A. Devdariani, E. Dalimier “Dipole transition-matrix elements of the one-electron heterodiatomic quasimolecules”
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Thank you for attention!