Algeb aic aspec s o he Riemann-Hilbe p oblem
o ma ix o hogonal polynomials1
Manuel Dom´ınguez de la Iglesia
Depa amen o de An´alisis Ma em´a ico, Uni e sidad de Se illa
In e na ional Symposium on O hogonal Polynomials and Special
Func ions: a Complex Analy ic Pe spec i e
Copenhague, June 12, 2012
1join wo k wi h F. A. G ¨unbaum and A. Ma ´ınez-Finkelsh ein
The RH p oblem o OP
The RH p oblem o MOP
Ou line
1The Riemann-Hilbe p oblem o o hogonal polynomials
The RHP o OPs
The Lax pai
Examples
2The Riemann-Hilbe p oblem o ma ix o hogonal polynomials
The RHP o MOPs
The Lax pai
Examples
Manuel Dom´ınguez de la Iglesia Algeb aic aspec s o he RHP o MOP
The RH p oblem o OP
The RH p oblem o MOP
The RHP o OPs
The Lax pai
Examples
Ou line
1The Riemann-Hilbe p oblem o o hogonal polynomials
The RHP o OPs
The Lax pai
Examples
2The Riemann-Hilbe p oblem o ma ix o hogonal polynomials
The RHP o MOPs
The Lax pai
Examples
Manuel Dom´ınguez de la Iglesia Algeb aic aspec s o he RHP o MOP
The RH p oblem o OP
The RH p oblem o MOP
The RHP o OPs
The Lax pai
Examples
O hogonal polynomials
Le dµbe a posi i e Bo el measu e suppo ed on R.
We will assume dµ(x) = ω(x)dx,ω≥0 and xiω, xjω0∈L1(R).
We can hen cons uc a amily o o hono mal polynomials (pn)ns. .
(pn,pm)ω=ZR
pn(x)pm(x)ω(x)dx =δn,m,n,m≥0
pn(x) = κn(xn+an,n−1xn−1+· · · ) = κnb
pn(x)
The monic polynomials b
pn(x) sa is y a h ee- e m ecu ence ela ion
xb
pn(x) = b
pn+1(x) + αnb
pn(x) + βnb
pn−1(x)
Manuel Dom´ınguez de la Iglesia Algeb aic aspec s o he RHP o MOP
The RH p oblem o OP
The RH p oblem o MOP
The RHP o OPs
The Lax pai
Examples
O hogonal polynomials
Le dµbe a posi i e Bo el measu e suppo ed on R.
We will assume dµ(x) = ω(x)dx,ω≥0 and xiω, xjω0∈L1(R).
We can hen cons uc a amily o o hono mal polynomials (pn)ns. .
(pn,pm)ω=ZR
pn(x)pm(x)ω(x)dx =δn,m,n,m≥0
pn(x) = κn(xn+an,n−1xn−1+· · · ) = κnb
pn(x)
The monic polynomials b
pn(x) sa is y a h ee- e m ecu ence ela ion
xb
pn(x) = b
pn+1(x) + αnb
pn(x) + βnb
pn−1(x)
Manuel Dom´ınguez de la Iglesia Algeb aic aspec s o he RHP o MOP

The RH p oblem o OP
The RH p oblem o MOP
The RHP o OPs
The Lax pai
Examples
O hogonal polynomials
Le dµbe a posi i e Bo el measu e suppo ed on R.
We will assume dµ(x) = ω(x)dx,ω≥0 and xiω, xjω0∈L1(R).
We can hen cons uc a amily o o hono mal polynomials (pn)ns. .
(pn,pm)ω=ZR
pn(x)pm(x)ω(x)dx =δn,m,n,m≥0
pn(x) = κn(xn+an,n−1xn−1+· · · ) = κnb
pn(x)
The monic polynomials b
pn(x) sa is y a h ee- e m ecu ence ela ion
xb
pn(x) = b
pn+1(x) + αnb
pn(x) + βnb
pn−1(x)
Manuel Dom´ınguez de la Iglesia Algeb aic aspec s o he RHP o MOP
The RH p oblem o OP
The RH p oblem o MOP
The RHP o OPs
The Lax pai
Examples
Solu ion o he RHP o o hogonal polynomials
We y o ind a 2 ×2 ma ix- alued unc ion Yn:C→C2×2such ha
1Ynis analy ic in C R
2Yn
+(x) = Yn
−(x)1ω(x)
0 1 when x∈R
3Yn(z)=(I2+O(1/z)) zn0
0z−nas z→ ∞
Fo n≥1 he unique solu ion o he RHP abo e is gi en by
Fokas-I s-Ki ae , 1990
Yn(z) = b
pn(z)C(b
pnω)(z)
−2πiγn−1b
pn−1(z)−2πiγn−1C(b
pn−1ω)(z)
whe e C( )(z) = 1
2πiRR
( )
−zd is he Cauchy ans o m and γn=κ2
n.
The exis ence and unici y is a consequence o he Mo e a’s heo em, Liou ille’s
heo em and de Yn(z) = 1.
Manuel Dom´ınguez de la Iglesia Algeb aic aspec s o he RHP o MOP
The RH p oblem o OP
The RH p oblem o MOP
The RHP o OPs
The Lax pai
Examples
Solu ion o he RHP o o hogonal polynomials
We y o ind a 2 ×2 ma ix- alued unc ion Yn:C→C2×2such ha
1Ynis analy ic in C R
2Yn
+(x) = Yn
−(x)1ω(x)
0 1 when x∈R
3Yn(z)=(I2+O(1/z)) zn0
0z−nas z→ ∞
Fo n≥1 he unique solu ion o he RHP abo e is gi en by
Fokas-I s-Ki ae , 1990
Yn(z) = b
pn(z)C(b
pnω)(z)
−2πiγn−1b
pn−1(z)−2πiγn−1C(b
pn−1ω)(z)
whe e C( )(z) = 1
2πiRR
( )
−zd is he Cauchy ans o m and γn=κ2
n.
The exis ence and unici y is a consequence o he Mo e a’s heo em, Liou ille’s
heo em and de Yn(z) = 1.
Manuel Dom´ınguez de la Iglesia Algeb aic aspec s o he RHP o MOP
The RH p oblem o OP
The RH p oblem o MOP
The RHP o OPs
The Lax pai
Examples
Solu ion o he RHP o o hogonal polynomials
We y o ind a 2 ×2 ma ix- alued unc ion Yn:C→C2×2such ha
1Ynis analy ic in C R
2Yn
+(x) = Yn
−(x)1ω(x)
0 1 when x∈R
3Yn(z)=(I2+O(1/z)) zn0
0z−nas z→ ∞
Fo n≥1 he unique solu ion o he RHP abo e is gi en by
Fokas-I s-Ki ae , 1990
Yn(z) = b
pn(z)C(b
pnω)(z)
−2πiγn−1b
pn−1(z)−2πiγn−1C(b
pn−1ω)(z)
whe e C( )(z) = 1
2πiRR
( )
−zd is he Cauchy ans o m and γn=κ2
n.
The exis ence and unici y is a consequence o he Mo e a’s heo em, Liou ille’s
heo em and de Yn(z) = 1.
Manuel Dom´ınguez de la Iglesia Algeb aic aspec s o he RHP o MOP
The RH p oblem o OP
The RH p oblem o MOP
The RHP o OPs
The Lax pai
Examples
The Lax pai I
We look o a pai o i s -o de di e ence/di e en ial equa ions o he o m
Yn+1(z) = En(z)Yn(z),d
dz Yn(z) = Fn(z)Yn(z)
P oblem. Typically, he coe icien Fn(z) is di icul o ob ain. We can a oid ha
by ans o ming he RHP in ano he RHP wi h cons an jump.
Conside he ans o ma ion
Xn(z) = Yn(z)ω1/20
0ω−1/2
We obse e ha Xnis in e ible and ha
Xn
+(x) = Xn
−(x)1 1
0 1
Tha means ha Xnhas a cons an jump
⇒En(z) and Fn(z) a e comple ely de e mined by hei beha io a z→ ∞.
Manuel Dom´ınguez de la Iglesia Algeb aic aspec s o he RHP o MOP

The RH p oblem o OP
The RH p oblem o MOP
The RHP o OPs
The Lax pai
Examples
The Lax pai II
I we addi ionally assume ha (ω1/2)0
ω1/2is a polynomial o deg ee m, hen
Xn+1(z) = z−αn1
2πiγ−1
n
−2πiγn0
| {z }
En(z)
Xn(z)
d
dz Xn(z) = −Bn(z)−1
2πiγ−1
nAn(z)
2πiAn−1(z)γn−1Bn(z)
| {z }
Fn(z)
Xn(z)
whe e An(z) and Bn(z) a e polynomials o deg ee m−1 and m espec i ely.
C oss-di e en ia ing he Lax pai yield
Compa ibili y condi ions
E0
n(z) + En(z)Fn(z) = Fn+1(z)En(z)
also known as s ing equa ions.
Manuel Dom´ınguez de la Iglesia Algeb aic aspec s o he RHP o MOP
The RH p oblem o OP
The RH p oblem o MOP
The RHP o OPs
The Lax pai
Examples
The Lax pai II
I we addi ionally assume ha (ω1/2)0
ω1/2is a polynomial o deg ee m, hen
Xn+1(z) = z−αn1
2πiγ−1
n
−2πiγn0
| {z }
En(z)
Xn(z)
d
dz Xn(z) = −Bn(z)−1
2πiγ−1
nAn(z)
2πiAn−1(z)γn−1Bn(z)
| {z }
Fn(z)
Xn(z)
whe e An(z) and Bn(z) a e polynomials o deg ee m−1 and m espec i ely.
C oss-di e en ia ing he Lax pai yield
Compa ibili y condi ions
E0
n(z) + En(z)Fn(z) = Fn+1(z)En(z)
also known as s ing equa ions.
Manuel Dom´ınguez de la Iglesia Algeb aic aspec s o he RHP o MOP
The RH p oblem o OP
The RH p oblem o MOP
The RHP o OPs
The Lax pai
Examples
Example I: He mi e polynomials
Conside ω(x) = e−x2⇒He mi e polynomials (Hn)n.
The ans o ma ion Xn(z) = Yn(z) e−z2/20
0ez2/2!gi es he ollowing Lax pai
Xn+1(z) = z1
2πiγ−1
n
−2πiγn0Xn(z),d
dz Xn(z) = −z−1
πiγ−1
n
4πiγn−1zXn(z)
The di e ence equa ion gi es (using βn=γn/γn+1) he TTRR
xb
Hn(x) = b
Hn+1(x) + βnb
Hn−1(x),
while he di e en ial equa ion gi es he ladde ope a o s
b
H0
n(x) = 2βnb
Hn−1(x),b
H0
n(x)−2xb
Hn(x) = −2b
Hn+1(x).
The compa ibili y condi ions a e
βn+1 −βn=1
2⇒βn=n
2
Manuel Dom´ınguez de la Iglesia Algeb aic aspec s o he RHP o MOP
The RH p oblem o OP
The RH p oblem o MOP
The RHP o OPs
The Lax pai
Examples
Example I: He mi e polynomials
Conside ω(x) = e−x2⇒He mi e polynomials (Hn)n.
The ans o ma ion Xn(z) = Yn(z) e−z2/20
0ez2/2!gi es he ollowing Lax pai
Xn+1(z) = z1
2πiγ−1
n
−2πiγn0Xn(z),d
dz Xn(z) = −z−1
πiγ−1
n
4πiγn−1zXn(z)
The di e ence equa ion gi es (using βn=γn/γn+1) he TTRR
xb
Hn(x) = b
Hn+1(x) + βnb
Hn−1(x),
while he di e en ial equa ion gi es he ladde ope a o s
b
H0
n(x) = 2βnb
Hn−1(x),b
H0
n(x)−2xb
Hn(x) = −2b
Hn+1(x).
The compa ibili y condi ions a e
βn+1 −βn=1
2⇒βn=n
2
Manuel Dom´ınguez de la Iglesia Algeb aic aspec s o he RHP o MOP
The RH p oblem o OP
The RH p oblem o MOP
The RHP o OPs
The Lax pai
Examples
Example I: He mi e polynomials
Conside ω(x) = e−x2⇒He mi e polynomials (Hn)n.
The ans o ma ion Xn(z) = Yn(z) e−z2/20
0ez2/2!gi es he ollowing Lax pai
Xn+1(z) = z1
2πiγ−1
n
−2πiγn0Xn(z),d
dz Xn(z) = −z−1
πiγ−1
n
4πiγn−1zXn(z)
The di e ence equa ion gi es (using βn=γn/γn+1) he TTRR
xb
Hn(x) = b
Hn+1(x) + βnb
Hn−1(x),
while he di e en ial equa ion gi es he ladde ope a o s
b
H0
n(x) = 2βnb
Hn−1(x),b
H0
n(x)−2xb
Hn(x) = −2b
Hn+1(x).
The compa ibili y condi ions a e
βn+1 −βn=1
2⇒βn=n
2
Manuel Dom´ınguez de la Iglesia Algeb aic aspec s o he RHP o MOP

The RH p oblem o OP
The RH p oblem o MOP
The RHP o OPs
The Lax pai
Examples
Example I: He mi e polynomials
Conside ω(x) = e−x2⇒He mi e polynomials (Hn)n.
The ans o ma ion Xn(z) = Yn(z) e−z2/20
0ez2/2!gi es he ollowing Lax pai
Xn+1(z) = z1
2πiγ−1
n
−2πiγn0Xn(z),d
dz Xn(z) = −z−1
πiγ−1
n
4πiγn−1zXn(z)
The di e ence equa ion gi es (using βn=γn/γn+1) he TTRR
xb
Hn(x) = b
Hn+1(x) + βnb
Hn−1(x),
while he di e en ial equa ion gi es he ladde ope a o s
b
H0
n(x) = 2βnb
Hn−1(x),b
H0
n(x)−2xb
Hn(x) = −2b
Hn+1(x).
The compa ibili y condi ions a e
βn+1 −βn=1
2⇒βn=n
2
Manuel Dom´ınguez de la Iglesia Algeb aic aspec s o he RHP o MOP
The RH p oblem o OP
The RH p oblem o MOP
The RHP o OPs
The Lax pai
Examples
Example II: F eud o hogonal polynomials
Conside ω(x) = e−x4⇒F eud polynomials (Pn)n.
Xn(z) = Yn(z) e−z4/20
0ez4/2!sa is ies he ollowing Lax pai
Xn+1(z) = z1
2πiγ−1
n
−2πiγn0Xn(z)
d
dz Xn(z) = −2z3−4βnz−2
πiγ−1
n(z2+βn+βn+1)
8πiγn−1(z2+βn+βn+1) 2z3+ 4βnzXn(z)
The ladde ope a o s a e
b
P0
n(x)+4βnxb
Pn(x) = 4(x2+βn+βn+1)βnb
Pn−1(x)
b
P0
n(x)+4x3b
Pn(x) = −4(x2+βn+βn+1)b
Pn+1(x)
The compa ibili y condi ions a e
n= 4βn(βn+1 +βn+βn−1)
Manuel Dom´ınguez de la Iglesia Algeb aic aspec s o he RHP o MOP
The RH p oblem o OP
The RH p oblem o MOP
The RHP o OPs
The Lax pai
Examples
Example II: F eud o hogonal polynomials
Conside ω(x) = e−x4⇒F eud polynomials (Pn)n.
Xn(z) = Yn(z) e−z4/20
0ez4/2!sa is ies he ollowing Lax pai
Xn+1(z) = z1
2πiγ−1
n
−2πiγn0Xn(z)
d
dz Xn(z) = −2z3−4βnz−2
πiγ−1
n(z2+βn+βn+1)
8πiγn−1(z2+βn+βn+1) 2z3+ 4βnzXn(z)
The ladde ope a o s a e
b
P0
n(x)+4βnxb
Pn(x) = 4(x2+βn+βn+1)βnb
Pn−1(x)
b
P0
n(x)+4x3b
Pn(x) = −4(x2+βn+βn+1)b
Pn+1(x)
The compa ibili y condi ions a e
n= 4βn(βn+1 +βn+βn−1)
Manuel Dom´ınguez de la Iglesia Algeb aic aspec s o he RHP o MOP
The RH p oblem o OP
The RH p oblem o MOP
The RHP o OPs
The Lax pai
Examples
Example II: F eud o hogonal polynomials
Conside ω(x) = e−x4⇒F eud polynomials (Pn)n.
Xn(z) = Yn(z) e−z4/20
0ez4/2!sa is ies he ollowing Lax pai
Xn+1(z) = z1
2πiγ−1
n
−2πiγn0Xn(z)
d
dz Xn(z) = −2z3−4βnz−2
πiγ−1
n(z2+βn+βn+1)
8πiγn−1(z2+βn+βn+1) 2z3+ 4βnzXn(z)
The ladde ope a o s a e
b
P0
n(x)+4βnxb
Pn(x) = 4(x2+βn+βn+1)βnb
Pn−1(x)
b
P0
n(x)+4x3b
Pn(x) = −4(x2+βn+βn+1)b
Pn+1(x)
The compa ibili y condi ions a e
n= 4βn(βn+1 +βn+βn−1)
Manuel Dom´ınguez de la Iglesia Algeb aic aspec s o he RHP o MOP
The RH p oblem o OP
The RH p oblem o MOP
The RHP o MOPs
The Lax pai
Examples
Solu ion o he RHP o MOP
Yn:C→C2N×2Nsuch ha
1Ynis analy ic in C R
2Yn
+(x) = Yn
−(x)INW(x)
0 INwhen x∈R
3Yn(z)=(I2N+O(1/z)) znIN0
0z−nINas z→ ∞
Fo n≥1 he unique solu ion o he RH p oblem abo e is gi en by
Yn(z) = b
Pn(z)C(b
PnW)(z)
−2πiγn−1b
Pn−1(z)−2πiγn−1C(b
Pn−1W)(z)!
whe e C(F)(z) = 1
2πiRR
F( )
−zd and γn=κ∗
nκn.
Manuel Dom´ınguez de la Iglesia Algeb aic aspec s o he RHP o MOP

The RH p oblem o OP
The RH p oblem o MOP
The RHP o MOPs
The Lax pai
Examples
Solu ion o he RHP o MOP
Yn:C→C2N×2Nsuch ha
1Ynis analy ic in C R
2Yn
+(x) = Yn
−(x)INW(x)
0 INwhen x∈R
3Yn(z)=(I2N+O(1/z)) znIN0
0z−nINas z→ ∞
Fo n≥1 he unique solu ion o he RH p oblem abo e is gi en by
Yn(z) = b
Pn(z)C(b
PnW)(z)
−2πiγn−1b
Pn−1(z)−2πiγn−1C(b
Pn−1W)(z)!
whe e C(F)(z) = 1
2πiRR
F( )
−zd and γn=κ∗
nκn.
Manuel Dom´ınguez de la Iglesia Algeb aic aspec s o he RHP o MOP
The RH p oblem o OP
The RH p oblem o MOP
The RHP o MOPs
The Lax pai
Examples
The Lax pai I
We look o a pai o i s -o de di e ence/di e en ial equa ions o he o m
Yn+1(z) = En(z)Yn(z),d
dz Yn(z) = Fn(z)Yn(z)
Goal: ob ain an in e ible ans o ma ion Yn→Xnsuch ha Xnhas a cons an
jump ac oss R. Conside Xn(z) = Yn(z)V(z) whe e
V(z) = T(z)0
0 T−∗(z)
whe e Tis an in e ible N×Nsmoo h ma ix unc ion.
This mo i a es o conside a ac o iza ion o he weigh in he o m
W(x) = T(x)T∗(x),x∈R.
This ac o iza ion is no unique since
T(x) = b
T(x)S(x),x∈R
b
T(x) is uppe iangula and S(x) is an a bi a y smoo h and uni a y ma ix.
Manuel Dom´ınguez de la Iglesia Algeb aic aspec s o he RHP o MOP
The RH p oblem o OP
The RH p oblem o MOP
The RHP o MOPs
The Lax pai
Examples
The Lax pai I
We look o a pai o i s -o de di e ence/di e en ial equa ions o he o m
Yn+1(z) = En(z)Yn(z),d
dz Yn(z) = Fn(z)Yn(z)
Goal: ob ain an in e ible ans o ma ion Yn→Xnsuch ha Xnhas a cons an
jump ac oss R. Conside Xn(z) = Yn(z)V(z) whe e
V(z) = T(z)0
0 T−∗(z)
whe e Tis an in e ible N×Nsmoo h ma ix unc ion.
This mo i a es o conside a ac o iza ion o he weigh in he o m
W(x) = T(x)T∗(x),x∈R.
This ac o iza ion is no unique since
T(x) = b
T(x)S(x),x∈R
b
T(x) is uppe iangula and S(x) is an a bi a y smoo h and uni a y ma ix.
Manuel Dom´ınguez de la Iglesia Algeb aic aspec s o he RHP o MOP
The RH p oblem o OP
The RH p oblem o MOP
The RHP o MOPs
The Lax pai
Examples
The Lax pai I
We look o a pai o i s -o de di e ence/di e en ial equa ions o he o m
Yn+1(z) = En(z)Yn(z),d
dz Yn(z) = Fn(z)Yn(z)
Goal: ob ain an in e ible ans o ma ion Yn→Xnsuch ha Xnhas a cons an
jump ac oss R. Conside Xn(z) = Yn(z)V(z) whe e
V(z) = T(z)0
0 T−∗(z)
whe e Tis an in e ible N×Nsmoo h ma ix unc ion.
This mo i a es o conside a ac o iza ion o he weigh in he o m
W(x) = T(x)T∗(x),x∈R.
This ac o iza ion is no unique since
T(x) = b
T(x)S(x),x∈R
b
T(x) is uppe iangula and S(x) is an a bi a y smoo h and uni a y ma ix.
Manuel Dom´ınguez de la Iglesia Algeb aic aspec s o he RHP o MOP
The RH p oblem o OP
The RH p oblem o MOP
The RHP o MOPs
The Lax pai
Examples
The Lax pai I
We look o a pai o i s -o de di e ence/di e en ial equa ions o he o m
Yn+1(z) = En(z)Yn(z),d
dz Yn(z) = Fn(z)Yn(z)
Goal: ob ain an in e ible ans o ma ion Yn→Xnsuch ha Xnhas a cons an
jump ac oss R. Conside Xn(z) = Yn(z)V(z) whe e
V(z) = T(z)0
0 T−∗(z)
whe e Tis an in e ible N×Nsmoo h ma ix unc ion.
This mo i a es o conside a ac o iza ion o he weigh in he o m
W(x) = T(x)T∗(x),x∈R.
This ac o iza ion is no unique since
T(x) = b
T(x)S(x),x∈R
b
T(x) is uppe iangula and S(x) is an a bi a y smoo h and uni a y ma ix.
Manuel Dom´ınguez de la Iglesia Algeb aic aspec s o he RHP o MOP

The RH p oblem o OP
The RH p oblem o MOP
The RHP o MOPs
The Lax pai
Examples
The Lax pai II
We addi ionally assume
T0(z) = G(z)T(z),
whe e Gis a ma ix polynomial o deg ee m(mos o ou examples)
Yn+1(z) = z−αn1
2πiγ−1
n
−2πiγn0
| {z }
En(z;G)
Yn(z)
d
dz Yn(z) = −Bn(z;G)−1
2πiγ−1
nAn(z;G)
2πiAn−1(z;G)γn−1B∗
n(z;G)
| {z }
Fn(z;G)
Yn(z)
whe e Anand Bna e ma ix polynomials o deg ee m−1 and m espec i ely.
C oss-di e en ia ing he Lax pai yield he compa ibili y condi ions
E0
n(z;G) + En(z;G)Fn(z;G) = Fn+1(z;G)En(z;G)
Manuel Dom´ınguez de la Iglesia Algeb aic aspec s o he RHP o MOP
The RH p oblem o OP
The RH p oblem o MOP
The RHP o MOPs
The Lax pai
Examples
The Lax pai II
We addi ionally assume
T0(z) = G(z)T(z),
whe e Gis a ma ix polynomial o deg ee m(mos o ou examples)
Yn+1(z) = z−αn1
2πiγ−1
n
−2πiγn0
| {z }
En(z;G)
Yn(z)
d
dz Yn(z) = −Bn(z;G)−1
2πiγ−1
nAn(z;G)
2πiAn−1(z;G)γn−1B∗
n(z;G)
| {z }
Fn(z;G)
Yn(z)
whe e Anand Bna e ma ix polynomials o deg ee m−1 and m espec i ely.
C oss-di e en ia ing he Lax pai yield he compa ibili y condi ions
E0
n(z;G) + En(z;G)Fn(z;G) = Fn+1(z;G)En(z;G)
Manuel Dom´ınguez de la Iglesia Algeb aic aspec s o he RHP o MOP
The RH p oblem o OP
The RH p oblem o MOP
The RHP o MOPs
The Lax pai
Examples
The Lax pai II
We addi ionally assume
T0(z) = G(z)T(z),
whe e Gis a ma ix polynomial o deg ee m(mos o ou examples)
Yn+1(z) = z−αn1
2πiγ−1
n
−2πiγn0
| {z }
En(z;G)
Yn(z)
d
dz Yn(z) = −Bn(z;G)−1
2πiγ−1
nAn(z;G)
2πiAn−1(z;G)γn−1B∗
n(z;G)
| {z }
Fn(z;G)
Yn(z)
whe e Anand Bna e ma ix polynomials o deg ee m−1 and m espec i ely.
C oss-di e en ia ing he Lax pai yield he compa ibili y condi ions
E0
n(z;G) + En(z;G)Fn(z;G) = Fn+1(z;G)En(z;G)
Manuel Dom´ınguez de la Iglesia Algeb aic aspec s o he RHP o MOP
The RH p oblem o OP
The RH p oblem o MOP
The RHP o MOPs
The Lax pai
Examples
The Lax pai III
I he e exis s a non- i ial ma ix- alued unc ion S, non-singula on C, smoo h
and uni a y on R, s. .
H(z) = T(z)S0(z)S∗(z)T−1(z)
is also a polynomial, hen e
T=TS sa is ies
W(x) = e
T(x)e
T
∗(x),x∈R,e
T
0(z) = e
G(z)e
T(z),z∈C,
wi h e
G(z) = G(z) + H(z) and he ma ix Xnsa is ies
d
dz Xn(z) = Fn(z;G)Xn(z) + Fn(z;H)Xn(z)−Xn(z)χ(z)0
0−χ∗(z)
wi h χ(z) = S0(z)S∗(z).
Consequences: We ha e a class o ladde ope a o s.
Manuel Dom´ınguez de la Iglesia Algeb aic aspec s o he RHP o MOP
The RH p oblem o OP
The RH p oblem o MOP
The RHP o MOPs
The Lax pai
Examples
In o de o use he eedom in he ma ix case by a uni a y ma ix unc ion
Swe ha e o impose addi ional cons ain s on he weigh W.
The ma ix Hcan be w i en as
H(x) = eAxχe−Ax=χ+ adA(χ)x+ ad2
A(χ)x2
2+· · · ,
whe e χ(x) = S0(x)S∗(x) is skew-He mi ian on R.
This ma ix equa ion was conside ed al eady by Du ´an-G ¨unbaum (2004),
when χis a cons an ma ix.
I deg H= 0 hen χ=iaIN,a∈R⇒No new ladde ope a o s.
I deg H= 1 hen
1A=L=
N
X
i=1
νiEi,i+1,and χ=iJ=i
N
X
i=1
(N−i)Ei,i
⇒adA(χ) = −Aand S(x) = eiJx
2A=L(IN+L)−1,and χ=iJ
⇒adA(χ) = −A+A2and S(x) = eiJx
Manuel Dom´ınguez de la Iglesia Algeb aic aspec s o he RHP o MOP

The RH p oblem o OP
The RH p oblem o MOP
The RHP o MOPs
The Lax pai
Examples
In o de o use he eedom in he ma ix case by a uni a y ma ix unc ion
Swe ha e o impose addi ional cons ain s on he weigh W.
The ma ix Hcan be w i en as
H(x) = eAxχe−Ax=χ+ adA(χ)x+ ad2
A(χ)x2
2+· · · ,
whe e χ(x) = S0(x)S∗(x) is skew-He mi ian on R.
This ma ix equa ion was conside ed al eady by Du ´an-G ¨unbaum (2004),
when χis a cons an ma ix.
I deg H= 0 hen χ=iaIN,a∈R⇒No new ladde ope a o s.
I deg H= 1 hen
1A=L=
N
X
i=1
νiEi,i+1,and χ=iJ=i
N
X
i=1
(N−i)Ei,i
⇒adA(χ) = −Aand S(x) = eiJx
2A=L(IN+L)−1,and χ=iJ
⇒adA(χ) = −A+A2and S(x) = eiJx
Manuel Dom´ınguez de la Iglesia Algeb aic aspec s o he RHP o MOP
The RH p oblem o OP
The RH p oblem o MOP
The RHP o MOPs
The Lax pai
Examples
In o de o use he eedom in he ma ix case by a uni a y ma ix unc ion
Swe ha e o impose addi ional cons ain s on he weigh W.
The ma ix Hcan be w i en as
H(x) = eAxχe−Ax=χ+ adA(χ)x+ ad2
A(χ)x2
2+· · · ,
whe e χ(x) = S0(x)S∗(x) is skew-He mi ian on R.
This ma ix equa ion was conside ed al eady by Du ´an-G ¨unbaum (2004),
when χis a cons an ma ix.
I deg H= 0 hen χ=iaIN,a∈R⇒No new ladde ope a o s.
I deg H= 1 hen
1A=L=
N
X
i=1
νiEi,i+1,and χ=iJ=i
N
X
i=1
(N−i)Ei,i
⇒adA(χ) = −Aand S(x) = eiJx
2A=L(IN+L)−1,and χ=iJ
⇒adA(χ) = −A+A2and S(x) = eiJx
Manuel Dom´ınguez de la Iglesia Algeb aic aspec s o he RHP o MOP
The RH p oblem o OP
The RH p oblem o MOP
The RHP o MOPs
The Lax pai
Examples
In o de o use he eedom in he ma ix case by a uni a y ma ix unc ion
Swe ha e o impose addi ional cons ain s on he weigh W.
The ma ix Hcan be w i en as
H(x) = eAxχe−Ax=χ+ adA(χ)x+ ad2
A(χ)x2
2+· · · ,
whe e χ(x) = S0(x)S∗(x) is skew-He mi ian on R.
This ma ix equa ion was conside ed al eady by Du ´an-G ¨unbaum (2004),
when χis a cons an ma ix.
I deg H= 0 hen χ=iaIN,a∈R⇒No new ladde ope a o s.
I deg H= 1 hen
1A=L=
N
X
i=1
νiEi,i+1,and χ=iJ=i
N
X
i=1
(N−i)Ei,i
⇒adA(χ) = −Aand S(x) = eiJx
2A=L(IN+L)−1,and χ=iJ
⇒adA(χ) = −A+A2and S(x) = eiJx
Manuel Dom´ınguez de la Iglesia Algeb aic aspec s o he RHP o MOP
The RH p oblem o OP
The RH p oblem o MOP
The RHP o MOPs
The Lax pai
Examples
In o de o use he eedom in he ma ix case by a uni a y ma ix unc ion
Swe ha e o impose addi ional cons ain s on he weigh W.
The ma ix Hcan be w i en as
H(x) = eAxχe−Ax=χ+ adA(χ)x+ ad2
A(χ)x2
2+· · · ,
whe e χ(x) = S0(x)S∗(x) is skew-He mi ian on R.
This ma ix equa ion was conside ed al eady by Du ´an-G ¨unbaum (2004),
when χis a cons an ma ix.
I deg H= 0 hen χ=iaIN,a∈R⇒No new ladde ope a o s.
I deg H= 1 hen
1A=L=
N
X
i=1
νiEi,i+1,and χ=iJ=i
N
X
i=1
(N−i)Ei,i
⇒adA(χ) = −Aand S(x) = eiJx
2A=L(IN+L)−1,and χ=iJ
⇒adA(χ) = −A+A2and S(x) = eiJx
Manuel Dom´ınguez de la Iglesia Algeb aic aspec s o he RHP o MOP
The RH p oblem o OP
The RH p oblem o MOP
The RHP o MOPs
The Lax pai
Examples
In o de o use he eedom in he ma ix case by a uni a y ma ix unc ion
Swe ha e o impose addi ional cons ain s on he weigh W.
The ma ix Hcan be w i en as
H(x) = eAxχe−Ax=χ+ adA(χ)x+ ad2
A(χ)x2
2+· · · ,
whe e χ(x) = S0(x)S∗(x) is skew-He mi ian on R.
This ma ix equa ion was conside ed al eady by Du ´an-G ¨unbaum (2004),
when χis a cons an ma ix.
I deg H= 0 hen χ=iaIN,a∈R⇒No new ladde ope a o s.
I deg H= 1 hen
1A=L=
N
X
i=1
νiEi,i+1,and χ=iJ=i
N
X
i=1
(N−i)Ei,i
⇒adA(χ) = −Aand S(x) = eiJx
2A=L(IN+L)−1,and χ=iJ
⇒adA(χ) = −A+A2and S(x) = eiJx
Manuel Dom´ınguez de la Iglesia Algeb aic aspec s o he RHP o MOP

The RH p oblem o OP
The RH p oblem o MOP
The RHP o MOPs
The Lax pai
Examples
In o de o use he eedom in he ma ix case by a uni a y ma ix unc ion
Swe ha e o impose addi ional cons ain s on he weigh W.
The ma ix Hcan be w i en as
H(x) = eAxχe−Ax=χ+ adA(χ)x+ ad2
A(χ)x2
2+· · · ,
whe e χ(x) = S0(x)S∗(x) is skew-He mi ian on R.
This ma ix equa ion was conside ed al eady by Du ´an-G ¨unbaum (2004),
when χis a cons an ma ix.
I deg H= 0 hen χ=iaIN,a∈R⇒No new ladde ope a o s.
I deg H= 1 hen
1A=L=
N
X
i=1
νiEi,i+1,and χ=iJ=i
N
X
i=1
(N−i)Ei,i
⇒adA(χ) = −Aand S(x) = eiJx
2A=L(IN+L)−1,and χ=iJ
⇒adA(χ) = −A+A2and S(x) = eiJx
Manuel Dom´ınguez de la Iglesia Algeb aic aspec s o he RHP o MOP
The RH p oblem o OP
The RH p oblem o MOP
The RHP o MOPs
The Lax pai
Examples
Fi s case A=L
New compa ibili y condi ions
Jαn−αnJ+αn=L+1
2(L2αn−αnL2),J−γ−1
nJγn=Lαn+αnL−2α2
n
New ladde ope a o s (0- h o de )
b
Pn(x)J−J
b
Pn(x)−x(
b
Pn(x)L−L
b
Pn(x)) + 2βn
b
Pn(x)−n
b
Pn(x) = 2(L−αn)βn
b
Pn−1(x)
b
Pn(x)(J−xL)−γ−1
n(J−xL∗)γn
b
Pn(x)+2βn+1
b
Pn(x)−(n+1)
b
Pn(x) = 2(αn−L)
b
Pn+1(x)
Fi s -o de di e en ial equa ion
(L−αn)
b
P
0
n(x)+(L−αn+xIN)(
b
Pn(x)L−L
b
Pn(x))−2βn
b
Pn(x) =
b
Pn(x)J−J
b
Pn(x)−n
b
Pn(x)
S u m-Liou ille ype di e en ial equa ion (Du ´an-G ¨unbaum, 2004)
b
P
00
n(x) + 2
b
P
0
n(x)(L−xIN) +
b
Pn(x)(L2
−2J) = (−2nIN+L2
−2J)
b
Pn(x)
Manuel Dom´ınguez de la Iglesia Algeb aic aspec s o he RHP o MOP
The RH p oblem o OP
The RH p oblem o MOP
The RHP o MOPs
The Lax pai
Examples
Fi s case A=L
New compa ibili y condi ions
Jαn−αnJ+αn=L+1
2(L2αn−αnL2),J−γ−1
nJγn=Lαn+αnL−2α2
n
New ladde ope a o s (0- h o de )
b
Pn(x)J−J
b
Pn(x)−x(
b
Pn(x)L−L
b
Pn(x)) + 2βn
b
Pn(x)−n
b
Pn(x) = 2(L−αn)βn
b
Pn−1(x)
b
Pn(x)(J−xL)−γ−1
n(J−xL∗)γn
b
Pn(x)+2βn+1
b
Pn(x)−(n+1)
b
Pn(x) = 2(αn−L)
b
Pn+1(x)
Fi s -o de di e en ial equa ion
(L−αn)
b
P
0
n(x)+(L−αn+xIN)(
b
Pn(x)L−L
b
Pn(x))−2βn
b
Pn(x) =
b
Pn(x)J−J
b
Pn(x)−n
b
Pn(x)
S u m-Liou ille ype di e en ial equa ion (Du ´an-G ¨unbaum, 2004)
b
P
00
n(x) + 2
b
P
0
n(x)(L−xIN) +
b
Pn(x)(L2
−2J) = (−2nIN+L2
−2J)
b
Pn(x)
Manuel Dom´ınguez de la Iglesia Algeb aic aspec s o he RHP o MOP
The RH p oblem o OP
The RH p oblem o MOP
The RHP o MOPs
The Lax pai
Examples
Fi s case A=L
New compa ibili y condi ions
Jαn−αnJ+αn=L+1
2(L2αn−αnL2),J−γ−1
nJγn=Lαn+αnL−2α2
n
New ladde ope a o s (0- h o de )
b
Pn(x)J−J
b
Pn(x)−x(
b
Pn(x)L−L
b
Pn(x)) + 2βn
b
Pn(x)−n
b
Pn(x) = 2(L−αn)βn
b
Pn−1(x)
b
Pn(x)(J−xL)−γ−1
n(J−xL∗)γn
b
Pn(x)+2βn+1
b
Pn(x)−(n+1)
b
Pn(x) = 2(αn−L)
b
Pn+1(x)
Fi s -o de di e en ial equa ion
(L−αn)
b
P
0
n(x)+(L−αn+xIN)(
b
Pn(x)L−L
b
Pn(x))−2βn
b
Pn(x) =
b
Pn(x)J−J
b
Pn(x)−n
b
Pn(x)
S u m-Liou ille ype di e en ial equa ion (Du ´an-G ¨unbaum, 2004)
b
P
00
n(x) + 2
b
P
0
n(x)(L−xIN) +
b
Pn(x)(L2
−2J) = (−2nIN+L2
−2J)
b
Pn(x)
Manuel Dom´ınguez de la Iglesia Algeb aic aspec s o he RHP o MOP
The RH p oblem o OP
The RH p oblem o MOP
The RHP o MOPs
The Lax pai
Examples
Final ema ks
Conclusions
1The ladde ope a o s me hod gi es mo e insigh abou he di e en ial
p ope ies o MOP and new phenomena
2This me hod wo ks o e e y weigh ma ix W. The co esponding
MOP sa is y di e en ial equa ions, bu no necessa ily o
S u m-Liou ille ype
Fu u e di ec ions
1Examples when supp(W)⊂[0,+∞) o supp(W)⊂[−1,1]
2Uni o m asymp o ics: s eepes descen analysis o RHP
(Dei -Zhou,1993) ex ended o MOPRL
Manuel Dom´ınguez de la Iglesia Algeb aic aspec s o he RHP o MOP

The RH p oblem o OP
The RH p oblem o MOP
The RHP o MOPs
The Lax pai
Examples
Final ema ks
Conclusions
1The ladde ope a o s me hod gi es mo e insigh abou he di e en ial
p ope ies o MOP and new phenomena
2This me hod wo ks o e e y weigh ma ix W. The co esponding
MOP sa is y di e en ial equa ions, bu no necessa ily o
S u m-Liou ille ype
Fu u e di ec ions
1Examples when supp(W)⊂[0,+∞) o supp(W)⊂[−1,1]
2Uni o m asymp o ics: s eepes descen analysis o RHP
(Dei -Zhou,1993) ex ended o MOPRL
Manuel Dom´ınguez de la Iglesia Algeb aic aspec s o he RHP o MOP