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LSpice
LSpice
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Single sum of squares of Clebsch--GordanClebsch–Gordan coefficients

Let $C^{j_3 m_3}_{j_1 m_1 j_2 m_2}$ be the standard Clebsch--GordanClebsch–Gordan coefficients of $SU(2)$$\operatorname{SU}(2)$. They obey the orthogonality relation $$ \sum_{j_3} \sum_{m_3} \left(C^{j_3 m_3}_{j_1 m_1 j_2 (m_3 - m_1)} \right)^2 = 1.$$ My question is about what can be said if I remove the sum over $j_3$. Does there exist a bound for $$ \max_{j_3} \sum_{m_3} \left(C^{j_3 m_3}_{j_1 m_1 j_2 (m_3 - m_1)} \right)^2 $$ in terms of $m_1$, $j_1$ and $j_2$? Of course it is $\leq 1$, but I am interested in whether this expression decays in $j_1$ and $j_2$, or some combination thereof. Positive or negative statements, or a reference, would be very useful. Thank you!

Single sum of squares of Clebsch--Gordan coefficients

Let $C^{j_3 m_3}_{j_1 m_1 j_2 m_2}$ be the standard Clebsch--Gordan coefficients of $SU(2)$. They obey the orthogonality relation $$ \sum_{j_3} \sum_{m_3} \left(C^{j_3 m_3}_{j_1 m_1 j_2 (m_3 - m_1)} \right)^2 = 1.$$ My question is about what can be said if I remove the sum over $j_3$. Does there exist a bound for $$ \max_{j_3} \sum_{m_3} \left(C^{j_3 m_3}_{j_1 m_1 j_2 (m_3 - m_1)} \right)^2 $$ in terms of $m_1$, $j_1$ and $j_2$? Of course it is $\leq 1$, but I am interested in whether this expression decays in $j_1$ and $j_2$, or some combination thereof. Positive or negative statements, or a reference, would be very useful. Thank you!

Single sum of squares of Clebsch–Gordan coefficients

Let $C^{j_3 m_3}_{j_1 m_1 j_2 m_2}$ be the standard Clebsch–Gordan coefficients of $\operatorname{SU}(2)$. They obey the orthogonality relation $$ \sum_{j_3} \sum_{m_3} \left(C^{j_3 m_3}_{j_1 m_1 j_2 (m_3 - m_1)} \right)^2 = 1.$$ My question is about what can be said if I remove the sum over $j_3$. Does there exist a bound for $$ \max_{j_3} \sum_{m_3} \left(C^{j_3 m_3}_{j_1 m_1 j_2 (m_3 - m_1)} \right)^2 $$ in terms of $m_1$, $j_1$ and $j_2$? Of course it is $\leq 1$, but I am interested in whether this expression decays in $j_1$ and $j_2$, or some combination thereof. Positive or negative statements, or a reference, would be very useful.

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onamoonlessnight
onamoonlessnight
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Let $C^{j_1 j_2}_{j_3 m_1 m_2}$$C^{j_3 m_3}_{j_1 m_1 j_2 m_2}$ be the standard Clebsch--Gordan coefficients of $SU(2)$. They obey the orthogonality relation $$ \sum_{j_3} \sum_{m_3} \left(C^{j_1 j_2}_{j_3 m_1 (m_3 - m_1)} \right)^2 = 1.$$$$ \sum_{j_3} \sum_{m_3} \left(C^{j_3 m_3}_{j_1 m_1 j_2 (m_3 - m_1)} \right)^2 = 1.$$ My question is about what can be said if I remove the sum over $j_3$. Does there exist a bound for $$ \max_{j_3} \sum_{m_3} \left(C^{j_1 j_2}_{j_3 m_1 (m_3 - m_1)} \right)^2 $$$$ \max_{j_3} \sum_{m_3} \left(C^{j_3 m_3}_{j_1 m_1 j_2 (m_3 - m_1)} \right)^2 $$ in terms of $m_1$, $j_1$ and $j_2$? Of course it is $\leq 1$, but I am interested in whether this expression decays in $j_1$ and $j_2$, or some combination thereof. Positive or negative statements, or a reference, would be very useful. Thank you!

Let $C^{j_1 j_2}_{j_3 m_1 m_2}$ be the standard Clebsch--Gordan coefficients of $SU(2)$. They obey the orthogonality relation $$ \sum_{j_3} \sum_{m_3} \left(C^{j_1 j_2}_{j_3 m_1 (m_3 - m_1)} \right)^2 = 1.$$ My question is about what can be said if I remove the sum over $j_3$. Does there exist a bound for $$ \max_{j_3} \sum_{m_3} \left(C^{j_1 j_2}_{j_3 m_1 (m_3 - m_1)} \right)^2 $$ in terms of $m_1$, $j_1$ and $j_2$? Of course it is $\leq 1$, but I am interested in whether this expression decays in $j_1$ and $j_2$, or some combination thereof. Positive or negative statements, or a reference, would be very useful. Thank you!

Let $C^{j_3 m_3}_{j_1 m_1 j_2 m_2}$ be the standard Clebsch--Gordan coefficients of $SU(2)$. They obey the orthogonality relation $$ \sum_{j_3} \sum_{m_3} \left(C^{j_3 m_3}_{j_1 m_1 j_2 (m_3 - m_1)} \right)^2 = 1.$$ My question is about what can be said if I remove the sum over $j_3$. Does there exist a bound for $$ \max_{j_3} \sum_{m_3} \left(C^{j_3 m_3}_{j_1 m_1 j_2 (m_3 - m_1)} \right)^2 $$ in terms of $m_1$, $j_1$ and $j_2$? Of course it is $\leq 1$, but I am interested in whether this expression decays in $j_1$ and $j_2$, or some combination thereof. Positive or negative statements, or a reference, would be very useful. Thank you!

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onamoonlessnight
onamoonlessnight
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Single sum of squares of Clebsch--Gordan coefficients

Let $C^{j_1 j_2}_{j_3 m_1 m_2}$ be the standard Clebsch--Gordan coefficients of $SU(2)$. They obey the orthogonality relation $$ \sum_{j_3} \sum_{m_3} \left(C^{j_1 j_2}_{j_3 m_1 (m_3 - m_1)} \right)^2 = 1.$$ My question is about what can be said if I remove the sum over $j_3$. Does there exist a bound for $$ \max_{j_3} \sum_{m_3} \left(C^{j_1 j_2}_{j_3 m_1 (m_3 - m_1)} \right)^2 $$ in terms of $m_1$, $j_1$ and $j_2$? Of course it is $\leq 1$, but I am interested in whether this expression decays in $j_1$ and $j_2$, or some combination thereof. Positive or negative statements, or a reference, would be very useful. Thank you!