Untitled

#load("casimir_eigenvalues_tmp.RData")
load("casimir_eigenvalues.RData")

Eigenvalues

The theoretical prediction for the eigenvalues of \(E^2\) is

\[ \lambda_k=k(k+2) \] with multiplicity

\[ \mu_k=(k+1)^2 \] with \(k=0,1,2,...\)

\(L^\dagger L\)

Here we plot the eigenvalues of \(L^\dagger L\)

## theoretical prediction
N<-Neig-1
i <- 0
k <- 0
Nnondeg <- 0
while (i<N){
  i <- i +(k+1)^2
  k <- k+1
  Nnondeg <- Nnondeg +1 
}
theo_l <- rep(0,N)
count <- 1
for (k in c(0:Nnondeg)){
  for (i in c(1:(k+1)^2) ){
    theo_l[count] <- k*(k+2)
    count <- count +1
  }
}

gg <- ggplot()+theme_bw()
gg <- gg + geom_point(aes( x=c(1:N)  , y=theo_l[1:N] , color="continuum",
                           shape="continuum") )
l<-list()

for (im in c(1:length(lambdaLdL)) ){

l[[im]]<-sort(abs(lambdaLdL[[im]]))[1:N]
#N<- length(l)

gg <- gg + geom_point(aes_string( x=c(1:N)  , y=l[[im]] ,
                           color=as.factor(rep(ms[im],N)) ,
                           shape=as.factor(rep(ms[im],N)) )
                      )

                       


}

## Warning: `aes_string()` was deprecated in ggplot2 3.0.0.
## ℹ Please use tidy evaluation ideoms with `aes()`

#gg <- gg + xlim(0,60) +ylim(0,25) 
gg <- gg + xlab("id") +ylab("$\\lambda$")
#library(plotly)
# ggplotly(gg)
ggplotly(gg)

## Warning: The shape palette can deal with a maximum of 6 discrete values because
## more than 6 becomes difficult to discriminate; you have 8. Consider
## specifying shapes manually if you must have them.

\(L^2\)

here we plot the eigenvalues of \(L^2\). Since \(L^2\) is not hermitian, the eigevalues will be complex, here we order them by absolute value

gg <- ggplot()+theme_bw()
gg <- gg + geom_point(aes( x=c(1:N)  , y=theo_l[1:N] , color="continuum",
                           shape="continuum") )
l<-list()
lr<-list()
for (im in c(1:length(lambdaL2)) ){


#l[[im]]<- lambdaL2[[im]][order(abs(lambdaL2[[im]]))[1:N] ]
l[[im]] <- lambdaL2[[im]]
lr[[im]]<- lambdaL2r[[im]]
 # gg <- gg + geom_point(aes_string( x=c(1:N)  , y=Re(l[[im]]) ,
 #                            color=as.factor(rep(ms[im],N)) ,
 #                            shape=as.factor(rep(ms[im],N)) )
 #                      )
# gg <- gg + geom_point(aes_string( x=c(1:N)  , y=Im(l[[im]]) ,
#                            color=as.factor(rep(paste0(ms[im],"im"),N)) ,
#                            shape=as.factor(rep(paste0(ms[im],"im"),N)) )
#                       )

gg <- gg + geom_point(aes_string( x=c(1:(N))  , y=abs(l[[im]]) ,
                           color=as.factor(rep(paste0(ms[im],"abs"),N)) ,
                           shape=as.factor(rep(paste0(ms[im],"abs"),N)) )
                      )
                       
# gg <- gg + geom_point(aes_string( x=c(1:N)  , y=Re(lr[[im]] ) ,
#                            color=as.factor(rep(paste0(ms[im],"Re"),N)) ,
#                            shape=as.factor(rep(paste0(ms[im],"Re"),N)) )
#                       )


}
gg <- gg + xlab("id") +ylab("$\\lambda$")
#library(plotly)
# ggplotly(gg)
ggplotly(gg)

## Warning: The shape palette can deal with a maximum of 6 discrete values because
## more than 6 becomes difficult to discriminate; you have 8. Consider
## specifying shapes manually if you must have them.

\((L+L^\dagger)^2\)

gg <- ggplot()+theme_bw()
gg <- gg + geom_point(aes( x=c(1:N)  , y=theo_l[1:N] , color="continuum",
                           shape="continuum") )
l<-list()

for (im in c(1:length(lambdaLre2)) ){


l[[im]]<- lambdaLre2[[im]][order(abs(lambdaLre2[[im]]))[1:N] ]

gg <- gg + geom_point(aes_string( x=c(1:N)  , y=Re(l[[im]]) ,
                           color=as.factor(rep(ms[im],N)) ,
                           shape=as.factor(rep(ms[im],N)) )
                      )
# gg <- gg + geom_point(aes_string( x=c(1:N)  , y=Im(l[[im]]) ,
#                            color=as.factor(rep(paste0(ms[im],"im"),N)) ,
#                            shape=as.factor(rep(paste0(ms[im],"im"),N)) )
#                       )
# 
# gg <- gg + geom_point(aes_string( x=c(1:N)  , y=abs(l[[im]]) ,
#                            color=as.factor(rep(paste0(ms[im],"abs"),N)) ,
#                            shape=as.factor(rep(paste0(ms[im],"abs"),N)) )
#                       )
                       


}
gg <- gg + xlab("id") +ylab("$\\lambda$")
#library(plotly)
# ggplotly(gg)
ggplotly(gg)

## Warning: The shape palette can deal with a maximum of 6 discrete values because
## more than 6 becomes difficult to discriminate; you have 8. Consider
## specifying shapes manually if you must have them.

\(L^\dagger L+R^\dagger R\)

gg <- ggplot()+theme_bw()
gg <- gg + geom_point(aes( x=c(1:N)  , y=theo_l[1:N] , color="continuum",
                           shape="continuum") )
l<-list()

for (im in c(1:length(lambdaLdLpRdR)) ){


l[[im]]<- lambdaLdLpRdR[[im]][order(abs(lambdaLdLpRdR[[im]]))[1:N] ]

gg <- gg + geom_point(aes_string( x=c(1:N)  , y=Re(l[[im]]) ,
                           color=as.factor(rep(ms[im],N)) ,
                           shape=as.factor(rep(ms[im],N)) )
                      )
gg <- gg + geom_point(aes_string( x=c(1:N)  , y=Im(l[[im]]) ,
                           color=as.factor(rep(paste0(ms[im],"im"),N)) ,
                           shape=as.factor(rep(paste0(ms[im],"im"),N)) )
                      )

gg <- gg + geom_point(aes_string( x=c(1:N)  , y=abs(l[[im]]) ,
                           color=as.factor(rep(paste0(ms[im],"abs"),N)) ,
                           shape=as.factor(rep(paste0(ms[im],"abs"),N)) )
                      )
                       


}
gg <- gg + xlab("id") +ylab("$\\lambda$")
#library(plotly)
# ggplotly(gg)
ggplotly(gg)

## Warning: The shape palette can deal with a maximum of 6 discrete values because
## more than 6 becomes difficult to discriminate; you have 22. Consider
## specifying shapes manually if you must have them.

\(L^2 +R^2\)

gg <- ggplot()+theme_bw()
gg <- gg + geom_point(aes( x=c(1:N)  , y=theo_l[1:N] , color="continuum",
                           shape="continuum") )
l<-list()

for (im in c(1:length(lambdaL2pR2)) ){


l[[im]]<- lambdaL2pR2[[im]][order(abs(lambdaL2pR2[[im]]))[1:N] ]

gg <- gg + geom_point(aes_string( x=c(1:N)  , y=Re(l[[im]]) ,
                           color=as.factor(rep(ms[im],N)) ,
                           shape=as.factor(rep(ms[im],N)) )
                      )
gg <- gg + geom_point(aes_string( x=c(1:N)  , y=Im(l[[im]]) ,
                           color=as.factor(rep(paste0(ms[im],"im"),N)) ,
                           shape=as.factor(rep(paste0(ms[im],"im"),N)) )
                      )

gg <- gg + geom_point(aes_string( x=c(1:N)  , y=abs(l[[im]]) ,
                           color=as.factor(rep(paste0(ms[im],"abs"),N)) ,
                           shape=as.factor(rep(paste0(ms[im],"abs"),N)) )
                      )
                       


}
gg <- gg + xlab("id") +ylab("$\\lambda$")
#library(plotly)
# ggplotly(gg)
ggplotly(gg)

## Warning: The shape palette can deal with a maximum of 6 discrete values because
## more than 6 becomes difficult to discriminate; you have 22. Consider
## specifying shapes manually if you must have them.

\((L+L^\dagger)^2 +(R+R^\dagger)^2\)

gg <- ggplot()+theme_bw()
gg <- gg + geom_point(aes( x=c(1:N)  , y=theo_l[1:N] , color="continuum",
                           shape="continuum") )
l<-list()

for (im in c(1:length(lambdaLre2pRre2)) ){


l[[im]]<- lambdaLre2pRre2[[im]][order(abs(lambdaLre2pRre2[[im]]))[1:N] ]

gg <- gg + geom_point(aes_string( x=c(1:N)  , y=l[[im]]-l[[im]][1] ,
                           color=as.factor(rep(ms[im],N)) ,
                           shape=as.factor(rep(ms[im],N)) )
                      )
# gg <- gg + geom_point(aes_string( x=c(1:N)  , y=Im(l[[im]]) ,
#                            color=as.factor(rep(paste0(ms[im],"im"),N)) ,
#                            shape=as.factor(rep(paste0(ms[im],"im"),N)) )
#                       )
# 
# gg <- gg + geom_point(aes_string( x=c(1:N)  , y=abs(l[[im]]) ,
#                            color=as.factor(rep(paste0(ms[im],"abs"),N)) ,
#                            shape=as.factor(rep(paste0(ms[im],"abs"),N)) )
#                       )
                       


}
gg <- gg + xlab("id") +ylab("$\\lambda$")
#library(plotly)
# ggplotly(gg)
ggplotly(gg)

## Warning: The shape palette can deal with a maximum of 6 discrete values because
## more than 6 becomes difficult to discriminate; you have 8. Consider
## specifying shapes manually if you must have them.

\((L+R)^\dagger (L+R)\)

gg <- ggplot()+theme_bw()
gg <- gg + geom_point(aes( x=c(1:N)  , y=theo_l[1:N] , color="continuum",
                           shape="continuum") )
l<-list()

for (im in c(1:length(lambdaLpRdLpR)) ){


l[[im]]<- lambdaLpRdLpR[[im]][order(abs(lambdaLpRdLpR[[im]]))[1:N] ]

gg <- gg + geom_point(aes_string( x=c(1:N)  , y=Re(l[[im]]) ,
                           color=as.factor(rep(ms[im],N)) ,
                           shape=as.factor(rep(ms[im],N)) )
                      )
gg <- gg + geom_point(aes_string( x=c(1:N)  , y=Im(l[[im]]) ,
                           color=as.factor(rep(paste0(ms[im],"im"),N)) ,
                           shape=as.factor(rep(paste0(ms[im],"im"),N)) )
                      )

gg <- gg + geom_point(aes_string( x=c(1:N)  , y=abs(l[[im]]) ,
                           color=as.factor(rep(paste0(ms[im],"abs"),N)) ,
                           shape=as.factor(rep(paste0(ms[im],"abs"),N)) )
                      )
                       


}
gg <- gg + xlab("id") +ylab("$\\lambda$")
#library(plotly)
# ggplotly(gg)
ggplotly(gg)

## Warning: The shape palette can deal with a maximum of 6 discrete values because
## more than 6 becomes difficult to discriminate; you have 22. Consider
## specifying shapes manually if you must have them.