... known^1.1

At zero temperature the expectation value $\langle...\rangle$ is taken with respect to the ground state of the system.

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... matrix^1.2

It is interesting to note that the normalization factor $N!/(N-m)!$ of the $m$-body correlation function comes from the number of ways to make groups of $m$-particles out of $N$ particles, but at the same time can be found from the properties of the field operators $\hat\Psi^\dagger\vert N\rangle=\sqrt{N}\vert N-1\rangle$, $\hat\Psi\vert N\rangle=\sqrt{N+1}\vert N+1\rangle$

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... wave^1.3

The normalization of the scattering solution is of no interest to us, so in following we will always omit the normalization factor.

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... sphere^1.4

Note that therefore the energy is purely kinetic and the interaction potential does not enter in an explicit way, instead it sets the boundary condition on the solution.

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... as^1.5

In the three-dimensional system we look for solutions with spherical symmetry. In a one-dimensional system it is equivalent to searching even solutions.

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... scattering^1.6

The textbook definition for the three-dimensional scattering length (1.44) can be recasted in a similar form $a_{3D} = \lim\limits_{k\to 0}\partial\delta/\partial k$. We prefer to have a definition in terms of a derivative, as it does not cause any ambiguity in the choice a free particle phase. In three dimensions the phase of sinus (1.42) in absence of the scattering potential is fixed to zero due to the condition (1.40), which is no longer so in $1D$ case, as it should be fixed to $\pi/2$. Instead the definition (1.63) takes into account the difference between the phase in presence of scatterer and in its absence. See also footnote on p. .

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... solution^1.7

It turns out that this property is general and can be used as an alternative to (1.63) definition of the one-dimensional scattering length.

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... obtain^1.8

We used property $\Delta (1/r) = -4\pi\delta(r)$, which can be easily obtained from the solution $f(\r ) = -\frac{1}{4\pi} \int\frac{\rho(\r ')}{\vert\r -\r '\vert}{d\vec r}'$ to the Poisson equation $\Delta f(\r ) = \rho(\r )$ substituting the point charge $\rho(\r ) =\delta(\r )$.

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... equation^1.9

Additional literature on the topic of pseudopotential description can be found in classical articles [Fer36],[HY57] and in books [BW52],p.74, [Hua87].

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...$a_{1D}$^1.10

As discussed in Sec. 1.3.3.4, the size of a hard-rod equals to the one-dimensional scattering length on HR potential (1.79).

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...$\vert\psi\rangle$^1.11

It is important to note that $\vert\psi\rangle$ is not a wave function and in this sense the derived below GPE (1.115) is not a ``non linear Schrödinger equation''. In particular its time evolution is driven by the chemical potential $\mu$ instead of the energy of the system $E$, as it happens for the solution of the Schrödinger equation.

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... has^1.12

This approximation is called polytropic.

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...1.12^1.13

Many theories produces results that fall into the class of equations of state described by formula (1.136). For example GP theory, ideal fermi gas, TG gas.

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... is^1.14

S. Stringari, unpublished.

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... LDA^1.15

It is interesting to note that the result (1.151) of an approximate solution for particles of infinite repulsion coincides with an exact result for particles interacting with $g/r^2$ interaction (Calogero-Sutherland model [Cal69,Sut71]) in the limit $g\to 0$. Indeed, as shown in [Sut71] the energy of such a gas equals $E/N = \frac{1}{2}\hbar\omega_z(1+\lambda(N-1))$, where $\lambda$ is related to the strength of interaction $g=2\lambda(\lambda-1)$. There are two different ways of taking the limit $g\to 0$: 1) $\lambda\to 0$, $E/N\to\frac{1}{2}\hbar\omega_z$ i.e. this limit corresponds to non-interacting bosons all staying in the lowest state of a harmonic oscillator 2) $\lambda\to 0$, $E/N\to\frac{1}{2}N\hbar\omega_z$, i.e. this limit preserves the singularity of the interaction while makes the potential energy vanishing.

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... density^1.16

In this section we keep a different notation for the linear density $\rho\equiv n_{1D} = N/L$

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...exact^2.1

Of course, as this method is a statistical one and all outputs are obtained within statistical errors which can be decreased by making a longer series of measurements.

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... as^2.2

The subscript ${\bf R}$ of the differential operator indicates that the derivative has to be taken for every component of ${\bf R}$.

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... function^2.3

for details of the calculation refer to Sec. 2.7

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...$\psi _T$^2.4

The definition (2.15) of the drift force can be understood in a classical analogy. In a classical system the probability distribution has a Boltzman form. The coordinate part of the distribution function is related to the potential energy $p({\bf R}) = const \exp(-U({\bf R}))$ (we put the fictitious temperature to one). The classical force is an antigradient of the potential energy ${\bf F}= - \nabla_{\bf R}U({\bf R}) = \nabla_{\bf R}\ln p({\bf R})$. If we approximate a quantum probability distribution by the square of a trial wave function (2.1), then the force equals exactly to (2.15).

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... distribution^2.1

The formula (2.31) should be understood in the statistical sense, the average of any value $A$ over the l.h.s. and r.h.s distributions are equal to each other in the limit when size of the population $N_W$ tends to infinity $\int A({\bf R}) f({\vec r},\tau)\,{\bf dR} = \lim\limits_{N_W\to\infty} \int A({\bf R}) \sum\limits_{i=1}^{N_W} C \delta ({\bf R - R_i}(\tau))\,{\bf dR}$

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... as^2.2

Note that for a repulsive gas $a_{1D}<0$ while for attractive $a_{1D}>0$.

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... function^3.1

Mean square radii are calculated using the pure estimator technique developed in [CB95]

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... satisfied^3.2

This characteristic for LDA parameter is universal, for example systems with different number of particles behave similarly if the combination $N\lambda a_\perp^2/a^2_{3D}$ is the same. It becomes clear from from equation (1.134) noting that $N\lambda a_\perp^2/a^2_{3D}= \Delta_1^2$.

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... by^4.1

See, also, footnote on p. 

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... lengths^7.1

Considering an homogenious condensate, we assume that its size is large enough, particularly that the gas is the Thomas-Fermi conditions. Then results for a homogenious gas will be approximately valid for a gas in a trap.

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... physics^8.1

See, e.g., [Voi95]

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...)^8.2

Note that Eqs. 1.69-8.5 are valid only if the condition $\vert a_{1D}\vert\gg a_\rho^3 n_{1D}^2$ is satisfied [Ols98].

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... equations^8.3

Details on the numerical solution of the integral equations and a table for the function $e(\gamma)$ can be downloaded from http://www.science.unitn.it/~astra/1Dfermions/.

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...$2m$^8.4

The coefficient and sign of the third term on the right hand side of Eq. 8.9 differ from Eq. A7 in Ref. [CEE+91].

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... gas^9.1

Note that for $k_F\vert a\vert\ll 1$ the nonanalytic correction to the ground-state energy due to the superfluid gap is exponentially small.

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...$\sigma(x)$^10.1

It is convenient to use following equality $\int\limits_{-\infty}^\infty\frac{e^{\pm ikx}}{c^2+a^2(\varkappa-k)^2} \frac{dk}{2\pi}=\frac{1}{2ac}e^{\pm i\varkappa x -\frac{c}{a}\vert x\vert}$

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... analytically^11.1

Compare with [GR80] $\int_L^\infty x^{\mu-1} \cos x dx = \frac{1}{2}[e^{-i\mu\pi/2}\Gamma(\mu, iL)+e^{i\mu\pi/2}\Gamma(\mu, -iL)]$

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... simplified^11.2

See [GR80] 3.761.7 $\int_0^\infty x^{\mu-1} \cos(ax) dx = \frac{\Gamma(\mu)}{a^\mu} \cos \frac{\mu\pi}{2}, a>0, 0<Re \mu<1$

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... obtain^11.3

In dimensionless units $y=kL$ formulae (B.6, B.7) look like $f(k,L,\alpha) = k^{\alpha-1}\int\limits_{kL}^\infty \frac{\cos\,y}{y^\alpha}\,dy$ and $f(k,L,\alpha) = \frac{1}{kL^\alpha} \left(-\sin\,kL+\frac{\alpha}{kL}\cos\,kL\r... ...\alpha+1) k^{\alpha-1} \int\limits_{kL}^\infty \frac{\cos\,y}{y^{\alpha+2}}\,dy$

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