Minimal gaugomaly mediation

Mixed anomaly and gauge mediation (“gaugomaly” mediation) gives a natural solution to the SUSY flavor problem with a conventional LSP dark matter candidate. We present a minimal version of gaugomaly mediation where the messenger masses arise directly from anomaly mediation, automatically generating a messenger scale of order 50 TeV. We also describe a simple relaxation mechanism that gives rise to realistic μ and Bμ terms. B is naturally dominated by the anomaly-mediated contribution from top loops, so the μ-Bμ sector only depends on a single new parameter. In the minimal version of this scenario the full SUSY spectrum is determined by two continuous parameters (the anomaly-and gauge-mediated SUSY breaking masses) and one discrete parameter (the number of messengers). We show that these simple models can give realistic spectra with viable dark matter.     

Dynamical generation of flavour

We propose the generation of Standard Model fermion hierarchy by the extension of renormalizable SO(10) GUT with O(N_g) family gauge symmetry. In this scenario, Higgs representations of SO(10) also carry family indices and are called Yukawons. Vacuum expectation values of these Yukawon fields break GUT and family symmetry and generate MSSM Yukawa couplings dynamically. We have demonstrated this idea using \({\mathbf {10}}\oplus {\mathbf {210}} \oplus {\mathbf {126}} \oplus {\overline {\mathbf {126}}}\) Higgs irrep, ignoring the contribution of 120-plet which is, however, required for complete fitting of fermion mass-mixing data. The effective MSSM matter fermion couplings to the light Higgs pair are determined by the null eigenvectors of the MSSM-type Higgs doublet superfield mass matrix \(\mathcal {H}\). A consistency condition on the doublet ([1,2,±1]) mass matrix (\(\text {Det}(\mathcal {H})=\) 0) is required to keep one pair of Higgs doublets light in the effective MSSM. We show that the Yukawa structure generated by null eigenvectors of \(\mathcal {H}\) are of generic kind required by the MSSM. A hidden sector with a pair of (S_ab; ?_ab) fields breaks supersymmetry and facilitates \(D_{O(N_{g})}\hspace *{-1pt}=\) 0. SUSY breaking is communicated via supergravity. In this scenario, matter fermion Yukawa couplings are reduced from 15 to just 3 parameters in MSGUT with three generations.     

The convergence behaviour of expansions in the classical collision theory

In the classical collision theory the scattering angle? depends on the impact parameterb and on the kinetic energyE _r of the relative motion. This angle?(b, E _r ) is expanded for two limiting cases: 1. Expansion in powers of the potentialV(r)/E _r (momentum approximation). 2. Expansion in powers of the impact parameterb (central collision approximation). The radius of convergence of the series depends onb andE _r . It will be given for the following potentialsV(r):

$$A\left( {\frac{a}{r}} \right)^\mu ;Ae^{ - \frac{r}{a}} ;A\frac{a}{r}e^{ - \frac{r}{a}} ;A\left( {\frac{a}{r}} \right)^2 e^{ - \left( {\frac{r}{a}} \right)^2 } .$$     

SUSY see-saw and NMSO(10)GUT inflation after BICEP2

Supersymmetric see-saw slow roll inflection point inflation occurs along a MSSM D-flat direction associated with gauge invariant combination of Higgs, slepton and right-handed sneutrino at a scale set by the right-handed neutrino mass \(M_{{\nu }^{c}} \sim 10^{6}\!\,-\,\!10^{13}\) GeV. The tensor to scalar perturbation ratio r～10^?3 can be achieved in this scenario. However, this scenario faced difficulty in being embedded in the realistic new minimal supersymmetric SO(10) grand unified theory (NMSO(10)GUT). The recent discovery of B-mode polarization by BICEP2, changes the prospects of NMSO(10)GUT inflation. Inflection point models become strongly disfavoured, as the trilinear coupling of SUSY see-saw inflation potential gets suppressed relative to the mass parameter favoured by BICEP2. Large values of r≈0.2 can be achieved with super-Planck scale inflaton values and mass scales of inflaton ≥10 ¹³ GeV. In NMSO(10)GUT, this can be made possible with an admixture of heavy Higgs doublet fields, i.e., other than MSSM Higgs field, which are present and have masses of order GUT scale.     

Kinetically modified non-minimal inflation with exponential frame function

We consider supersymmetric (SUSY) and non-SUSY models of chaotic inflation based on the \(\phi ^n\) potential with \(n=2\) or 4. We show that the coexistence of an exponential non-minimal coupling to gravity \(f_\mathcal{R}=\mathrm{e}^{c_\mathcal{R}\phi ^{p}}\) with a kinetic mixing of the form \(f_{\mathrm{K}}=c_{\mathrm{K}}f_\mathcal{R}^m\) can accommodate inflationary observables favored by the Planck and Bicep2/Keck Array results for \(p=1\) and 2, \(1\le m\le 15\) and \(2.6\times 10^{-3}\le r_{\mathcal {R}\mathrm{K}}=c_\mathcal{R}/c_{\mathrm{K}}^{p/2}\le 1,\) where the upper limit is not imposed for \(p=1\). Inflation is of hilltop type and it can be attained for subplanckian inflaton values with the corresponding effective theories retaining the perturbative unitarity up to the Planck scale. The supergravity embedding of these models is achieved employing two chiral gauge singlet supefields, a monomial superpotential and several (semi)logarithmic or semi-polynomial Kähler potentials.     

A New Set of Limiting Gibbs Measures for the Ising Model on a Cayley Tree

For the Ising model (with interaction constant J>0) on the Cayley tree of order k≥2 it is known that for the temperature T≥T _c,k =J/arctan?(1/k) the limiting Gibbs measure is unique, and for T<T _c,k there are uncountably many extreme Gibbs measures. In the Letter we show that if \(T\in(T_{c,\sqrt{k}}, T_{c,k_{0}})\), with \(\sqrt{k} then there is a new uncountable set \({\mathcal{G}}_{k,k_{0}}\) of Gibbs measures. Moreover \({\mathcal{G}}_{k,k_{0}}\ne {\mathcal{G}}_{k,k'_{0}}\), for k ₀≠k′₀. Therefore if \(T\in (T_{c,\sqrt{k}}, T_{c,\sqrt{k}+1})\), \(T_{c,\sqrt{k}+1} then the set of limiting Gibbs measures of the Ising model contains the set {known Gibbs measures}\(\cup(\bigcup_{k_{0}:\sqrt{k}.     

On the nature of the liquid-to-glass transition equation

Within the model of delocalized atoms, it is shown that the parameter δT_g, which enters the glasstransition equation qτ_g = δT_g and characterizes the temperature interval in which the structure of a liquid is frozen, is determined by the fluctuation volume fraction \({f_g} = {\left( {{{\Delta {V_e}} \mathord{\left/ {\vphantom {{\Delta {V_e}} V}} \right. \kern-\nulldelimiterspace} V}} \right)_{T = {T_g}}}\) frozen at the glass-transition temperature T_g and the temperature T_g itself. The parameter δT_g is estimated by data on f_g and T_g. The results obtained are in agreement with the values of δT_g calculated by the Williams–Landel–Ferry (WLF) equation, as well as with the product qτ_g—the left-hand side of the glass-transition equation (q is the cooling rate of the melt, and τ_g is the structural relaxation time at the glass-transition temperature). Glasses of the same class with f_g ≈ const exhibit a linear correlation between δT_g and T_g. It is established that the currently used methods of Bartenev and Nemilov for calculating δT_g yield overestimated values, which is associated with the assumption, made during deriving the calculation formulas, that the activation energy of the glass-transition process is constant. A generalized Bartenev equation is derived for the dependence of the glass-transition temperature on the cooling rate of the melt with regard to the temperature dependence of the activation energy of the glasstransition process. A modified version of the kinetic glass-transition criterion is proposed. A conception is developed that the fluctuation volume fraction f = ΔV_e/V can be interpreted as an internal structural parameter analogous to the parameter ξ in the Mandelstam–Leontovich theory, and a conjecture is put forward that the delocalization of an active atom—its critical displacement from the equilibrium position—can be considered as one of possible variants of excitation of a particle in the Vol’kenshtein–Ptitsyn theory. The experimental data used in the study refer to a constant cooling rate of q = 0.05 K/s (3 K/min).     

Asymptotic Behavior of the Selberg Zeta Functions for Degenerating Families of Hyperbolic Manifolds

Let {M _k } be a degenerating sequence of finite volume, hyperbolic manifolds of dimension d, with d = 2 or d = 3, with finite volume limit M _∞. Let \({Z_{M_{k}} (s)}\) be the associated sequence of Selberg zeta functions, and let \({{\mathcal{Z}}_{k} (s)}\) be the product of local factors in the Euler product expansion of \({Z_{M_{k}} (s)}\) corresponding to the pinching geodesics on M _k . The main result in this article is to prove that \({Z_{M_{k}} (s)/{\mathcal{Z}}_{k} (s)}\) converges to \({Z_{M_{\infty}} (s)}\) for all \({s \in \mathbf{C}}\)with Re(s) > (d ? 1)/2. The significant feature of our analysis is that the convergence of \({Z_{M_{k}} (s)/{\mathcal{Z}}_{k} (s)}\) to \({Z_{M_{\infty}} (s)}\) is obtained up to the critical line, including the right half of the critical strip, a region where the Euler product definition of the Selberg zeta function does not converge. In the case d = 2, our result reproves by different means the main theorem in Schulze (J Funct Anal 236:120–160, 2006).     

On the Exact Evaluation of the Face-Centred Cubic Lattice Green Function

The mathematical properties of the lattice Green function

Open image in new window

are investigated, where w=w ₁+iw ₂ lies in a complex plane which is cut from w=?1 to w=3, and {? ₁,? ₂,? ₃} is a set of integers with ? ₁+? ₂+? ₃ equal to an even integer. In particular, it is proved that G(2n,0,0;w), where n=0,1,2,…, is a solution of a fourth-order linear differential equation of the Fuchsian type with four regular singular points at w=?1,0,3 and ∞. It is also shown that G(2n,0,0;w) satisfies a five-term recurrence relation with respect to the integer variable n. The limiting function

$G^{-}(2n,0,0;w_1)\equiv\lim_{\epsilon\rightarrow0+}G(2n,0,0;w_1-\mathrm{i}\epsilon) =G_{\mathrm{R}}(2n,0,0;w_1)+\mathrm{i}G_{\mathrm {I}}(2n,0,0;w_1) ,\nonumber $

where w ₁∈(?1,3), is evaluated exactly in terms of ₂ F ₁ hypergeometric functions and the special cases G ^?(2n,0,0;0), G ^?(2n,0,0;1) and G(2n,0,0;3) are analysed using singular value theory. More generally, it is demonstrated that G(? ₁,? ₂,? ₃;w) can be written in the form

Open image in new window

where Open image in new window are rational functions of the variable ξ, K(k _?) and E(k _?) are complete elliptic integrals of the first and second kind, respectively, with

$k_{-}^2\equiv k_{-}^2(w)={1\over2}- {2\over w} \biggl(1+{1\over w} \biggr)^{-{3\over2}}- {1\over2} \biggl(1-{1\over w} \biggr ) \biggl(1+{1\over w} \biggr)^{-{3\over2}} \biggl(1-{3\over w} \biggr)^{1\over2}\nonumber $

and the parameter ξ is defined as

$\xi\equiv\xi(w)= \biggl(1+\sqrt{1-{3\over w}} \,\biggr)^{-1} \biggl(-1+\sqrt{1+{1\over w}} \,\biggr) .\nonumber $

This result is valid for all values of w which lie in the cut plane. The asymptotic behaviour of G ^?(2n,0,0;w ₁) and G(2n,0,0;w ₁) as n→∞ is also determined. In the final section of the paper a new ₂ F ₁ product form for the anisotropic face-centred cubic lattice Green function is given.

     

QED with vector and axial vector types of interaction and dynamical generation of particle masses

We consider a model of electrodynamics with two types of interaction, the vector \((e\bar \psi (\gamma ^\mu A_\mu )\psi )\) and axial vector \((e_A \bar \psi (\gamma ^\mu \gamma ^5 B_\mu )\psi )\) interactions, i.e., with two types of vector gauge fields, which corresponds to the local nature of the complete massless-fermion symmetry group U(1) ? U _A (1). We present a phenomenological model with spontaneous symmetry breaking through which the fermion and the axial vector field B_μ acquire masses. Based on an approximate solution of the Dyson equation for the fermion mass operator, we demonstrate the phenomenon of dynamical chiral symmetry breaking when the field B_μ has mass. We show the possibility of eliminating the axial anomalies in the model under consideration when introducing other types of fermions (quarks) within the standard-model fermion generations. We consider the polarization operator for the field B_μ and the procedure for removing divergences when calculating it. We demonstrate the emergence of a mass pole in the propagator of the particles that correspond to the field B_03BC when chiral symmetry is broken and consider the problems of regularizing closed fermion loops with axial vector vertices in connection with chiral symmetry breaking.     

Symmetries of the pseudo-diffusion equation and related topics

We show in details how to determine and identify the algebra g = {A_i} of the infinitesimal symmetry operators of the following pseudo-diffusion equation (PSDE) LQ ≡ \(\left[ {\frac{\partial }{{\partial t}} - \frac{1}{4}\left( {\frac{{{\partial ^2}}}{{\partial {x^2}}} - \frac{1}{{{t^2}}}\frac{{{\partial ^2}}}{{\partial {p^2}}}} \right)} \right]\) Q(x, p, t) = 0. This equation describes the behavior of the Q functions in the (x, p) phase space as a function of a squeeze parameter y, where t = e ^2y. We illustrate how G _i(λ) ≡ exp[λA _i] can be used to obtain interesting solutions. We show that one of the symmetry generators, A ₄, acts in the (x, p) plane like the Lorentz boost in (x, t) plane. We construct the Anti-de-Sitter algebra so(3, 2) from quadratic products of 4 of the A _i, which makes it the invariance algebra of the PSDE. We also discuss the unusual contraction of so(3, 1) to so(1, 1)? h². We show that the spherical Bessel functions I ₀(z) and K ₀(z) yield solutions of the PSDE, where z is scaling and “twist” invariant.     

Rare kaon decay $$K^ + \to \pi ^ + \nu \bar \nu $$ in S U(3)C ⊗ S U(3)L ⊗ U(1)N models

The rare kaon decay \(K^ + \to \pi ^ + \nu \bar \nu \) is considered in the framework of models based on the SU(3) _C ? SU(3) _L ? U(1) _N (3-3-1) gauge group. In the 3-3-1 model with right-handed neutrinos, the lower bound of the Z’ mass is derived at 3 TeV, and that in the minimal version, at 1.7 TeV.     

Sbottom signature of supersymmetric golden region

The experimental search limit on the Higgs boson mass points to a “golden region” in the Minimal Supersymmetric Standard Model parameter space in which the fine-tuning in the electroweak sector is minimized. One of the sbottoms is relatively light since its mass parameter is related to that of the stop. The decay products of the light sbottom typically include a W boson and a b jet. We studied the pair production of the light sbottoms at the Large Hadron Collider and examined its discovery potential via collider signature of 4 jets + 1 lepton + missing E _T . Such channel differs from conventional sbottom searches of either \(\tilde{b}_{1} \rightarrow b \chi_{2}^{0} \rightarrow b \ell^{+}\ell^{-} \chi_{1}^{0}\) or \(\tilde{b}_{1} \rightarrow t \chi_{1}^{-}\) with reconstructed top quark. We analyzed the Standard Model backgrounds for this channel and developed a set of cuts to identify the signal and suppress the backgrounds. We showed that with 100 fb^?1 integrated luminosity, a significance level of 5–6 σ could be reached for the light sbottom discovery at the LHC.     

Verstärkung einer He-Ne-Gasentladung für die Laserwellenlänge λ=6328 AE

The optical gain of He-Ne discharges for the laser wave-length of 6328 AE is investigated experimentally. The measurements are performed in two independent methods, which both give the same results. The gain of the He-Ne discharge is measured for a number of discharge tubes with different tube-lengthsl and tube-diametersD. The experiments show that the maximum gain? ₀ is a function of tube-length and-diameter:?G ₀(l,D) ?

$$\hat G_0 (l,D) \cong \left[ {1 + 0,5\left( {\frac{{D_0 }}{D}} \right)^{1,4} } \right]^{{l \mathord{\left/ {\vphantom {l {l_0 }}} \right. \kern-\nulldelimiterspace} {l_0 }}} $$     

New minimal <Emphasis Type="Italic">SO</Emphasis>(10) GUT: A theory for all epochs

The supersymmetric SO(10) theory (NMSO(10)GUT) based on the \({{\mathbf {210}+\mathbf {126} +\overline {\mathbf {126}}}}\) Higgs system proposed in 1982 has evolved into a realistic theory capable of fitting the known low energy particle physics data besides providing a dark matter candidate and embedding inflationary cosmology. It dynamically resolves longstanding issues such as fast dimension five-operator mediated proton decay in SUSY GUTs by allowing explicit and complete calculation of crucial threshold effects at M_SUSY and M_GUT in terms of fundamental parameters. This shows that SO(10) Yukawas responsible for observed fermion masses as well as operator dimension-five-mediated proton decay can be highly suppressed on a ‘Higgs dissolution edge’ in the parameter space of GUTs with rich superheavy spectra. This novel and generically relevant result highlights the need for every realistic UV completion model with a large /infinite number of heavy fields coupled to the light Higgs doublets to explicitly account for the large wave function renormalization effects on emergent light Higgs fields. The NMSGUT predicts large-soft SUSY breaking trilinear couplings and distinctive sparticle spectra. Measurable or near measurable level of tensor perturbations – and thus large inflaton mass scale – may be accommodated within the NMSGUT by supersymmetric see-saw inflation based on an LHN flat direction inflaton if the Higgs component contains contributions from heavy Higgs components. Successful NMSGUT fits suggest a renormalizable Yukawon ultraminimal gauged theory of flavour based upon the NMSGUT Higgs structure.     

Random Walk with Barycentric Self-interaction

We study the asymptotic behaviour of a d-dimensional self-interacting random walk (X _n ) _n∈? (?:={1,2,3,…}) which is repelled or attracted by the centre of mass \(G_{n} = n^{-1} \sum_{i=1}^{n} X_{i}\) of its previous trajectory. The walk’s trajectory (X ₁,…,X _n ) models a random polymer chain in either poor or good solvent. In addition to some natural regularity conditions, we assume that the walk has one-step mean drift

$\mathbb{E}[X_{n+1} - X_n \mid X_n - G_n = \mathbf{x}] \approx\rho\|\mathbf{x}\|^{-\beta}\hat{ \mathbf{x}}$

for ρ∈? and β≥0. When β<1 and ρ>0, we show that X _n is transient with a limiting (random) direction and satisfies a super-diffusive law of large numbers: n ^?1/(1+β) X _n converges almost surely to some random vector. When β∈(0,1) there is sub-ballistic rate of escape. When β≥0 and ρ∈? we give almost-sure bounds on the norms ‖X _n ‖, which in the context of the polymer model reveal extended and collapsed phases.

Analysis of the random walk, and in particular of X _n ?G _n , leads to the study of real-valued time-inhomogeneous non-Markov processes (Z _n ) _n∈? on [0,∞) with mean drifts of the form

$ \mathbb{E}[ Z_{n+1} - Z_n \mid Z_n = x ] \approx\rho x^{-\beta} - \frac {x}{n},$

(0.1)

where β≥0 and ρ∈?. The study of such processes is a time-dependent variation on a classical problem of Lamperti; moreover, they arise naturally in the context of the distance of simple random walk on ? ^d from its centre of mass, for which we also give an apparently new result. We give a recurrence classification and asymptotic theory for processes Z _n satisfying (0.1), which enables us to deduce the complete recurrence classification (for any β≥0) of X _n ?G _n for our self-interacting walk.

     

Rules of correspondence in atomic physics

The Smirnov method of analytic continuation (B.M. Smirnov, Sov. Phys. JETP 20, 345 (1964)) has been justified and developed for atomic physics. It has been shown that the polarizability of alkali atoms α, their van der Waals interaction constant C ₆, and the oscillator strength of the transition to the first P state f ₀₁ are related to the parameter 〈r ²〉 and gap in the spectrum \(\frac{3}{2}\frac{f}{\Delta } \approx \frac{3}{2}\alpha \Delta \approx {\left( {3{C_6}\Delta } \right)^{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$2$}}}} \approx \left\langle {{r^2}} \right\rangle \). The average square of the coordinate of the valence electron 〈r ²〉 in the first approximation has a hydrogen dependence \({J_1} = \frac{1}{{2{v^2}}}.\) on the filling factor ν, which is defined in terms of the first ionization potential: xxxxxxxxx     

Suche nach Strahlungsprozessen zweiter Ordnung beim Zerfall von Ag109m

The decay of an excited state by the emission of twoγ-quanta (γ γ-transitions) or two conversion electrons (e e-transitions) or oneγ-quantum and one conversion electron (γ e-transitions) is expected as a second order radiation process. The decay of Ag^109m was examined for such events using a special arrangement of two NaJ-scintillation counters in coincidence. The energies of coincident quanta were displayed on the two axes of an “X-Y”-Oscilloscope respectively. For the ratio ofγ γ-transitions to one-quantum transitions an upper limit of\(\frac{{W_{\gamma \gamma } }}{{W_\gamma }} \leqq 1,9 \cdot 10^{ - 5} \) was obtained. Furthermore theγ-spectrum in coincidence withK X-rays was studied. From these measurementse e- andγ e-transition rates can be calculated for the case ofK shell conversion. The results obtained are:

$$\frac{{W_{^e K^e K} }}{{W_\gamma }} = \left( {8,1_{ - 1,7}^{ + 0,6} } \right) \cdot 10^{ - 3} and\frac{{W_{\gamma ^e K} }}{{W_\gamma }}< 1,5 \cdot 10^{ - 3} .$$     

Krein formula with compensated singularities for the ND-mapping and the generalized Kirchhoff condition at the Neumann Schrödinger junction

The Neumann Schrödinger operator \(\mathcal{L}\) is considered on a thin 2D star-shaped junction, composed of a vertex domain Ω_int and a few semi-infinite straight leads ω _m , m = 1, 2, ..., M, of width δ, δ ? diam Ω_int, attached to Ω_int at Γ ? ?Ω_int. The potential of the Schrödinger operator l ^ω on the leads vanishes, hence there are only a finite number of eigenvalues of the Neumann Schrödinger operator L _int on Ω_int embedded into the open spectral branches of l ^ω with oscillating solutions χ _±(x, p) = \(e^{ \pm iK_ + x} e_m \) of l ^ω χ _± = p ² χ _±. The exponent of the open channels in the wires is

$K_ + (\lambda ) = p\sum\limits_{m = 1}^M {e^m } \rangle \langle e^m = \sqrt \lambda P_ + $

, with constant e _m , on a relatively small essential spectral interval Δ ? [0, π ² δ ^?2). The scattering matrix of the junction is represented on Δ in terms of the ND mapping

$\mathcal{N} = \frac{{\partial P_ + \Psi }}{{\partial x}}(0,\lambda )\left| {_\Gamma \to P_ + \Psi _ + (0,\lambda )} \right|_\Gamma $

as

$S(\lambda ) = (ip\mathcal{N} + I_ + )^{ - 1} (ip\mathcal{N} - I_ + ), I_ + = \sum\limits_{m = 1}^M {e^m } \rangle \langle e^m = P_ + $

. We derive an approximate formula for \(\mathcal{N}\) in terms of the Neumann-to-Dirichlet mapping \(\mathcal{N}_{\operatorname{int} } \) of L _int and the exponent K _? of the closed channels of l ^ω . If there is only one simple eigenvalue λ ₀ ∈ Δ, L _intφ0 = λ _0φ0 then, for a thin junction, \(\mathcal{N} \approx |\vec \phi _0 |^2 P_0 (\lambda _0 - \lambda )^{ - 1} \) with

$\vec \phi _0 = P_ + \phi _0 = (\delta ^{ - 1} \int_{\Gamma _1 } {\phi _0 (\gamma )} d\gamma ,\delta ^{ - 1} \int_{\Gamma _2 } {\phi _0 (\gamma )} d\gamma , \ldots \delta ^{ - 1} \int_{\Gamma _M } {\phi _0 (\gamma )} d\gamma )$

and \(P_0 = \vec \phi _0 \rangle |\vec \phi _0 |^{ - 2} \langle \vec \phi _0 \),

$S(\lambda ) \approx \frac{{ip|\vec \phi _0 |^2 P_0 (\lambda _0 - \lambda )^{ - 1} - I_ + }}{{ip|\vec \phi _0 |^2 P_0 (\lambda _0 - \lambda )^{ - 1} + I_ + }} = :S_{appr} (\lambda )$

. The related boundary condition for the components P ₊Ψ(0) and P ₊Ψ′(0) of the scattering Ansatz in the open channel \(P_ + \Psi (0) = (\bar \Psi _1 ,\bar \Psi _2 , \ldots ,\bar \Psi _M ), P_ + \Psi '(0) = (\bar \Psi '_1 , \bar \Psi '_2 , \ldots , \bar \Psi '_M )\) includes the weighted continuity (1) of the scattering Ansatz Ψ at the vertex and the weighted balance of the currents (2), where

$\frac{{\bar \Psi _m }}{{\bar \phi _0^m }} = \frac{{\delta \sum\nolimits_{t = 1}^M { \bar \Psi _t \bar \phi _0^t } }}{{|\vec \phi _0 |^2 }} = \frac{{\bar \Psi _r }}{{\bar \phi _0^r }} = :\bar \Psi (0)/\bar \phi (0), 1 \leqslant m,r \leqslant M$

(1)

,

$\sum\limits_{m = 1}^M {\bar \Psi '_m } \bar \phi _0^m + \delta ^{ - 1} (\lambda - \lambda _0 )\bar \Psi /\bar \phi (0) = 0$

(1)

. Conditions (1) and (2) constitute the generalized Kirchhoff boundary condition at the vertex for the Schrödinger operator on a thin junction and remain valid for the corresponding 1D model. We compare this with the previous result by Kuchment and Zeng obtained by the variational technique for the Neumann Laplacian on a shrinking quantum network.