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while the fundamental equations of motion , both in classical and quantum mechanics , are invariant with respect to time reversal , the world around us is full of irreversible processes . on a microscopic level , it is enough to recall spontaneous decays of particles , nuclei , inelastic scattering processes , etc . and , of course , an irreversible process _ par excellence _ is the wave packet reduction which is the core of copenhagen description of a measuring process performed on a quantum system .    a description of such processes in quantum mechanics is typically associated with enlarging the state hilbert space , conventionally referred to as coupling the system to a heat bath @xcite . as a rule , it is assumed that ( i ) the bath is a system with infinite number of degrees of freedom , ( ii ) the bath hamiltonian has a continuous spectrum , and ( iii ) the presence ( or absence ) of irreversible modes is determined by the energies involved rather than the coupling strength . while this all is true in many cases , situations may exist where the system has neither of the listed properties . this was the motivation which led uzy smilansky to formulation a simple model @xcite of a quantum graph coupled to a heat bath consisting of a single harmonic oscillator which exhibits an irreversible behavior . the easiest way to describe the model is to phrase it in pde terms , then its hamiltonian is a schrdinger operator , @xmath0 on @xmath1 , with a singular interaction supported by the line @xmath2 the strength of which depends on the coordinate @xmath3 . the system governed by the hamiltonian undergoes an abrupt spectral transition at the critical value @xmath4 ; while for @xmath5 the hamiltonian is positive above this value its spectrum covers the whole real axis . the mechanism of this effect is easy to understand . the first and the last term in represent hamiltonian of a one - dimensional system with a point interaction which has a single eigenvalue equal to @xmath6 @xcite . this negative contribution to the energy competes with the oscillator potential with the balance flipping exactly at the said critical value of @xmath7 . the model was subsequently an object of investigations , generalizations and modifications in a series of mathematically rigorous papers . the discrete spectrum in the subcritical case and its behavior as @xmath8 were analyzed in @xcite , the absolute continuity of the other spectral component was established in @xcite , an extension to the situation when the ` heat bath ' consists of two or more oscillators was discussed in @xcite . other modifications consisted of replacing the oscillator by a potential well of a different shape @xcite or by replacing the line by a more general graph @xcite corresponding to the original smilansky proposal . it is also possible to have the motion in the @xmath9 direction restricted to an interval with periodic boundary conditions @xcite ; in the first named of these papers time evolution of wave packets is investigated confirming the idea that spectral change in the supercritical case corresponds to the possibility of ` escape to infinity ' . let us also mention a modification in which the singular interaction is replaced by a regular potential channel which gets deeper and more narrow as the distance from a fixed point increases @xcite .    in connection with the last mentioned system let us point out that similar parameter dependent spectral transitions can be also observed in other systems , an example being the one described by the hamiltonian @xmath10 on @xmath11 @xcite . in this case the change is even more dramatic , from the spectrum which is below bounded and purely discrete to the whole real line , the transition occurring at the critical value @xmath12 , the ground state eigenvalue of the appropriate ( an)harmonic oscillator . to our knowledge , spectral transitions of this type were for the first time noted in @xcite . our object of interest in the present paper is the original smilansky hamiltonian . despite the fact that many of its properties were demonstrated in the papers mentioned above , there is still a room for new insights . the first topic here concerns the discrete spectrum of @xmath13 and comes from the observation that it is relatively easy to find the eigenvalues and eigenfunctions numerically turning the task into a matrix problem of a sufficiently large dimension . another question to be discussed here concerns _ resonances _ in the system described by the hamiltonian . while the absolute continuity of the spectrum above the threshold and the existence of the wave operators were established in @xcite , the possible resonance character of the scattering seems to have escaped attention . in our recent paper @xcite we have demonstrated how one can find poles of the analytically continued resolvent and derived asymptotic expansions for them , here we focus on the physical meaning of the resonances and illustrate their behavior by numerical results . to make the paper self - contained we summarize first the known facts about the operator that are indicated briefly in the introduction . it is obvious that we can consider the positive values of the coupling constant @xmath7 as the opposite case is obtained via a mirror transformation with respect to the axis @xmath2 .    _ ( a ) spectral transition : _ in the subcritical case , @xmath14 , the particle remains localized in the vicinity of the @xmath9 axis , while for @xmath15 it can according to @xcite escape to infinity along the singular ` channel ' in the @xmath3 direction . in spectral terms , this corresponds to the switch from a positive spectrum of @xmath13 to a below unbounded one which occurs at @xmath16 . _ ( b ) the continuous spectrum _ of @xmath13 covers the interval @xmath17 for @xmath14 , the positive semi - axis for @xmath18 , and the whole real axis if @xmath15 . moreover , this spectrum is in fact purely absolutely continuous ( * ? ? ? vii.2 ) , in particular , the operator is purely absolutely continuous in the supercritical case when @xmath19 .    _ ( c ) eigenvalue existence and location : _ for any @xmath20 the discrete spectrum of @xmath13 is nonempty , simple , and finite , being contained in @xmath21 , i.e. the subcritical operator is positive . the eigenvalues are decreasing functions of the coupling constant @xmath7 . on the other hand , there are no eigenvalues @xmath22 , and in the supercritical case , @xmath15 , the point spectrum of @xmath13 is empty , in other words , there are no eigenvalues embedded in the continuum .    _ ( d ) near critical asymptotics : _ the number of eigenvalues in the subcritical regime increases with the coupling constant and explodes as it approaches the critical value @xmath16 behaving asymptotically as @xcite @xmath23 where @xmath24 conventionally denotes the number of eigenvalues of the operator @xmath25 below @xmath26 with the multiplicity taken into account .    _ ( e ) weak - coupling asymptotics : _ for small enough @xmath7 there is a single eigenvalue which behaves as @xmath27 as @xmath28 @xcite . in fact , due to the mirror symmetry of the spectrum with respect to @xmath7 the error term can be replaced by @xmath29 . to find the discrete spectrum we note first that the solutions to the eigenvalue problem can be expanded using the ` transverse ' base naturally spanned by the functions @xmath30 corresponding to the oscillator eigenvalues @xmath31 ; here @xmath32 are hermite polynomials with conventional normalization . furthermore , the task can be simplified by noting that the problem has a mirror symmetry w.r.t . which allows us to decompose @xmath13 into the trivial odd part @xmath33 and the even part @xmath34 which is equivalent to the operator on @xmath35 , where @xmath36 , with the symbol and the domain consisting of @xmath37 functions satisfying the boundary conditions @xmath38 substituting the ansatz @xmath39 with @xmath40 into and using orthonormality of the basis , we get for solution with the energy @xmath41 the equation @xmath42 where @xmath43 is the coefficient vector and @xmath44 is an operator in @xmath45 , in other words , an infinite matrix with the elements @xmath46 note that the matrix @xmath44 is in fact tridiagonal because @xmath47 in other words , @xmath44 is a jacobi matrix . note that using a general sequence @xmath48 instead of @xmath49 in the ansatz   one can reformulate the problem as an analogue of the birman - schwinger principle as first proposed by m.  solomyak in @xcite , see a detailed discussion in  @xcite . the same approach works also for some modifications of the original smilansky model mentioned in the introduction where , however , the structure of the appropriate matrix problem is different and may be more complicated .    to find the values of @xmath41 for which the secular equation is satisfied for a given @xmath7 we solve a truncated matrix problem increasing the cut - off until a numerical stability is reached . the convergence is slower the closer is the value of @xmath41 to the threshold value @xmath50 . we present results for the sizes up to @xmath51 .     as functions of @xmath7,width=453 ] the results are plotted below in a series of figures . they are , of course , in agreement with the theoretical predictions listed in the previous section , but they provide additional insights . to begin with , fig . [ ev_plot ] shows eigenvalues of @xmath13 as functions of the coupling constant . we see that in the most part of the subcritical region , more than @xmath52 of it , the hamiltonian has a single eigenstate . the second eigenvalue appears only at @xmath53 , the next thresholds are @xmath54 . this does not give a good picture of the spectral behavior near the critical values . instead we can plot the eigenvalue as functions of @xmath7 using a logarithmic scale . the result is shown in fig . [ log_plot ] . we see that their number indeed grows as we approach the transition point filling the allowed interval so that @xmath55 , but that the discrete spectrum is asymptotically equidistantly distributed . the same result allows us to illustrate the solomyak asymptotic formula : it is obvious from fig . [ solombound ] that the leading term of the expansion gives a good approximation as long as @xmath56 . the opposite asymptotical situation is the weak coupling . in fig . [ weakfig ] we plot the ground - state eigenvalue as a function of @xmath7 . in the double logarithmic scale the graph is practically indistinguishable from a straight line which allows us to read out the power of the leading term as well as the coefficient @xmath57 appearing in the leading term of .    since , in contrast to points ( a)(d ) of the previous section the weak - coupling asymptotics is a new observation it is worth to show a way how it can be derived . to this purpose we introduce a symbol for the energy gap @xmath58 and denote by @xmath59 the normalized eigenfunction of @xmath44 given by corresponding to zero eigenvalue . without loss of generality we may assume that @xmath60 is non - negative . writing the lines of the corresponding infinite system of equation explicitly , we get @xmath61          & \frac{\sqrt{k}\lambda}{2\sqrt{2}}\,c^\lambda_{k-1 } + \sqrt{k + \mu_\lambda}\,c^\lambda_k         + \frac{\sqrt{k+1}\lambda}{2\sqrt{2}}\,c^\lambda_{k+1 } = 0\ , , \qquad k\ge 1\ , . \end{split}\ ] ] using elementary inequalities @xmath62 and that @xmath63 , we get from the second relation in the estimate @xmath64 which implies @xmath65 and combining it with the normalization assumption , @xmath66 , we arrive at @xmath67 using @xmath66 once more , we conclude that @xmath68 then inequality together with the normalization yield @xmath69 . in combination with   and the second relation of   for @xmath70 this implies further @xmath71 using now   and   we obtain from the first relation in   that @xmath72 which is nothing else but the formula . in the counterpart to this paper @xcite we give a modified version of this argument which allows , at least in principle , to compute higher terms of this asymptotic expansion .     for @xmath73,width=453 ]    before proceeding further , let us also look at the eigenfunctions . in fig . [ eigenfig ] we show a contour plot of the first six eigenfunctions of @xmath13 for @xmath73 , in other words , @xmath74 . in the side plots we feature restrictions of the eigenfunctions on the @xmath3-axis . as expected the eigenfunctions are concentrated in the vicinity of the ` escape channel ' , i.e. the negative part of the @xmath3-axis . the plots also show that they obey the usual restriction , commonly referred to as courant nodal domain theorem ; the inequality becomes sharp from the third eigenfunction . the pattern is interesting , the odd eigenfunctions ( except the ground state ) have closed nodal lines only , the even ones have an unbounded nodal line . one wonders whether this property holds for the discrete spectrum generally . our second main point is to show that smilansky model also exhibits interesting resonance behavior . having said that , one has to explain first what is meant by a resonance . there are different definitions , the two main being the _ resolvent resonances _ associated with poles in the analytic continuation of the resolvent over the cut(s ) corresponding to the continuous spectrum , and _ scattering resonances _ associated with singularities of the scattering matrix . while these two can often be identified , this property has to be checked in each particular case ; recall that the former are determined by a single operator , the latter come from comparison of the free and interacting dynamics . from the differential equation point of view finding poles of the analytically continued resolvent means to solve the eigenvalue equation for looking for a complex ` eigenvalue ' corresponding to an ` eigenfunction ' which does not belong to @xmath75 being exponentially increasing with @xmath76 . modifying the argument of the previous section which led to the infinite matrix problem the task is again reduced to the equation . a subtle point in this case , as with other ` guiding channel ' systems with infinite number of transverse modes @xcite producing infinite number of cuts in the continuous spectrum , is that the riemann surface of energy has infinite number of sheets , in fact an uncountable one . we are interested in those of them neighboring with the physical sheet which in the present context means that looking for resonances on the @xmath77-th sheet we have to flip sign of the first @xmath78 signs of the square roots in ; for a more thorough discussion of this problem we refer the reader to  @xcite . let us turn to scattering resonances . suppose that the incident wave comes in the @xmath79-th channel from the left with the energy @xmath80 . we use the ansatz @xmath81 \sum_{n=0}^\infty t_{mn}\ , \mathrm{e}^{-ip_nx } \psi_n(y ) ,      & \quad x > 0 . \end{cases}\ ] ] where we have denoted @xmath82 . let us note that the on - shell scattering matrix with the elements has the size @xmath83 where @xmath84 $ ] , the other ` reflection ' and ` transmission ' amplitudes correspond to the evanescent modes present in the scattering solution . the sign choice of the square roots signifying the riemann sheet choice is the same as in the previous case . it is straightforward to compute from here the boundary values @xmath85 and @xmath86 . the continuity requirement at @xmath2 together with the orthonormality of the basis @xmath87 yields @xmath88 furthermore , substituting the boundary values from the ansatz into @xmath89 and integrating the obtained expression with @xmath90 we obtain @xmath91 the requirement of solvability of this system leads to the same condition as before which allows us to conclude that the _ resolvent and scattering resonances in smilansky model coincide_.    the next question concerns the existence of resonances and their behavior as the coupling constant @xmath7 changes . there are some standard mechanisms producing resonances . one is a perturbation of embedded eigenvalues of the hamiltonian , however , we have seen that in the present case there are no such eigenvalues . another possibility comes from perturbation of the singularities associated with the thresholds of the transverse channels , i.e. the branching points of the energy surface cuts    the weak - coupling analysis follows the route as for the discrete spectrum and shows that for small @xmath7 a resonance pole splits of each threshold according to the asymptotic expansion . this indeed leads to emergence of complex poles in the vicinity of thresholds which behave as @xmath92 as one can readily check repeating the argument of the previous section , in fact , the weak coupling of the bound state is included if we put @xmath93 in . for @xmath94 we get genuine complex poles the distance for the corresponding threshold is proportional to @xmath95 and the trajectory asymptote is the ` steeper ' the larger @xmath79 is . a more detailed discussion of the weak - coupling behavior of these resonances and also of resonances lying on the sheets that are not adjacent to the physical sheet can be again found in @xcite .     runs through the interval @xmath96 . resonances that do not depart from thresholds are marked by dashed lines , width=377 ]    however , this tells us nothing about the pole behavior for larger @xmath7 . to get a more complete picture , we again analyze the problem numerically . as described above , we look for complex values of @xmath41 for which the secular equation is satisfied for a given @xmath7 . now , however , @xmath44 is modified by flipping sign of the first @xmath78 signs of the square roots . then , a truncated matrix problem ia solved , tuning the cut - off until a numerical stability is reached . the results are shown in fig . [ resofig ] where we demonstrate resonances splitting from the lowest thresholds , from the second to the fifth one together with their weak - coupling asymptotes according to . we see that the behavior beyond the weak - coupling regime is different , the pole trajectories return to the real axis and they may even end up as an isolated eigenvalue . this is not surprising , however , since a similar behavior is known from other resonance models @xcite being first observed in the classical lee - friedrichs model @xcite . what is even more interesting , the numerical analysis shows that the system has other resonances that do not emerge from thresholds and appear when the coupling surpasses a critical value . in fig . [ resofig ] we show two such trajectories located at the second and third sheet of the energy surfaces and emerging at @xmath97 and @xmath98 , respectively . one can conjecture naturally that these are not the only ones and that the system governed by smilansky hamiltonian has a rich family of resonances of different kinds which deserve a separate investigation . this research was supported by the czech science foundation ( gar ) within the project 14 - 06818s . 99    s.  albeverio , f.  gesztesy , r.  hegh - krohn , h.  holden : _ solvable models in quantum mechanics _ , 2nd edition with an appendix by p.  exner , ams chelsea publishing , providence , r.i . , 2005 . d.  barseghyan , p.  exner : a regular version of smilansky model , _ j. math * 55 * ( 2014 ) , 042104 . d.  barseghyan , p.  exner , a.  khrabustovskyi , m.  tater : spectral analysis of a class of schrdinger operators exhibiting a parameter - dependent spectral transition , _ j. phys . a : math . theor . _ * 49 * ( 2016 ) , 165302 . davies : _ open quantum systems _ , academic press , london 1976 . evans , m.  solomyak : smilansky s model of irreversible quantum graphs . i : the absolutely continuous spectrum , _ j. phys . a : math . gen . _ * 38 * ( 2005 ) , 46114627 . evans , m.  solomyak : smilansky s model of irreversible quantum graphs . ii : the point spectrum , _ j. phys . a : math . * 38 * ( 2005 ) , 76617675 . exner : _ open quantum systems and feynman integrals _ ; fundamental theories of physics , vol .  6 , d.  reidel , dordrecht 1985 . exner : solvable models of resonances and decays , proceedings of the conference `` mathematical physics , spectral theory and stochastic analysis '' ( goslar 2011 ; m.  demuth , w.  kirsch , eds . ) , birkhuser , basel 2013 ; pp .  165227 . exner , h.  kovak : _ quantum waveguides _ , springer , cham 2015 . p.  exner , v.  lotoreichik , m.  tater : on resonances and bound states of smilansky hamiltonian , _ nanosystems : phys . chem . math . _ ( 2016 ) , to appear ; ` arxiv :  1607.00540  [ math - ph ] ` i.  guarneri : irreversible behaviour and collapse of wave packets in a quantum system with point interactions , _ j. phys . a : math . * 44 * ( 2011 ) , 485304 g.  hhler : ber die exponentialnherung beim teilchenzerfall , _ zs . phys . _ * 152 * ( 1958 ) , 546565 . s.  naboko , m.  solomyak : on the absolutely continuous spectrum in a model of an irreversible quantum graph , _ proc . * 92 * ( 2006 ) , 251272 . m.  reed , b.  simon : _ methods of modern mathematical physics . i : functional analysis . _ , academic press , new york 1980 . g.  rozenblum , m.  solomyak : on a family of differential operators with the coupling parameter in the boundary condition , _ j. comput . _ * 208 * ( 2007 ) , 5771 . u.  smilansky : irreversible quantum graphs , _ waves in random media _ * 14 * ( 2004 ) , s143153 . m.  solomyak : on the discrete spectrum of a family of differential operators , _ funct . * 38 * ( 2004 ) , 217223 . m.  solomyak : on a mathematical model of irreversible quantum graphs , _ st . petersbg . j. _ * 17 * ( 2006 ) , 835864 . m.  solomyak : on the limiting behaviour of the spectra of a family of differential operators , _ j. phys . a : math . gen . _ * 39 * ( 2006 ) , 1047710489 . m.  znojil : quantum exotic : a repulsive and bottomless confining potential , _ j. phys . a : math . gen . _ * 31 * ( 1998 ) , 33493355 .
we analyze the hamiltonian proposed by smilansky to describe irreversible dynamics in quantum graphs and studied further by solomyak and others . 
 we derive a weak - coupling asymptotics of the ground state and add new insights by finding the discrete spectrum numerically . 
 furthermore , we show that the model has a rich resonance structure .    ` smilansky model , discrete spectrum , weak coupling , resonances 35j10 , 35p15 , 65n25 , 81q15 , 81q35 `