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the quest for understanding ultra - energetic events such as gamma - ray bursts , active - galactic nuclei , quasars etc . passes through understanding of the compact massive objects which are thought to be their central engine  the rotating black holes , magnetars , binary neutron stars , etc .. for this , one needs to make a clear difference between theoretical , numerical and observational black hole ( bh ) and also , to keep account of the strengths and the weaknesses of the methods  numerical and observational  when working with the different types of objects .    in the case of black holes candidates in the role of central engines , the theoretical energy extraction processes of ultra - relativistic jets ( such as the penrose process or the blandford - znajek process ) should depend on the rotational bh parameter ( @xmath0 ) , with a predicted threshold for effective jet launching to be around @xmath1 , with more conservative estimate at @xmath2 @xcite . observationally , the measured rotational parameters seem to follow this requirement  a list of measured @xmath0 can be found in table 1 in @xcite . from it , one can see that in general , the measurements correspond to the expectations . there is , however , a catch . the measurement of the spins of the black hole candidates is not independent  it depends critically on the measurement of their masses , which comes with serious incertitude @xcite . one such example is the black hole candidate gro j0422 + 32 . its mass was estimated to @xmath3 @xcite but further analysis showed that its mass can be as low as @xmath4 @xcite  thus putting it in the mass - interval of neutron stars ( @xmath5 ) ) . in the same article , a number of other black hole candidates have their masses re - examined and corrected , which poses a serious question about all the results on black hole candidates . additionally , in @xcite it was demonstrated that numerous black hole candidates can be fitted by super - spinars ( objects with @xmath6 ) , thus showing that the x - ray spectrum is not enough to fix the spin of the black hole candidate . considering the importance of the mass - spin measurement for the modeling of astrophysical jets , it is clear that there is an acute need for a better understanding of the theory of the rotating black hole . the vacuum solution of the einstein equations which may be used for description of a rotating massive compact object ( black hole or super - spinar ) is the kerr metric . the kerr metric depends only on 2 parameters  the rotation @xmath7 to denote the theoretical rotational parameter , while the observational will be @xmath0 . usually the mass of the kerr metric is taken for a constant so @xmath8 . ] and the mass @xmath9 . while there is a known theoretical problem in using the kerr metric for objects different from rotating black holes ( the problem of finding interior  matter  solution matching the exterior vacuum kerr solution ) , it is thought that it approximates well enough any distant rotating compact massive object , the precise type of the object being specified by the boundary conditions imposed additionally on the given by the metric spacetime to fix its physical content , like black holes , super - spinars , etc . . confirming that the observational spectra indeed coincide with the kerr _ black hole _ spectra , is an important task facing astronomy and astrophysics .    for a long time , the theory of perturbations was the only viable method of dealing with different type of matter in the kerr metric , or to study evolution of several interacting black holes . in the past decade , the perturbation theory has been replaced by the much more sophisticated , but also computationally expensive and complicated numerical relativity simulations method . the theory of perturbations , however , still has place in the study of this type of objects , allowing a clearer view of the purely gravitational properties of the system . moreover , the important role of perturbative approach has been confirmed by full - gr numerical calculations . in the theory of perturbations , the characteristic ring - down of the black hole , as a result of perturbation , is dominating in later epochs through the complex frequencies called the quasi - normal modes ( qnms ) . the qnms were used for the first wave - templates for gravitational detectors . they were also observed in the late - time evolution of grmhd simulations . finally , qnms were used in full gr simulations as a marker of the end of the merger phase @xcite .    in @xcite we presented the quasinormal and quasibound spectra of the kerr metric obtained by using a novel approach of solving the governing equations in terms of the confluent heun functions and imposing the required boundary conditions directly on them . while this method allowed us to obtain both well - known old and also some new results , the novelty of the method makes its numerical stability uncharted . in the current paper , we study in detail the numerical stability of those spectra which is crucial for further use of the method and the results . in particular , we demonstrate the numerical stability with respect to deviations in the actual radial infinity and the position in the complex plane of the radial variable , proving that the so obtained quasinormal and quasibound spectra are indeed stable , as expected from the theory , and we show a counter - example of a spurious spectrum which despite being a root of the system of equations under consideration , does not satisfy the criteria for numerical stability and is thus non - physical . the linearized perturbations of the kerr metric are described by the teukolsky master equation , from which under the substitution @xmath10 one obtains the teukolsky radial equation ( tre ) and the teukolsky angular equation ( tae ) @xcite ( where @xmath11 for integer spins and @xmath12 is the complex frequency ) . the tae and tre describe on the same footing perturbations with different spin ( scalar waves , fermions , electromagnetic and gravitational waves ) without taking into account of the origin of the perturbation . their numerical solutions have provoked the creation of numerical methods aimed at solving the non - trivial transcendental system of differential equations through different approximations , the most well - known being the method of continued fractions and that of direct integration performed by leaver and andresson ( @xcite , + @xcite . for a review on the methods for studying the qnm spectra , see + @xcite . the tre and the tae are coupled differential equations for whose solutions one needs to impose the appropriate boundary conditions on the boundaries of the physical interval of interest . depending on the boundary conditions , one obtains as solutions of system the quasinormal spectra ( black hole boundary conditions , see below ) , the quasibound spectra ( resonant waves ) or total - transmission modes ( waves moving in one direction , i.e. entirely to the left or to the right ) . all of them are discrete spectra of complex , possibly infinite number of frequencies ( numbered by @xmath13 ) , which dominate the late - time evolution of the object s response to perturbations . unlike normal modes , qnms do not form a complete set , because the mathematical problem is not self - adjoint . therefore , one can not analyze the behavior of the system entirely in terms of qnms . the qnms , however , can be used to discern a black hole from other compact massive object such observational test was proposed in @xcite , and for a theoretical estimate of the em output for non - rotating bh  @xcite .    explicitly , the tae and the tre are :    @xmath14    and    @xmath15    where @xmath16 , @xmath17 , @xmath18 and @xmath19 . we fix @xmath20 for the spin of the perturbation and @xmath21 are the two horizons of the bh . the problem has two known parameter : the mass @xmath9 and the rotation @xmath7 and two unknown parameters : the frequency @xmath12 and the constant of separation of the angular equation @xmath22 . depending on the sign of the imaginary part of the frequency one can consider both perturbations damping with time , @xmath23 , ( the qnm case ) and also perturbations growing exponentially with time , @xmath24 ( the qbm case ) .    for details on the different classes of solutions of the tae and the tre em perturbations ( @xmath20 ) , see @xcite . here we will present only the equations we use to solve the system .    * the equation governing the spectrum of the tae is * : @xmath25    where @xmath26 , @xmath27 and the parameters are :    for the case @xmath28 : @xmath29 and    for the case @xmath30 : @xmath31    * the equation governing the spectrum of the tre is * : @xmath32 where @xmath33 and the parameters are :    @xmath34 , @xmath35 , @xmath36 , + @xmath37 , + @xmath38.$ ]    here the @xmath39 is the confluent heun function  the unique local frobenius solutions of the second - order linear ordinary differential equation of the fuchsian type @xcite , + @xcite with 2 regular singularities ( @xmath40 ) and one irregular ( @xmath41 ) , normalized to 1 at @xmath42 ( @xcite , + @xcite ) . the singularities of the tae are the poles of the sphere ( regular ) and infinity ( irregular ) , those of the tre are the two horizons : @xmath43 ( regular ) and @xmath44 ( irregular ) . in order to obtain the spectrum , one needs to impose specific boundary conditions .    for the angular equation , one requires angular regularity . the wronskian of two solutions regular around one of the poles , @xmath45 and @xmath46 , should become equal to zero , i.e. @xmath47=0 $ ] , i.e.    @xmath48=\frac{\text{heunc}'(\alpha_1,\beta_1,\gamma_1,\delta_1,\eta_1,\left ( \cos \left ( \pi/6   \right )   \right ) ^{2})}{\text{heunc}(\alpha_1,\beta_1,\gamma_1,\delta_1,\eta_1,\left ( \cos \left ( \pi/6   \right )   \right ) ^{2})}+\notag\\ \frac{\text{heunc}'(\alpha_2,\beta_2,\gamma_2,\delta_2,\eta_2,\left ( \sin \left ( \pi/6   \right )   \right ) ^{2})}{\text{heunc}(\alpha_2,\beta_2,\gamma_2,\delta_2,\eta_2,\left ( \sin \left ( \pi/6   \right )   \right ) ^{2})}+ p=0 \label{wr1}\end{aligned}\ ] ]    where the derivatives are with respect to @xmath49 , and we use @xmath50 ( the qnm should not depend on the value of the parameter @xmath51 in the spectral conditions ) . the parameter p  can be found in @xcite .    for the radial equation , one can impose the black hole boundary conditions ( bhbc ) : waves going simultaneously into the event horizon ( @xmath52 ) and into infinity ; or the quasibound boundary conditions ( qbbc ) : waves going simultaneously out of the event horizon ( @xmath52 ) and out of infinity @xcite . those two conditions lead to :     bhbc : for @xmath28 , the only valid solution is @xmath53 , while for @xmath54 , the solution @xmath53 is valid for frequencies for which @xmath55 . also , it requires that : @xmath56 ( the direction of steepest descent ) . qbbc : for @xmath28 , the only valid solution is @xmath57 , while for @xmath54 , the solution @xmath57 is valid for frequencies for which @xmath55 . also , it requires that : @xmath58 . the total - transmission modes ( ttm ) do not appear in the em case , in contrast to the gravitational case .    here , we will use the @xmath59-method @xcite in its most direct form @xmath60 to vary @xmath61 , and afterwards we will check whether the bhbc or the qbbc are satisfied . in order to obtain the spectrum @xmath62 , we will set the mass to @xmath63 and the radial infinity to @xmath64 ( where not stated otherwise ) and we will solve eqs . and . the system of two equations featuring the confluent heun functions will be solved by the two - dimensional mller algorithm , developed by the team @xcite . the algorithm is realized in maple code and the numbers presented below are obtained using maple 18 and 2015 . the software floating point number ( the parameter  digits  ) is set to 64 ( unless stated otherwise ) , the precision of the algorithm  to 25 digits . note that the numerical error due to the integration is hidden in the evaluation of the confluent heun function which depends on the maple parameter  digits `` . with current algorithms , one is not able to set the final number of stable digits in the heunc evaluation , but one can request the integration to be performed with certain precision , controlled by ' ' digits `` . in general , with our choice of the parameters of the root - finding algorithms , we can expect at least 15 digits of precision of the final result for the qnm / qbm , a number which can be increased by increasing further ' ' digits  , but increasing also the computational cost of the calculations . the numerical results of the qnm and qbm spectra are presented in + @xcite . here we will focus only on the tests of the numerical stability of those modes . we will consider two types of deviations : deviation in the absolute value of the radial variable and deviations in the argument of radial variable . the qnm / qbm spectra should not depend on the radial variable at all , because @xmath12 is a separation constant of the teukolsky master equation . if the spectra depend on @xmath65 , then either the actual numerical infinity is too low and we need to go further away from the horizon ( i.e. increasing @xmath66 ) , or the spectrum is not qnm / qbm , but a numerical artifact . one should keep in mind that the qnm / qbm are expected to be stable on @xmath61 only in a limited region around the direction of steepest descent @xcite , which in principle can be a curve and not a straight line , as in the first approximation .    as discussed in @xcite , we obtained 3 types of spectra : the qnm , the qbm and a spurious spectrum . the last one is not independent on the value of the radial variable , as we demonstrate below . thus , it can not be considered a physical solution of the spectral problem , even though its @xmath67 are formal roots of the spectral system .      to study numerically the stability of the qnm / qbm spectra , we will work only with the @xmath68 case ( no rotation ) because it is computationally simpler . the results are readily extended to the case with rotation ( @xmath69 ) .    for the non - rotating case , the spectra are found directly from the radial equation @xmath70 ( @xmath71 ) for the qnm ( qbm )  since in this case the angular solutions are known : @xmath72 , @xmath73 . this means that we can work with the one - dimensional mller algorithm , used very successfully in the case of the jet - spectra of the black hole and naked singularities @xcite . the main difficulty encountered in our previous works was the limited precision of the algorithms evaluating the confluent heun functions . in maple version 18 and higher , those algorithms have been redesigned to allow arbitrary precision for heunc . this facilitated significantly the current stability study and because of it , now it is possible to obtain the qnm and the qbm with arbitrary digits of significance . the tre , just as the tae , has 16 classes of solutions , valid in different sectors of the complex plane . while for the qnm the solution @xmath74 covers the needed sectors of the complex plane , for the qbm the above solution @xmath75 requires a significant increase in the parameter digits compared to @xmath74 to obtain the same precision . for this reason , we used one of the other , better convergent , solutions of the tre which are linearly independent of @xmath74 and have the same roots . the new solution differs from @xmath75 only because in it @xmath76 . similarly , one may find another solution which produces the qnm and it differs from @xmath74 by change @xmath77 . over all , there are 4 solutions from which one can obtain the qnm spectra and 4 for the qbm spectra , the only difference being the convergence of the root - finding algorithm . as mentioned above the qnm / qbm are a discrete set of frequencies numbered @xmath78 in the direction of increasing absolute value of their imaginary part , thus denoted here as @xmath79 . one can see the qnm / qbm frequencies with @xmath80 on fig . [ fig1 ] a ) . on fig . [ fig1 ] b ) we have plotted the boundary conditions corresponding to the 2 types of spectra .    the study of the stability in this case can be seen on fig . [ fig2]-[fig4 ] . on them , we focused on the following cases :    1 . stability of the modes with respect to the parameter digits in maple + the results are on fig . [ fig2 ] a ) where on the x - axis we have plotted the  digits  in use and on the y - axis the real part of the difference between two consecutive frequencies evaluated for those digits : @xmath81 for 4 different modes @xmath13 . clearly the difference quickly falls to a very small number and for digits>60 for @xmath82 , one has @xmath83 . the general tendency is that one needs higher digits for higher modes . 2 . stability with respect to the absolute value of the radial variable + on fig . [ fig2 ] b ) we have plotted @xmath84)|(|r|)$ ] in log - scale on the y - axis , where @xmath85)=\omega[n](r)-\omega[n](r')$ ] . from the plot it is clear that the frequencies depend on @xmath66 only near the horizon , with higher modes depending on @xmath65 stronger . for the modes on the plot ( @xmath86 ) , @xmath87 corresponds to the actual radial infinity after which the modes stabilize and remain the same with further increase of @xmath66 ( with the difference between two consecutive evaluations @xmath88 ) . 3 .   stability with respect to the argument of the radial variable + the qbm and the qnm spectra are stable to the precision required by the root - finding algorithm in a very large interval of epsilon . if we consider @xmath89 , we find that the qnm frequencies with positive real parts appear for @xmath90 ( i.e. @xmath91 ) and remain stable in an interval on @xmath59 depending on @xmath13  for @xmath92 , @xmath93 $ ] and for @xmath94 $ ] , for @xmath95 , @xmath96 $ ] and for @xmath97 $ ] , for @xmath98 , @xmath99 $ ] and for @xmath100 $ ] . the symmetry with respect to the imaginary axis in the dependence on @xmath59 is clear  we see that for @xmath90 the frequencies with the positive real parts disappear and on their place , the root - finding algorithm finds the qnm frequencies with the negative real parts . for the qbm frequencies we observe the inverse behavior those with the negative real part appear for @xmath101 . this symmetry is due to the fact that the @xmath102 frequencies of each type are obtained with the same differential equation in a different region of the complex plane . + additionally , one can see that the interval on @xmath59 moves to the left with the increase of @xmath13 , which , combined with the need of increasing of the digits makes the detailed study of the @xmath59-dependence for the modes with @xmath103 computationally expensive . on fig . [ fig2 ] c ) the dependence @xmath104 for @xmath92 is plotted , but it is representative for the behavior of all modes . generally , what we observe is a plateau - like region of the @xmath59-interval where the frequencies remain stable with more than 20 digits , followed by a region where the frequencies start changing smoothly , but in a very steep manner . all the numerical data point to the stability of the so - obtained frequencies in certain intervals on @xmath59 and @xmath66 , just as expected by the theory . * near - extremal regime ( @xmath105 ) *    let us define the angular velocity of the kbh : @xmath106 , with @xmath107 being the inner and the outer bh horizon ; the area of the kbh horizon : @xmath108 and the hawking temperature : @xmath109 . hod in @xcite derived two analytical formulas for the frequency @xmath12 in the near - extremal regime in terms of @xmath110 and @xmath111 , i.e. @xmath112 and @xmath113 , where @xmath114 is a complex number . as part of our test of the validity of our numerical results , we tested how well this theoretical formula approximates the so - obtained frequencies . the results can be seen on fig . [ fig3 ] .    clearly , @xmath115 fits the numerical results better . on those plots we have used @xmath116 , although for the case @xmath117 the best fit is for @xmath118 . the deviation from the theoretical formula is less than @xmath119 for @xmath120 , which is surprising considering the fact that the confluent heun function in this regime is expected to become numerically unstable . the fact that @xmath121 does not approximate the frequencies better , even though @xmath122 , may be due to the low @xmath13 we are using , since the hod s original approximation was for infinitely damped frequencies ( i.e. @xmath13 very high ) . * qnms in physical units *    finally , we would like to calculate the qnm frequencies in physical units for two observationally well - studied objects . from @xcite we know that : lmc x-1 has @xmath123 , @xmath124 and cygnus x-1  @xmath125 , @xmath126 . using the formula provided in the appendix , we find the physical frequencies of the first two qnms to be :    @xmath127 , + @xmath128 . clearly , frequencies of less than a khz damping in a manner of milliseconds , are inaccessible to even the best earth radio telescopes because of the ionospheric absorption and scattering . a possible lunar or space - based radio telescope could listen below @xmath129mhz , but even then , hearing the ringing of stellar black holes would be extremely difficult because of their matter - rich surrounding which will absorb the em spectrum . such observations can be made only if the qnms are up - scattered by some unknown mechanism to frequencies observable on the earth . this is an open question facing theoretical and observational astrophysics .      on fig . [ fig4 ] , one can see an example of a spurious spectrum corresponding to @xmath130 . once again we have a discrete infinite spectrum , although the points on the figure are very closely spaced , and thus the central part looks like a line ( see fig . [ for @xmath131 those frequencies change , like in the qnm / qbm case , however the change is less pronounced . from fig . [ fig4]b , it is clear that even though the spurious spectrum is very similar to the qnm / qbm , it does nt follow the same boundary conditions , with some points corresponding to infinity bhbc ( qnm ) ( @xmath132 ) and the rest  to infinity qbbc ( qbm ) . because they are roots to both @xmath57 and @xmath53 , the boundary condition on @xmath133 is ambiguous in those points the functions @xmath75 and @xmath74 stop being linearly independent ( up to certain numerical precision ) .    those roots have been confirmed with the 3 methods of heunc evaluation in maple  the direct integration , the series expansion and the taylor expansion . the study on the stability in this case is on fig . [ fig4]-[fig6 ] :    1 . stability of the modes with respect to the parameter digits + this case repeats our study of the digits dependence on qnm / qbm , so we did not plot this dependence again . as expected , the modes with a bigger imaginary part require higher value for  digits  . 2 .   stability with respect to the absolute value of the radial variable + the results are on fig . [ fig5 ] a ) and b ) , where we plotted 5 consecutive spurious modes . on fig . [ fig5 ] a ) one can see how the absolute value of the frequency corresponding to each mode changes with the increase of @xmath66 ( the actual infinity ) . one can see that it decreases , and upon a closer inspection , it seems like the imaginary part of the modes tends to a constant while the real part tends to zero . we could not reach that probable limit up to @xmath134 , where the calculations become computationally too expensive . one can see hints of such a limit also on fig . [ fig5 ] b ) , where a complex plot of 5 consecutive modes in the interval for @xmath135 is plotted . on it @xmath64 is to the right and there is a hint of convergence for the different modes . for the moment , however , such a limit can not be definitely confirmed numerically . 3 .   stability with respect to the argument of the radial variable + the study of the dependence of the modes on @xmath136 using the @xmath59-method can be seen on fig . [ fig6 ] . on it , we have varied @xmath137 for one mode with @xmath138 ( 0.044511+.297566i ) and one  with @xmath131 ( 0.023201 + 0.110030i ) . the interval for @xmath59 means that the argument is moved in a complete circle in the unit - sphere . as can be seen from the figure , the complex plot of those modes also draws a circle . one can see also the boundary conditions in this case . as mentioned in @xcite , the spurious modes are obtained from both equations ( @xmath75 and @xmath74 ) for the same value of argument  the left branch for @xmath139 , the right  for @xmath140 . on the plots we have used only points calculated from @xmath74 but the values obtained from @xmath75 coincide with them to a precision of @xmath141 . 4 .   stability with respect to rotation + on fig . [ fig5 ] c ) we plotted the behavior of one spurious mode with respect to an increase of the rotation from @xmath142 . one can see the characteristic for the confluent heun functions loops , which occur also for the qnm modes . since those modes evolve with rotation all the way to the naked singularity regime , it is important to be able to distinguish them from the qnm / qbm modes by checking the @xmath66 dependence of the resulting frequencies and the boundary conditions . this problem is the most serious for the qnm mode with @xmath143 which lies near a spurious mode , ( see fig . [ fig1 ] ) , but considering how densely spaced is the spurious modes spectrum , it could occur also near other modes ( for @xmath144 $ ] ) . it is important to note that those modes are roots not only to the radial equation , but to the spectral system . they can be seen to propagate for @xmath69 and even in the naked singularity case  for @xmath145 while the qnm / qbm spectra do disappear for @xmath145 or at least for the moment can not be located by the root - finding algorithm . because of their persistence in the numerical results , the only way to recognize the spurious spectra is through the radial stability checks we performed . it is important when discussing the numerical stability of a spectrum to make the difference between the numerical stability of the frequencies , the stability of the obtained spectrum and the stability of the spacetime with the given metric . above , we emphasized that we discussed the numerical stability of the results as an essential test of how trustable they are . when dealing with numerical solutions , we always work with certain precision and in a certain numerical domain , where the numerical algorithm is applicable . the latter depends on both the theory of the function in question and the numerical routines evaluating it .    in our case , the function we are working with is the confluent heun function . because of the irregularity of the singular point at infinity , there are no currently known series expansions working at infinity , and the available asymptotics of the solutions of the confluent heun equation is not properly developed , too . thus , one has limited options when dealing with infinity . the method of the continued fraction is one possible way to deal with the problem . with it , one moves the singularities , so that : @xmath146 . in this way , the series is convergent from @xmath42 to the next nearest singularity @xmath147 which corresponds to infinity . with this method , one obtains a continued fractions for the coefficients of the recurrence , from which the @xmath148variable is missing . the limits of applicability of this method are only partially known  for example , it does not work for purely imaginary modes . what is more important , this method does not address the fact that there are 16 classes of solutions of the confluent heun type valid in different parts of the complex plane . there is also the problem with the branch cuts spectra needed for the full description of the perturbation ( @xcite ) . finally , observations of the effect of the stokes and anti - stokes lines were noted in works utilizing the phase - amplitude method by andersson @xcite . all those details point to the need of proper development of the theory of the confluent heun equation and its solutions , in order to have clarity about the area of applicability of the different methods and their physical interpretation . in our case , to test the numerical stability of our results , we compare the output of the 3 methods of evaluation of the confluent heun function currently implemented in maple  direct integration , taylor expansion and continuation of the series expansion around @xmath147 , and then we check if small deviations in the parameters lead to significant changes in the frequencies . we also plot the stokes and anti - stokes contours , as well as a 3rd complex plot of the function around a root for @xmath68 . all those tests point to numerical stability of the qnm and qbm spectra and to instability of the third one . it is important to note that the qnm problem in black holes is an analogue to the one of finding the poles of the s - matrix of the schrodinger equation . in quantum mechanics , those poles depending on the boundary conditions may correspond to bound states , virtual states or resonant states . all of them correspond to different boundary conditions in which one may deal with solutions with exponentially decreasing or increasing asymptotics ( bound states or resonances ) . in astrophysics , because of the existence of the horizon on one of the boundaries of the s - matrix and , furthermore , the fact that all the frequencies are complex , one can not apply directly the same terminology and methods to the problem . for this reason , we work with quasinormal modes and quasibound modes , even though the bound states in quantum mechanics are purely imaginary modes . in quantum mechanics , the branch cut of the s - matrix also plays an important and well - known role . in the theory of qnm we have currently a very limited number of works on this approach ( @xcite , + @xcite ) . apart from the numerical stability , one also may discuss the physical stability of the spectrum . here with spectrum  , we refer to the solutions of the spectral problem defined by the exact solutions of the differential equations and the boundary conditions we impose on them . as known from the theory of the differential equations , different boundary conditions describe different physical objects , even though they are governed by the same differential equation . in this context , we can divide the different spectra into stable and unstable . for example , the spectrum obtained from black hole boundary conditions in our work has been found to be physically stable , i.e. corresponding to perturbations which damp with time . on the other hand , the spectrum corresponding to the quasibound modes is clearly physically unstable , because the perturbations grow with time . there could be an other explanation , however . as mentioned in @xcite , the ranges of the boyer - lindquist coordinates for the kerr metric are : @xmath149 . in the case of the standard substitution @xmath10 , the perturbation will damp with time for @xmath23 and @xmath150 . for @xmath151 the perturbation will grow exponentially for the same condition . for the qbm spectra , the situation reverses and for @xmath24 the perturbations will damp with time for @xmath151 and grow exponentially for @xmath150 . considering the full range of the variables is not unusual . for example in @xcite , the standard extension @xmath152 , equivalent to @xmath153 was already considered . also , it is known that the metric is invariant under the changes @xmath154 , meaning that the inversion of time can be considered as inversion of the direction of rotation . therefore , both qnm and qbm are valid frequencies in the whole range for the variables of the problem . finally , the question of the stability of the physical object described by the metric is more complicated and has a long history ( for example @xcite ) . in many of those studies , linear perturbation theory was used with certain ambiguity in the definitions . when one speaks of the stability of the schwarzschild black hole , one implies that the boundary conditions will be those of a black hole ( bhbc ) . the schwarzschild metric , however , just like the kerr metric , in linear perturbation theory , allows one to impose other types of boundary conditions  both the inverse bh conditions which we call qbbc or the total - transmission modes boundary conditions . in those cases , however , one deviates from the classical description of a black hole since the waves are not entering the horizon , but moving away from it . this problem becomes even more complicated when we work with the kerr metric . the kerr metric according to the theory has 2 horizons ( @xmath155 ) and one physical singularity ( @xmath156 ) , which is singularity of the scalar invariants of the metric . this definition usually ignores the differential invariants where the horizons are also visible . what is more important , the singularities visible for the differential equation when @xmath157 are only the two horizons and infinity . because of this , following the theory of ordinary differential equations , one needs to impose the boundary conditions on two of those singularities in order to specify the singular spectral problem . usually , one regards the inner horizon as physically not interesting and imposes the boundary conditions on the outer event horizon and infinity . but there have been studies in which one imposes the boundary conditions on two horizons , which have shown that the region between the two horizons is _ unstable _ with respect to linear perturbations . this poses the question of the physical interpretation of the different regions of the kerr metric and their analytical extensions . in any case , there are 3 different types of boundary conditions  bhbc , qbbc and ttm  which correspond to different physical situations occurring on the background of the kerr metric and their study has its theoretical value . the quasinormal modes are a well - known and seriously studied problem in gravitational physics . they have been extensively used in different studies , but one significant problem has remained and it is related to the theory of the confluent heun functions and their roots . in series of articles , we have studied different physical configurations using the confluent heun functions in maple and we demonstrated that this method can lead to interesting scientific results , however , this novel approach also requires the development of new numerical tools and methods that are able to deal with the complexity of the heun differential equations and their solutions .    in this article , by using the exact solutions of the tre and tae , we were able to impose different boundary conditions on them , describing qnm and qbm . by checking the numerical stability of the so obtained spectra in both @xmath66 and @xmath61 , we were able to sift out the physical solutions from the spurious ones , with clear theoretical justification . this proves that the new approach indeed leads to a better understanding of the problem in question and also points to the huge potential which any improvement in the theory of the heun functions offers to physics . the authors would like to thank dr . edgardo cheb - terrab for useful discussions of the algorithms evaluating the heun functions in maple , and for continuing the improvement of those algorithms in the latest versions of maple . the authors would like to thank d. sc . svetlana pacheva for critical reading of the manuscript and the constructive comments . this article was supported by the foundation `` theoretical and computational physics and astrophysics '' , by the bulgarian national scientific fund under contracts do-1 - 872 , do-1 - 895 , do-02 - 136 , and sofia university scientific fund , contract 185/26.04.2010 , grants of the bulgarian nuclear regulatory agency for 2013 , 2014 and 2015 . p.f . posed the problem of the evaluation of the em qnms of rotating bhs as a continuation of previous studies of the applications of the confluent heun functions in astrophysics . he proposed the epsilon method and supervised the project . d.s . is responsible for the numerical results , their analysis and the plots and tables presented here . both the authors discussed the results at all stages . the manuscript was prepared by d.s . and edited by p.f .. table [ table_n0_om ] presents some of the values obtained for the em qnm , converted to physical units using the relations : @xmath158    note that in those formulas a factor of @xmath159 is missing because the em qnms were obtained for @xmath160 and not for @xmath161 . then if @xmath9 is the mass of the object in physical units ( we use @xmath162 ) , @xmath163  the mass of the sun ( @xmath164 $ ] ) and @xmath165 , c=2.99792458\,10 ^ 8[m / s]$ ] , one obtains @xmath166,\tau^{phys}\approx \frac{0.4925\,10^{-5}\,m / m_{\odot}}{\im(\omega)}[s].\ ] ]     & & & + & @xmath167[hz ] & @xmath168[ms ] & @xmath169[hz ] & @xmath170[ms ] & @xmath171[hz ] & @xmath172[ms ] + 0 & 802.1512449166 & 0.5325890917 & 802.1512449166 & 0.5325890917 & 802.1512449167 & 0.5325890917 + 0.2 & 804.9393652797 & 0.5343356142 & 849.8315682698 & 0.5388677452 & 763.6902591869 & 0.5299212818 + 0.6 & 829.2637578502 & 0.5526525743 & 996.9258848852 & 0.5772488810 & 704.6451920585 & 0.5313970112 + 0.98 & 884.6086875757 & 0.6427140687 & 1445.8841670353 & 1.2178343064 & 661.8628389523 & 0.5373275077 +   + & & & + & @xmath167[hz ] & @xmath168[ms ] & @xmath169[hz ] & @xmath170[ms ] & @xmath171[hz ] & @xmath172[ms ] + 0 & 472.3043572607 & 0.0638131626 & 472.3043572599 & 0.0638131626 & 472.3043572609 & 0.0638131626 + 0.2 & 479.7880705182 & 0.0641624606 & 549.5382706431 & 0.0658306413 & 415.5213067645 & 0.0624052214 + 0.6 & 530.3546588895 & 0.0677548111 & 792.4033820584 & 0.0741706889 &  &  + 0.98 & 408.1590806664 & 0.0756272782 &  &  &  & +     & & & + & @xmath173 & @xmath174 & @xmath175 + 0 & 2.0000000000 + @xmath176 & 2.0000000000 + @xmath177 & 2.0000000000 + @xmath178 + 0.2 & @xmath179 & @xmath180 & @xmath181 + 0.6 & @xmath182 & @xmath183 & @xmath184 + 0.98&@xmath185 & @xmath186 & @xmath187 +   + & & & + & @xmath173 & @xmath174 & @xmath175 + 0 & 2.0000000000+@xmath188i & 2.0000000000 + 4.82 @xmath189i & 2.0000000000 + 4.83 @xmath190 + 0.2&@xmath191 & @xmath192 & @xmath193 + 0.6&@xmath194 & @xmath195 &  + 0.98&@xmath196&@xmath197& +      , _ a numerically accurate investigation of black - 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the proper understanding of the electromagnetic counterpart of gravity - waves emitters is of serious interest to the multimessenger astronomy . in this article , we study the numerical stability of the quasinormal modes ( qnm ) and quasibound modes ( qbm ) obtained as solutions of the teukolsky angular equation and the teukolsky radial equation with appropriate boundary conditions . 
 we use the epsilon - method for the system featuring the confluent heun functions to study the stability of the spectra with respect to changes in the radial variable . 
 we find that the qnm and qbm are stable in certain regions of the complex plane , just as expected , while the third `` spurious '' spectrum was found to be numerically unstable and thus unphysical . 
 this analysis shows the importance of understanding the numerical results in the framework of the theory of the confluent heun functions , in order to be able to distinguish the physical spectra from the numerical artifacts .